Smoothing, scattering, and a conjecture of Fukaya
Abstract.
In 2002, Fukaya [19] proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture [47] by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold and the multi-valued Morse theory on the base of an SYZ fibration , and the other between deformation theory of the mirror and the same multi-valued Morse theory on . In this paper, we prove a reformulation of the main conjecture in Fukaya’s second correspondence, where multi-valued Morse theory on the base is replaced by tropical geometry on the Legendre dual . In the proof, we apply techniques of asymptotic analysis developed in [7, 9] to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in [8]. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.
1. Introduction
Two decades ago, in an attempt to understand mirror symmetry using the SYZ conjecture [47], Fukaya [19] proposed two correspondences:
-
•
Correspondence I: between the theory of pseudo-holomorphic curves (instanton corrections) on a Calabi–Yau manifold and the multi-valued Morse theory on the base of an SYZ fibration , and
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•
Correspondence II: between deformation theory of the mirror and the same multi-valued Morse theory on the base .
In this paper, we prove a reformulation of the main conjecture [19, Conj 5.3] in Fukaya’s Correspondence II, where multi-valued Morse theory on the SYZ base is replaced by tropical geometry on the Legendre dual . Such a reformulation of Fukaya’s conjecture was proposed and proved in [7] in a local setting; the main result of the current paper is a global version of the main result in loc. cit. A crucial ingredient in the proof is a precise link between tropical geometry on an integral affine manifold with singularities and smoothing of maximally degenerate Calabi–Yau varieties.
The main conjecture [19, Conj. 5.3] in Fukaya’s Correspondence II asserts that there exists a Maurer–Cartan element of the Kodaira–Spencer dgLa associated to deformations of the semi-flat part of that is asymptotically close to a Fourier expansion ([19, Eq. (42)]), whose Fourier modes are given by smoothings of distribution-valued 1-forms defined by moduli spaces of gradient Morse flow trees which are expected to encode counting of non-trivial (Maslov index 0) holomorphic disks bounded by Lagrangian torus fibers (see [19, Rem. 5.4]). Also, the complex structure defined by this Maurer–Cartan element can be compactified to give a complex structure on . At the same time, Fukaya’s Correspondence I suggests that these gradient Morse flow trees arise as adiabatic limits of loci of those Lagrangian torus fibers which bound non-trivial (Maslov index 0) holomorphic disks. This can be reformulated as a holomorphic/tropical correspondence, and much evidence has been found [18, 20, 39, 40, 12, 11, 38, 10, 4].
The tropical counterpart of such gradient Morse flow trees are given by consistent scattering diagrams, which were invented by Kontsevich–Soibelman [36] and extensively used in the Gross–Siebert program [29] to solve the reconstruction problem in mirror symmetry, namely, the construction of the mirror from smoothing of a maximally degenerate Calabi–Yau variety . It is therefore natural to replace the distribution-valued 1-form in each Fourier mode in the Fourier expansion [19, Eq. (42)] by a distribution-valued 1-form associated to a wall-crossing factor of a consistent scattering diagram. This was exactly how Fukaya’s conjecture [19, Conj. 5.3] was reformulated and proved in the local case in [7].
In order to reformulate the global version of Fukaya’s conjecture, however, we must also relate deformations of the semi-flat part with smoothings of the maximally degenerate Calabi–Yau variety . This is because consistent scattering diagrams were used by Gross–Siebert [28] to study the deformation theory of the compact log variety (whose log structure is specified by slab functions), instead of . For this purpose, we consider the open dense part
where is the generalized moment map in [43] and is an open dense subset such that contains the tropical singular locus and all codimension cells of .
Equipping with the trivial log structure, there is a semi-flat dgBV algebra governing its smoothings, and the general fiber of a smoothing is given by the semi-flat Calabi–Yau that appeared in Fukaya’s original conjecture [19, Conj. 5.3]. However, the Maurer–Cartan elements of cannot be compactified to give complex structures on . On the other hand, in our previous work [8] we constructed a Kodaira–Spencer–type pre-dgBV algebra which controls the smoothing of . A key observation is that a twisting of by slab functions is isomorphic to the restriction of to (Lemma 5.10).
Our reformulation of the global Fukaya conjecture now claims the existence of a Maurer–Cartan element of this twisted semi-flat dgBV algebra that is asymptotically close to a Fourier expansion whose Fourier modes give rise to the wall-crossing factors of a consistent scattering diagram. This conjecture follows from (the proof of) our main result, stated as Theorem 1.1 below, which is a combination of Theorem 4.18, the construction in §5.3.2 and Theorem 5.24:
Theorem 1.1.
There exists a solution to the classical Maurer–Cartan equation (4.11) giving rise to a smoothing of the maximally degenerate Calabi–Yau log variety over , from which a consistent scattering diagram can be extracted by taking asymptotic expansions.
A brief outline of the proof of Theorem 1.1 is now in order. First, recall that the pre-dgBV algebra which governs smoothing of the maximally degenerate Calabi–Yau variety was constructed in [8, Thm. 1.1 & §3.5], and we also proved a Bogomolov–Tian–Todorov–type theorem [8, Thm. 1.2 & §5] showing unobstructedness of the extended Maurer–Cartan equation (4.10), under the Hodge-to-de Rham degeneracy Condition 4.16 and a holomorphic Poincaré Lemma Condition 4.14 (both proven in [28, 17]). In Theorem 4.18, we will further show how one can extract from the extended Maurer–Cartan equation (4.10) a smoothing of , described as a solution to the classical Maurer–Cartan equation (4.11)
together with a holomorphic volume form which satisfies the normalization condition
| (1.1) |
where is a nearby vanishing torus in the smoothing.
Next, we need to tropicalize the pre-dgBV algebra . However, the original construction of in [8] using the Thom–Whitney resolution [49, 14] is too algebraic in nature. Here, we construct a geometric resolution exploiting the affine manifold structure on . Using the generalized moment map [43] and applying the techniques of asymptotic analysis (in particular the notion of asymptotic support) in [7], we define the sheaf of monodromy invariant tropical differential forms on in §5.1. According to Definition 5.5, a tropical differential form can be regarded as a distribution-valued form supported on polyhedral subsets of . Using the sheaf , we can take asymptotic expansions of elements in , and hence connect differential geometric operations in dgBV/dgLa with tropical geometry. In this manner, we can extract local scattering diagrams from Maurer–Cartan solutions as we did in [7], but we need to glue them together to get a global object.
To achieve this, we need the aforementioned comparison between and the semi-flat dgBV algebra which governs smoothing of the semi-flat part equipped with the trivial log structure. The key Lemma 5.10 says that the restriction of to the semi-flat part is isomorphic to precisely after we twist the semi-flat operator by elements corresponding to the slab functions associated to the initial walls of the form:
here the sum is over vertices in codimension one cells ’s which intersect with the essential singular locus (defined in §3.3), is a distribution-valued -form supported on a component of containing , is a holomorphic vector field and ’s are the slab functions associated to the initial walls. We remark that slab functions were used to specify the log structure on as well as the local models for smoothing in the Gross–Siebert program; see §2 for a review.
Now, the Maurer–Cartan solution obtained in Theorem 4.18 defines a new operator on which squares to zero. Applying the above comparison of dgBV algebras (Lemma 5.10) and the gauge transformation from Lemma 5.11, we show that, after restricting to , there is an isomorphism
for some element , where ‘s’ stands for scattering terms. From the description of , the element , to any fixed order , is written locally as a finite sum of terms supported on codimension one walls/slabs (Definitions 5.13 and 5.14. For the purpose of a brief discussion in this introduction, we will restrict ourselves to a wall below, though the same argument applies to a slab; see §5.3.2 for the details. In a neighborhood of each wall , the operator is gauge equivalent to via some vector field , i.e.
Employing the techniques for analyzing the gauge which we developed in [7, 9, 37], we see that the gauge will jump across the wall, resulting in a wall-crossing factor satisfying
where are the two chambers separated by . Then from the fact that the volume form is normalized as in (1.1), it follows that is closed under the semi-flat BV operator , and hence we deduce that the wall-crossing factor lies in the tropical vertex group. This defines a scattering diagram on the semi-flat part associated to . Finally, we prove consistency of the scattering diagram in Theorem 5.24. We emphasize that the consistency is over the whole even though the diagram is only defined on , because the Maurer–Cartan solution is globally defined on .
Remark 1.2.
Our notion of scattering diagrams (Definition 5.17) is a little bit more relaxed than the usual notion defined in [36, 29] in two aspects: One is that we do not require the generator of the exponents of the wall-crossing factor to be orthogonal to the wall.111It seems reasonable to relax this orthogonality condition because one cannot require such a condition in more general settings [5, 37]. The other is that we allow possibly infinite number of walls/slabs approaching strata of the tropical singular locus. See the paragraph after Definition 5.17 for more details. In practice, this simply means that we are considering a larger gauge equivalence class (or equivalently, a weaker gauge equivalence), which is natural from the point of view of both the Bogomolov–Tian–Todorov Theorem and mirror symmetry (in the A-side, this amounts to flexibility in the choice of the almost complex structure). We also have a different, but more or less equivalent, formulation of the consistency of a scattering diagram; see Definition 5.21 and §5.3.1.
Along the way of proving Fukaya’s conjecture, besides figuring out the precise relation between the semi-flat part and the maximally degenerate Calabi–Yau log variety , we also find the correct description of the Maurer–Cartan solutions near the singular locus, namely, they should be extendable to the local models prescribed by the log structure (or slab functions), as was hinted by the Gross–Siebert program. This is related to a remark by Fukaya [19, Pt. (2) after Conj. 5.3].
Another important point is that we have established in the global setting an interplay between the differential-geometric properties of the tropical dgBV algebra and the scattering (and other combinatorial) properties of tropical disks, which was speculated by Fukaya as well ([19, Pt. (1) after Conj. 5.3]) although he considered holomorphic disks instead of tropical ones.
Furthermore, by providing a direct linkage between Fukaya’s conjecture with the Gross–Siebert program [27, 28, 29] and Katzarkov–Kontsevich–Pantev’s Hodge theoretic viewpoint [33] through (recall from [8] that a semi-infinite variation of Hodge structures can be constructed from , using the techniques of Barannikov–Kontsevich [3, 2] and Katzarkov–Kontsevich–Pantev [33]), we obtain a more transparent understanding of mirror symmetry through the SYZ framework.
Remark 1.3.
A future direction is to apply the framework in this paper and the works [7, 8] to develop a local-to-global approach to understand genus mirror symmetry. In view of the ideas of Seidel [46] and Kontsevich [35], and also recent breakthroughs by Ganatra–Pardon–Shende [25, 26, 24] and Gammage–Shende [22, 23], we expect that there is a sheaf of algebras on the A-side mirror to (the enhancement of) that can be constructed by gluing local models. More precisely, a large volume limit of a Calabi–Yau manifold can be specified by removing from it a normal crossing divisor which represents the Kähler class of . This gives rise to a Weinstein manifold , and produces a mirror pair at the large volume/complex structure limits. In [23], Gammage–Shende constructed a Lagrangian skeleton from a combinatorial structure called fanifold, which can be extracted from the integral tropical manifold equipped with a polyhedral decomposition (here we assume that the gluing data is trivial). They also proved an HMS statement at the large limits. We expect that an A-side analogue of can be constructed from the Lagrangian skeleton in , possibly together with a nice and compatible SYZ fibration on , via gluing of local models. A local-to-global comparsion on the A-side and isomorphisms between the local models on the two sides should then yield an isomorphism of Frobenius manifolds.
Acknowledgement
We thank Kenji Fukaya, Mark Gross and Richard Thomas for their interest and encouragement, and also Helge Ruddat for useful comments on an earlier draft of this paper. We are very grateful to the anonymous referees for numerous constructive and extremely detailed comments/suggestions which have helped to greatly enhanced the exposition of the whole paper.
K. Chan was supported by grants of the Hong Kong Research Grants Council (Project No. CUHK14301420 & CUHK14301621) and direct grants from CUHK. N. C. Leung was supported by grants of the Hong Kong Research Grants Council (Project No. CUHK14301619 & CUHK14306720) and a direct grant (Project No. 4053400) from CUHK. Z. N. Ma was supported by National Science Fund for Excellent Young Scholars (Overseas). These authors contributed equally to this work.
List of notations
| , | §2.1 | lattice, for any -module |
|---|---|---|
| , | §2.1 | dual lattice of , for any -module |
| Def. 2.2 | integral tropical manifold equipped with a polyhedral | |
| decomposition | ||
| §2.1 | lattice generated by integral tangent vectors along | |
| §2.1 | relative interior of a polyhedron | |
| §2.1 | open neighborhood of | |
| §2.1 | lattice generated by normal vectors to | |
| §2.1 | fan structure along | |
| §2.1 | complete fan in constructed from | |
| §2.1 | is a cone in corresponding to | |
| §2.2 | lattice of integral tangent vectors of at | |
| , | Def. 2.9 | monodromy polytope of , dual monodromy polytope of |
| Def. 2.5 | sheaf of affine functions on | |
| Def. 2.5 | sheaf of piecewise affine functions on with respect to | |
| Def. 2.6 | sheaf of multi-valued piecewise affine functions on | |
| with respect to | ||
| Def. 2.7 | strictly convex multi-valued piecewise linear function | |
| §2.3 | localization of the fan at | |
| §2.3 | local affine scheme associated to used for open gluing | |
| §2.3 | group of piecewise multiplicative maps on | |
| Def. 2.15 | number encoding the change of across through | |
| §2.3 | closed stratum of associated to | |
| §2.4 | cone defined by the strictly convex function | |
| representing | ||
| §2.4 | monoid of integral points in | |
| §2.4 | parameter for a toric degeneration | |
| §2.4 | line bundle on having slab functions as sections | |
| §2.4 | local slab function associate to in the chart | |
| §2.4 | toric morphism induced from the monodromy polytope | |
| §2.4 | toric monoid describing the local model of toric degeneration | |
| near | ||
| §2.4 | toric monoid isomorphic to | |
| §2.4 | normal fan of a polytope | |
| §3.1 | generalized moment map |
| §3.2 | coordinate chart on | |
| (resp. ) | §3.3 | (resp. essential) tropical singular locus in |
| Def. 3.6 | surjective map with | |
| §4 | good cover (Condition 4.1) of with being Stein | |
| §4 | -order local smoothing model of | |
| Def. 4.2 | sheaf of -order holomorphic relative log polyvector fields on | |
| Def. 4.2 | sheaf of -order holomorphic log de Rham differentials on | |
| §4.1 | sheaf of -order holomorphic relative log de Rham differentials on | |
| Def. 4.2 | -order relative log volume form on | |
| §4.1 | BV operator on | |
| Def. 4.8 | local sheaf of -order polyvector fields | |
| Def. 4.9 | local sheaf of -order de Rham forms | |
| Def. 4.13 | global sheaf of -order polyvector fields from gluing of ’s | |
| Def. 4.13 | global sheaf of -order de Rham forms from gluing of ’s | |
| Def. 5.6 | global sheaf of tropical differential forms on | |
| §5.2.1 | semi-flat locus | |
| §5.2.1 | sheaf of -order semi-flat holomorphic relative vector fields | |
| §5.2.1 | sheaf of -order semi-flat holomorphic log de Rham forms | |
| eqt. (5.2) | sheaf of -order semi-flat holomorphic tropical vertex Lie algebras | |
| Def. 5.9 | sheaf of -order semi-flat polyvector fields | |
| Def. 5.9 | sheaf of -order semi-flat log de Rham forms | |
| Def. 5.12 | sheaf of -order semi-flat tropical vertex Lie algebras | |
| Def. 5.13 | wall equipped with a wall-crossing factor | |
| Def. 5.14 | slab equipped with a wall-crossing factor | |
| Def. 5.17 | scattering diagram | |
| §5.3.1 | complement of joints in the semi-flat locus | |
| §5.3.1 | the embedding | |
| §5.3.1 | -order wall-crossing sheaf associated to |
Notation 1.4.
We usually fix a rank lattice together with a strictly convex -dimensional rational polyhedral cone . We call the universal monoid. We consider the ring , a monomial element of which is written as for , and the maximal ideal . Then is an Artinian ring, and we denote by the completion of . We further equip , and with the natural monoid homomorphism , , which gives them the structure of a log ring (see [29, Definition 2.11]); the corresponding log analytic spaces are denoted as , and respectively.
Furthermore, we let , and (here ) be the spaces of log de Rham differentials on , and respectively, where we write for ; these are equipped with the de Rham differential satisfying . We also denote by , and , respectively, the spaces of log derivations, which are equipped with a natural Lie bracket . We write for the element with action , where is the natural pairing between and .
2. Gross–Siebert’s cone construction of maximally degenerate Calabi–Yau varieties
This section is a brief review of Gross–Siebert’s construction of the maximally degenerate Calabi–Yau variety from the affine manifold and its log structures from slab functions [27, 28, 29].
2.1. Integral tropical manifolds
We first recall the notion of integral tropical manifolds from [29, §1.1]. Given a lattice of rank , a rational convex polyhedron is a convex subset in given by a finite intersection of rational (i.e. defined over ) affine half-spaces. We usually drop the attributes “rational” and “convex” for polyhedra. A polyhedron is said to be integral if all its vertices lie in ; a polytope is a compact polyhedron. The group of integral affine transformations acts on the set of polyhedra in . Given a polyhedron , let be the smallest affine subspace containing , and denote by the corresponding lattice. The relative interior refers to taking the interior of in . There is an identification for the tangent space at . Write . Then a face of is the intersection of with a supporting hyperplane. Codimension one faces are called facets.
Let be the category whose objects are integral polyhedra and morphisms consist of the identity and integral affine isomorphisms onto faces (i.e. an integral affine morphism which is an isomorphism onto its image and identifies with a face of ). An integral polyhedral complex is a functor from a finite category to such that every face of still lies in the image of , and there is at most one arrow for every pair . By abuse of notation, we usually drop the notation and write to represent an integral polyhedron in the image of the functor. From an integral polyhedral complex, we obtain a topological space via gluing of the polyhedra along faces. We further assume that:
-
(1)
the natural map is injective for each , so that can be identified with a closed subset of called a cell, and a morphism can be identified with an inclusion of subsets;
-
(2)
a finite intersection of cells is a cell; and
-
(3)
is an orientable connected topological manifold of dimension without boundary which in addition satisfies the condition that .
Remark 2.1.
The condition will be used only in Theorem 4.18 to ensure that , where is the degenerate Calabi–Yau variety that we are going to construct.222In his recent work [15], Felten was able to prove Theorem 4.18 without assuming that . This corresponds to the condition that for smooth Calabi–Yau manifolds.
The set of -dimensional cells is denoted by , and the -skeleton by . For every , we define its open star by
which is an open subset of containing . A fan structure along is a continuous map such that
-
•
,
-
•
for every , the restriction is an integral affine submersion onto its image (meaning that it is induced by some epimorphism for some vector subspace ), and
-
•
the collection of cones forms a complete finite fan .
Two fan structures along are equivalent if they differ by composition with an integral affine transformation of k. If is a fan structure along and , then and there is a fan structure along induced from via the composition:
where is the quotient map.
Definition 2.2 ([29], Def. 1.2).
An integral tropical manifold is an integral polyhedral complex together with a fan structure along each such that whenever , the fan structure induced from is equivalent to .
Taking sufficiently small and mutually disjoint open subsets for and for , there is an integral affine structure on . We will further choose the open subsets ’s and ’s so that the affine structure is defined outside a closed subset of codimension two in , as in [27, §1.3]. This affine structure allows us to use parallel transport to identify the tangent spaces for different points outside the closed subset. For every we choose a maximal cell and consider the lattice of normal vectors (we suppress the dependence on because we will see that is monodromy invariant under the monodromy transformation given by any two vertices of and any two maximal cells containing ). We can identify with via , and write the fan structure as .
Example 2.3.
We take a -dimensional example from [1, Ex. 6.74] to illustrate the above definitions. Let be the convex hull of the points
so is a -simplex. Take (as a topological space) to be the boundary of . The polyhedral decomposition is defined so that the integral points are vertices as shown in Figure 1.
Then we define affine coordinate charts on as follows. On , we take which maps homeomorphically onto its image. At a vertex treated as a vector in 3, we let , where is the natural projection onto the quotient. By [1, Prop. 6.81], this gives an integral affine manifold with singularities. The affine structure can be extended to the complement of a subset consisting of points lying on the six edges of , with each edge containing points (colored in red in Figure 1). The fan structure can be defined similarly.
Locally near each singular point contained in an edge , the affine structure is described as a gluing of two affine charts and as in [30, §3.2]. The change of coordinates from to is given by the restriction of the map from to itself defined by
The fan structure is given as and the fan is the toric fan for . Figure 2 below illustrates the situation.
With the structure of an integral tropical manifold, the corners and edges in Figure 1 are flattened via the affine coordinate charts, and we can view as the 2-sphere equipped with a polyhedral decomposition and with affine singularities. Such an affine structure with singularities also appears in the base of an SYZ fibration of a K3 surface.
Example 2.4.
A -dimensional example can be constructed as in [1, Ex. 6.74]. Take to be the convex hull of the points
which gives a -simplex. Take (as a topological space) to be the boundary of . There are five -dimensional maximal cells intersecting along ten -dimensional facets. The polyhedral decomposition on each facet is as in Figure 3.
The affine structure can be extended to the complement of codimension 2 closed subset whose intersection with a triangle in Figure 3 is a -shaped locus. Locally near each of these triangles, it looks like Figure 4a.
has ten -dimensional faces, each of which is an edge with affine length . The polyhedral decomposition divides each edge into intervals as we can see in Figure 3. Locally near each of these length intervals, there are three -cells of intersecting along it. The locus on each -cell intersects on the interval as shown in Figure 4b.
Definition 2.5 ([27], Def. 1.43).
An integral affine function on an open subset is a continuous function on which is integral affine on for and on for . We denote by (or simply ) the sheaf of integral affine functions on .
A piecewise integral affine function (abbrev. as PA-function) on is a continuous function on which can be written as on for every , where and is a piecewise linear function on with respect to the fan . The sheaf of PA-functions on is denoted by .
There is a natural inclusion , and we let be the quotient:
Locally, an element is a collection of piecewise affine functions such that on each overlap , the difference is an integral affine function on .
Definition 2.6 ([27], Def. 1.45 and 1.47).
The sheaf is called the sheaf of multi-valued piecewise affine functions (abbrev. as MPA-funtions) of the pair . A section is said to be convex (resp. strictly convex) if for any vertex , there is a convex (resp. strictly convex) representative on . (Here, convexity (resp. strict convexity) means if we take any maximal cone with the affine function defined by requiring , we always have (resp. ) for ).
The set of all convex multi-valued piecewise affine functions gives a sub-monoid of under addition, denoted as ; we let be the dual monoid.
Definition 2.7 ([27], Def. 1.48).
The polyhedral decomposition is said to be regular if there exists a strictly convex multi-valued piecewise linear function .
We always assume that is regular with a fixed strictly convex .
2.2. Monodromy, positivity and simplicity
To describe monodromy, we consider two maximal cells and two of their common vertices . Taking a path going from to through , and then from back to through , we obtain a monodromy transformation . As in [27, §1.5], we are interested in two cases. The first case is when is connected to via a bounded edge . Let be the unique primitive vector pointing to along . For an integral tangent vector , the monodromy transformation is given by
| (2.1) |
for some , where is the natural pairing between and . The second case is when and are separated by a codimension one cell . Let be the unique primitive covector which is positive on . The monodromy transformation is given by
| (2.2) |
for some , where is the smallest face of containing . In particular, if we fix both , one obtains the formula
| (2.3) |
for some integer .
Definition 2.8 ([27], Def. 1.54).
We say that is positive if for all and with .
Following [27, Definition 1.58], we package the monodromy data into polytopes associated to for . The simplest case is when , whose monodromy polytope is defined by fixing a vertex and setting
| (2.4) |
where refers to taking the convex hull. It is well-defined up to translation and independent of the choice of . The normal fan of in is a refinement of the normal fan of . Similarly, when , one defines the dual monodromy polytope by fixing and setting
| (2.5) |
Again, this is well-defined up to translation and independent of the choice of . The fan in is a refinement of the normal fan of . For , a combination of monodromy and dual monodromy polytopes is needed. We let and . For each , we choose a vertex and let
Similarly, for each , we choose and let
These are well-defined up to translation and independent of the choices of and respectively.
Definition 2.9 ([27], Def. 1.60).
We say is simple if, for every , there are disjoint non-empty subsets
(where depends on ) such that
-
(1)
for and , if and only if and for some ;
-
(2)
is independent (up to translation) of and will be denoted by ; similarly, is independent (up to translation) of and will be denoted by ;
-
(3)
if is the standard basis in , then
are elementary simplices (i.e. a simplex whose only integral points are its vertices) in and respectively.
We need the following stronger condition in order to apply [28, Thm. 3.21] in a later stage:
Definition 2.10.
We say is strongly simple if it is simple, and for every , both and are standard simplices.
Example 2.11.
Consider the -dimensional example in Example 2.3. Following [1, Ex. 6.82(1)], we may choose the two adjacent vertices in Figure 1 to be and which bound a -cell . The two adjacent maximal cells are given by where and where . The tangent lattice can be identified with equipped with the basis , . If we let be a loop going from to through and going back to through , we have
for . Therefore, we have , and . This is an example of a positive and strongly simple (Definitions 2.8 and 2.10).
Example 2.12.
Next we consider the two types of -vertex in Example 2.4.
We begin with -vertex of type in Figure 4a. Following [1, Ex. 6.82(2)], the three vertices can be chosen to be
and , are -cells of lying in the affine hyperplanes with dual vector and respectively. If we identify with via parallel transport and choose the basis of as
then the monodromy transformations are given by
where is the loop going from to through and going back to through , with indices of ’s taken modulo . In this case, we have , is a -simplex and is a -simplex.
For the -vertex of type II in Figure 4b, we can choose
which are the end-points of a -cell . We choose the three maximal cells , and intersecting at to be the -cells lying in affine hyperplanes defined by , where
Let be the loop going from to through and then going back to through , with indices taken to be modulo . Then the corresponding monodromy transformations are given by
with respect to the basis
In this case, , is a -simplex and is a -simplex.
Both examples are positive and strongly simple.
Throughout this paper, we always assume that is positive and strongly simple. In particular, both and are standard simplices of positive dimensions, and (resp. ) is an internal direct summand of (resp. ).
2.3. Cone construction by gluing open affine charts
In this subsection, we recall the cone construction of the maximally degenerate Calabi–Yau , following [27] and [29, §1.2]. For this purpose, we take and to be the positive real axis in Notation 1.4. Throughout this paper, we will work in the category of analytic schemes.
We will construct as a gluing of affine analytic schemes parametrized by the vertices of . For each vertex , we consider the fan and take the analytic affine toric variety
where means analytification of the algebraic affine scheme given by . Here, the monoid structure for a general fan is given by
and we set in taking (by abuse of notation, we use to stand for both the fan and the monoid associated to a fan if there is no confusion); in other words, the ring is defined explicitly as
where denotes the support of the fan .
To glue these affine analytic schemes together, we need affine subschemes associated to with and natural open embeddings for . First, for such that , we consider the localization of at defined by
here recall that is the cone in (see the definition of a fan structure before Definition 2.2). This defines a new complete fan in consisting of convex, but not necessarily strictly convex, cones. If contains another vertex , we can identify the fans and as follows: for each maximal , we identify the maximal cones and by identifying the tangent spaces using parallel transport through . Patching these identifications for all together, we get a piecewise linear transformation from to , identifying the fans and and hence the corresponding monoids. This defines the affine analytic scheme
up to a unique isomorphism. Notice that can be identified (non-canonically) with the fan in , so actually
where is a complex torus.
For any , there is a map of monoids given by
(though there is no fan map from to in general), and hence a ring map
This gives an open inclusion of affine schemes
and hence a functor defined by
for .
We can further introduce twistings of the gluing of the affine analytic schemes . Toric automorphisms of are in bijection with the set of -valued piecewise multiplicative maps on with respect to the fan . Explicitly, for each maximal cone with , there is a monoid homomorphism such that if also contains , then . Denote by the multiplicative group of -valued piecewise multiplicative maps on . The group a priori depends on the choice of ; however, for different choices, say and , the groups can be identified via the identification . For , there is a natural restriction map given by restricting to those maximal cells with .
Definition 2.13 ([29], Def. 1.18).
A choice of open gluing data (for the cone construction) for is a set of elements such that
-
(1)
for all , and
-
(2)
if , then
Two choices of open gluing data are said to be cohomologous if there exists a system , with for each , such that whenever .
The set of cohomology classes of choices of open gluing data is a group under multiplication, denoted as . For , we will denote also by the corresponding toric automorphism on which is explicitly given by for . If is a choice of open gluing data, then we can define an -twisted functor by setting on objects and on morphisms. This defines the analytic scheme
Gross–Siebert [27] showed that as schemes when are cohomologous.
Remark 2.14.
Given , one can define a closed stratum of dimension by gluing together the -dimensional toric strata corresponding to the cones in , for all . Abstractly, it is isomorphic to the toric variety associated to the polyhedron . Also, for every pair , there is a natural inclusion . One can alternatively construct by gluing along the closed strata ’s according to the polyhedral decomposition; see [27, §2.2].
We recall the following definition from [27], which serves as an alternative set of combinatorial data for encoding .
Definition 2.15 ([27], Def. 3.25 and [29], Def. 1.20).
Let and with . For a vertex , we define
where are the two unique maximal cells such that , is an element projecting to the generator in pointing to , and is the parallel transport of to through . is independent of the choice of .
Let and be the two unique maximal cells such that . Let be the unique primitive generator pointing to . For any two vertices , we have the formula
| (2.6) |
relating monodromy data to the open gluing data, where is as discussed in (2.2). The formula (2.6) describes the interaction between monodromy and a fixed . We shall further impose the following lifting condition from [27, Prop. 4.25] relating and monodromy data:
Condition 2.16.
We say a choice of open gluing data satisfies the lifting condition if for any two vertices with , we have whenever .
2.4. Log structures
We need to equip the analytic scheme with log structures. The main reference is [27, §3 - 5].
Definition 2.17.
Let be an analytic space, a log structure on is a sheaf of monoids together with a homomorphism of sheaves of (multiplicative) monoids such that is an isomorphism. The ghost sheaf of a log structure is defined as the quotient sheaf , whose monoid structure is written additively.
Example 2.18.
Let be an analytic space and be a closed analytic subspace of pure codimension one. We denote by the inclusion. Then the sheaf of monoids
together with the natural inclusion defines a log structure on .
We write if we want to emphasize the log structure on . A general way to define a log structure is to take an arbitary homomorphism of sheaves of monoids
and then define the associated log structure by
In particular, this allows us to define log structures on an analytic space by pulling back those on another analytic space via a morphism . More precisely, given a log structure on , the pullback log structure on is defined to be the log structure associated to the composition . For more details of the theory of log structures, readers are referred to, e.g., [27, §3].
Example 2.19.
Taking a toric monoid (i.e. for a cone ), we can define by sending , where is the constant sheaf with stalk . From this we obtain a log structure on the analytic toric variety . Note that this is a special case of Example 2.18, where we take and to be the toric boundary divisor.
Before we describe the log structures on , let us first specify a ghost sheaf over . Recall that the polyhedral decomposition is assumed to be regular, namely, there exists a strictly convex multi-valued piecewise linear function . For any , we take a strictly convex representative of on , and define
where . For any , we take an integral affine function on such that vanishes on , and agrees with on all of for any . This induces a map by sending , whose composition with the quotient map gives a map of cones that corresponds to the monoid homomorphism . The ’s glue together to give the ghost sheaf over . There is a well-defined section given by gluing for each .
One may then hope to find a log structure on which is log smooth and with ghost sheaf given by . However, due to the presence of non-trivial monodromies of the affine structure, this can only be done away from a complex codimension subset not containing any toric strata. Such log structures can be described by sections of a coherent sheaf supported on the scheme-theoretic singular locus . We now describe the sheaf and some of its sections called slab functions; readers are referred to [27, §3 and 4] for more details.
For every , we consider , where is the toric variety associated to the polytope . From the fact that the normal fan of is a refinement of the normal fan of the -dimensional simplex (as in §2.2), we have a toric morphism
| (2.7) |
Now, corresponds to on . We let on , and define
| (2.8) |
Sections of can be described explicitly. For each , we consider the open subscheme of and the local trivialization
whose sections over are given by . Given where corresponding to , these local sections obey the change of coordinates given by
| (2.9) |
where and are part of the open gluing data . The section is said to be normalized if takes the value at the -dimensional toric stratum corresponding to a vertex , for all . We will restrict ourselves to normalized sections of . The complex codimension subset is taken to be the zero locus of on .
Only a subset of normalized sections of corresponds to log structures. For every vertex and containing , we choose a cyclic ordering of codimension one cells containing according to an orientation of . Let be the positively oriented normal to . Then the condition for to define a log structure is given by
| (2.10) |
where the group structure on is additive and that on is multiplicative. If is a normalized section satisfying this condition, we call the ’s slab functions.
Theorem 2.20 ([27], Thm. 5.2).
From now on, we always assume that is compact. To describe the log structure in Theorem 2.20, we first construct some local smoothing models: For each vertex , we represent the strictly convex piecewise linear function in a small neighborhood of by a strictly convex piecewise linear (so that ) and set
The element gives rise to a regular function on . We have a natural identification
through which we view as the toric boundary divisor in that corresponds to the holomorphic function , and as a local model for smoothing .
Using these local models, we can now describe the log structure around a point . On a neighborhood of , the local smoothing model is given by composing the two inclusions and . The natural monoid homomorphism defined by sending determines a log structure on which restricts to one on the toric boundary divisor . We further twist the inclusion as
| (2.11) |
here, for each , is chosen as an invertible holomorphic function on , where we denote , and such that they satisfy the relations
| (2.12) |
Then pulling back the log structure on via produces a log structure on which is log smooth.
These local choices of ’s are also required to be determined by the slab functions ’s, up to equivalences. Here, we shall just give the formula relating them; see [27, Thm. 3.22] for details. For any containing and two maximal cells such that , we take generating with some such that . Then the required relation is given by
| (2.13) |
which is independent of the choices of and .
By abuse of notation, we also let be the -th order thickening of over in the model under the above embedding. Then there is a natural divisorial log structure on over coming from restriction of the log structure on over (i.e. Example 2.18, which is the same as the one given by Example 2.19 in this case). Restricting to reproduces the log structure we constructed above, which is the log structure of over the log point locally around . We have a Cartesian diagram of log spaces
| (2.14) |
Next we describe the log structure around a singular point for some . Viewing where is a section of , we let and write . For every , we have the data ’s, ’s, and described in Definition 2.9 because is simple. Since the normal fan of is a refinement of , we have a natural toric morphism
| (2.15) |
and the identification . By the proof of [27, Thm. 5.2], is completely determined by the gluing data and the associated monodromy polytope where . In particular, we have and for . Locally, if we write by choosing some , then, for each , there exists an analytic function on such that for .
According to [28, §2.1], for each , we have , which gives
| (2.16) |
By convention, we write . By rearranging the indices ’s, we can assume that and . We introduce the convention that for and for . Then the local smoothing model near is constructed as , where
| (2.17) |
, and the distinguished element defines a family
by sending . The central fiber is given by , where
| (2.18) |
is equipped with the monoid structure
We have the ring isomorphism induced by the monoid isomorphism defined by sending .
We also fix some isomorphism coming from the identification of with the fan in . Taking a sufficiently small neighborhood of such that if , we define a map by composing with the map described on generators by
| (2.19) |
here is the -th coordinate function of , is the -th coordinate function of chosen so that is non-degenerate on ; also, each is an invertible holomorphic functions on , and they satisfy the equations (2.12) and (2.13) where we replace by
Letting be the -th order thickening of over in the model under the above embedding, we have a natural divisorial log structure on over induced from the inclusion (i.e. Example 2.18). Restricting it to gives the log structure of over the log point locally around .
3. A generalized moment map and the tropical singular locus on
In this section, we recall the construction of a generalized moment map from [43, Prop. 2.1]. Then we construct some convenient charts on the base tropical manifold and study its singular locus.
3.1. A generalized moment map
From this point onward, we will assume the vanishing of an obstruction class associated to the open gluing data , namely, , where the obstruction class is written multiplicatively (see [27, Thm. 2.34]). Under this assumption, one can construct an ample line bundle on as follows: For each polytope , by identifying (a closed stratum of described in Remark 2.14) with the projective toric variety associated to , we obtain an ample line bundle on . When the assumption holds, then there exists an isomorphism , for every pair , such that the isomorphisms ’s satisfy the cocycle condition, i.e. for every triple .333In fact, the vanishing of the obstruction class corresponds exactly to the validity of the cocycle condition. In particular, the degenerate Calabi–Yau is projective.
Sections of correspond to the lattice points . More precisely, given , there is a unique such that , and this determines a section of by toric geometry. This section extends uniquely as to such that . Further extending by to other cells gives a section of corresponding to , called a (-order) theta function. Now for a vertex , we can trivialize over using as the holomorphic frame. Then, for lying in a cell that contains , is of the form , where is a constant multiple of .
Under the above projectivity assumption, one can define a generalized moment map
| (3.1) |
following [43, Prop. 2.1]: First of all, the theta functions defines an embedding of . Restricting to each closed toric stratum , the only non-zero theta functions are those corresponding to . Also, there is an embedding of real tori such that the composition of with the inclusion is equivariant. The map is then defined by setting
| (3.2) |
which can be understood as a composition of maps
where is the standard moment map for and is the Lie algebra homomorphism induced by .
Fixing a vertex , we can naturally embed for all containing . Furthermore, we can patch the ’s into a linear map so that for each which contains . In particular, on the local chart associated with , we have the local description of the generalized moment map .
3.2. Construction of charts on
For any , we have
For later purposes, we would like to relate sufficiently small open convex subsets with Stein (or strongly -completed, as defined in [13]) open subsets . To do so, we need to introduce a specific collection of (non-affine) charts on .
Recall that there are natural maps and . By choosing a piecewise linear splitting , we have an identification of monoids , which induces the biholomorphism
where is a complex torus. Fixing a set of generators of the monoid , which is not necessarily a minimal set, we can define an embedding as an analytic subset using the functions ’s. We consider the real torus and its action on defined by , together with an embedding of real tori via , so that is -equivariant.
We consider the moment map defined by
| (3.3) |
which is obtained by composing the standard moment map , with the projection , . By [21, §4.2], induces a homeomorphism between the quotient and . Taking product with the log map (which is induced from the standard log map defined by ), we obtain a map ,444It depends on the choices of the splitting and the generators , but we omit these dependencies from our notations. and the following diagram
| (3.4) |
where is a homeomorphism which serves as a chart.
The homeomorphism exists because if we fix a vertex , then we can equip with an action by the real torus such that both and induce homeomorphisms from the quotient onto the images. The restriction of to , where is the zero cone, is a homeomorphism onto , which is nothing but (a generalized version of) the Legendre transform (see [21, §4.2] for the explicit formula); also, this homeomorphism is independent of the choices of the splitting and the generators .
The dependences of the chart on the choices of the splitting and the generators can be described as follows. First, if we choose another piecewise linear splitting , then there is a piecewise linear map recording the difference between and . The two corresponding coordinate charts and are then related by a homeomorphism such that
where for some point and runs through , via the formula . Second, if we choose another set of generators ’s, then the corresponding maps are related by a continuous map which maps each cone back to itself. This is because both induce a homeomorphism between and .
Now suppose that . We want to see how the charts , can be glued together in a compatible manner. We first make a compatible choice of splittings. So we fix a vertex and a piecewise linear splitting . We then choose a piecewise linear splitting such that is mapped into for any . Together with the natural maps and , we obtain an isomorphism . By composing together , and the natural monoid homomorphism , we get a splitting .
Using these choices of splittings, we have a biholomorphism
which fits into the following diagram
| (3.5) |
Here, the bottom left horizontal map is induced from a splitting obtained by composing with the splitting , and then identifying with the image lattice . The appearance of in the diagram is due to the twisting of by the open gluing data when it is glued to .
We also have to make a compatible choice of the generators and . First note that the restriction of to the open subset depends only on the subcollection of which contains those ’s that belong to some cone . We choose the set of generators for , with , to be the projection of through the natural map . Each can be expressed as for some , through the splitting . Notice that if , then we have and hence . By tracing through the biholomorphism in (3.5) and taking either the modulus or the log map, we have a map
satisfying
| (3.6) |
where . Here, is the part of the open gluing data associated to , and is the unique element representing the linear map defined by . For instance, the holomorphic function is identified with in , resulting in the expression on the right hand side. We have , where we use the splitting to obtain an isomorphism and an identification of the domains of the two maps and .
Lemma 3.1.
There is a base of open subsets of such that the preimage is Stein for any .
Proof.
First of all, it is well-known that analytic spaces associated to affine varieties are Stein. So is Stein for any . Now we fix a point . It suffices to show that there is a local base of such that the preimage is Stein for each . We work locally on . Consider the diagram (3.4) and write , where is the origin. By [13, Ch. 1, Ex. 7.4], the preimage under the log map is Stein for any convex which contains . Again by [13, Ch. 1, Ex. 7.4], any subset
where ’s are holomorphic functions, is Stein. By taking ’s to be the functions ’s associated to the set of all non-zero generators in and sufficiently small, we have a subset
of such that the preimage is Stein. Therefore, we can construct a local base of such that the preimage is Stein for any . Finally, since a product of Stein open subsets is Stein, we obtain our desired local base by taking the products of these subsets. ∎
3.3. The tropical singular locus of
We now specify a codimension singular locus of the affine structure using the charts introduced in (3.4) for such that . Given the chart that maps to , we define the tropical singular locus by requiring that
| (3.7) |
where is the normal fan of the polytope , and is the zero cone in ; here, is the element in representing the linear map , which is independent of the vertex . A subset of the form in (3.7) is called a stratum of in . The locus is independent of the choices of the splittings ’s and generators used to construct the charts ’s.
Remark 3.2.
Our definition of the singular locus is similar to those in [27, 29]; the only difference is that our locus is a collection of polyhedra in , instead of . Note that is homeomorphic to by the Legendre transform. This modification is needed for our construction of the contraction map below, where we need to consider the convex open subsets in , instead of those in .
Lemma 3.3.
For and a stratum in , the intersection of the closure in with is a union of strata of in .
Proof.
We consider the map described in equation (3.6) and take a neighborhood of a point in , where is some sufficiently small neighborhood of in . By shrinking if necessary, we may assume that , where is some element in . Writing , where are the components of according to the chosen decomposition , the equality follows from the compatibility of the open gluing data in Definition 2.13. If intersects the open subset , then must be the dual cone of some face in . The intersection is of the form
for some ( is absorbed by ), where is the dual cone of in , and hence we have . Therefore, the intersection of with in the open subset is given by , which is a union of strata. ∎
The tropical singular locus is naturally equipped with a stratification, where a stratum is given by for some cone of for some . We use the notation to denote the set of -dimensional strata of . The affine structure on introduced right after Definition 2.2 in §2.1 can be naturally extended to as in [29].
If we consider for some and , the corresponding monodromy transformation is non-trivial if and only if and , where is as in Definition 2.9. Therefore, the part of the singular locus lying in is determined by the subsets ’s. We may further define the essential singular locus to include only those strata contained in with non-trivial monodromy around them. We observe that the affine structure can be further extended to .
More explicitly, we have a projection
in which can be treated as a direct summand as in §2.2. So we can consider the pullback of the fan via the map , and realize as a refinement of this fan. Similarly, we have and the fan in under pullback via . The intersection can be described by replacing with the condition , with a stratum denoted by . This gives a stratification on .
Lemma 3.4.
For and a stratum in , the intersection of the closure in with is a union of strata of in .
Proof.
Given , we take a change of coordinate map together with a neighborhood as in the proof of Lemma 3.3. We need to show that for some cone . Let be the monodromy polytopes of , and be those of such that is the face of parallel to for . Then we have direct sum decompositions and . We can further choose an inclusion
in other words, for every , any in Definition 2.9 is not containing . For every and any , the element is zero for any two vertices of . We have the identification
As a result, any cone of codimension greater than intersecting will be a pullback of a cone under the projection to . Consider the commutative diagram of projection maps
| (3.8) |
we see that, in the open subset , every cone of codimension greater than coming from pullback via is a further pullback via . As a consequence, it must be of the form in . ∎
3.3.1. Contraction of to
We would like to relate the amoeba with the tropical singular locus introduced above.
Assumption 3.5.
We assume the existence of a surjective contraction map which is isotopic to the identity and satisfies the following conditions:
-
(1)
and the restriction is a homeomorphism.
-
(2)
maps into the essential singular locus .
-
(3)
For each , we have .
-
(4)
For each with , we have a decomposition
of the intersection into connected components ’s, where each is contractible and is the unique component containing the vertex .
-
(5)
For each and each point , is a connected compact subset.
-
(6)
For each and each point , there exists a local base around such that is Stein for every , and for any , we have for sufficiently small .
When , we can take because from [27, Ex. 1.62], we see that is a finite collection of points, with at most one point lying in each closed stratum , and the amoeba is exactly the image of under the generalized moment map .
When , the amoeba can possibly be of codimension one and we need to construct a contraction map as shown in Figure 4.
For , again from [27, Ex. 1.62], we see that if , then there is exactly one and , and is a line segment of affine length . In this case, consists of only one point, given by the intersection of the zero locus with . Taking to be the primitive vector in starting at that points into , we can write . Applying the log map , we see that . Therefore, for an edge , we can define to be the identity on .
On a codimension one cell such that (see Figure 5), we consider the log map , and take a sufficiently large polytope (colored purple in Figure 5) so that is a disjoint union of legs. We first contract each leg to the tropical singular locus (colored blue in Figure 5) along the normal direction to the tropical singular locus. Next, we contract the polytope to the -dimensional stratum of . Notice that the restriction of to the tropical singular locus is not the identity but rather a contraction onto itself. Once the contraction map is constructed for all codimension one cells , we can then extend it continuously to the whole of so that it is a diffeomorphism on for every maximal cell . The map is chosen such that the preimage for every point is a convex polytope in 2. Therefore, given any open subset which contains , we can find some convex open neighborhood of giving the Stein open subset . By taking in the chart as in the proof of Lemma 3.1, we have the open subset that satisfies condition (5) in Assumption 3.5.
In general, we need to construct inductively for each , so that the preimage is convex in the chart and the codimension one amoeba is contracted to the codimension 2 tropical singular locus . The reason for introducing such a contraction map is that we can modify the generalized moment map to one which is more closely related with tropical geometry:
Definition 3.6.
We call the composition the modified moment map.
3.3.2. Monodromy invariant differential forms on
Outside of the essential singular locus , we have a nice integral affine manifold , on which we can talk about the sheaf of (-valued) de Rham differential forms. But in fact, we can extend its definition to as well using monodromy invariant differential forms.
We consider the inclusion and the natural exact sequence
| (3.9) |
where denotes the sheaf of integral cotangent vectors on . For any , the stalk at a point can be described using the chart in (3.4). Using the description in §3.3, we have for some . Taking a vertex , we can consider the monodromy transformations ’s around the strata ’s that contain in their closures. We can identify the stalk as the subset of invariant elements of under all such monodromy transformations. Since is a cone, we have . Using the natural projection map , we have the identification . There is a direct sum decomposition , depending on a decomposition . This gives the map
| (3.10) |
in a sufficiently small neighborhood , locally defined up to a translation in . We need to describe the compatibility between the map associated to a point and that to a point such that .
The first case is when . We let for some . Then, after choosing suitable translations in for the maps and , we have the following commutative diagram:
| (3.11) |
The second case is when . Making use of the change of charts in equation (3.6), and the description in the proof of Lemma 3.4, we write
for some cone of positive codimension. In , we may assume is the pullback of a cone via as in equation (3.8). Since , we have and hence . Therefore, from , we obtain , inducing the map . As a result, we still have the commutative diagram (3.11) for a point sufficiently close to .
Definition 3.8.
Given as above, the stalk of at is defined as the stalk of the pullback of the sheaf of smooth de Rham forms on , which is equipped with the de Rham differential . This defines the complex of monodromy invariant smooth differential forms on . A section is a collection of elements , such that each can be represented by in a small neighborhood for some smooth form on , and satisfies the relation in for every .
Example 3.9.
In the -dimensional case in Example 2.11, we consider a singular point
for some . In this case, we can take to be the -dimenisonal stratum in and we have . Taking a generator of , we get an invariant affine coordinate which is the normal affine coordinate of . The stalk is then identified with the pullback of the space of germs of smooth differential forms from via . In particular, .
For the -vertex of type II in Example 2.12, the situation is similar to the -dimensional case. For , we still have , and in this case, are the two invariant affine coordinates. We can identify as the pullback of the space of germs of smooth differential forms from via .
For the -vertex of type I in Example 2.12, we use the identification via for the -dimensional cell separating two maximal cells and . In this case, is as shown (in blue color) in Figure 5 and is the fan of . If is the -dimensional stratum of , we have and as an invariant affine coordinate. If is a point on a leg of the -vertex, we have with coming from a generator of and coming from a generator of .
It follows from the definition that is a resolution. We shall also prove the existence of a partition of unity.
Lemma 3.10.
Given any and a sufficiently small neighborhood , there exists with compact support in such that and near . (Since is a subsheaf of the sheaf of continuous functions on , we can talk about the value for and .)
Proof.
If , the statement is a standard fact. So we assume that for some . As above, we can write . Furthermore, since is a cone in the fan , has as a direct summand, and the description of is compatible with the direct sum decomposition of . We may further assume that and is a simplex.
If is not the smallest cone (i.e. the one consisting of just the origin in ), we have a decomposition and the natural projection . Then, locally near , we can write the normal fan as for some normal fan of a lower dimensional simplex. For any vector tangent to at and the corresponding affine function locally near , we always have . This allows us to construct a bump function along the -direction. So we are reduced to the case when is the smallest cone in the fan .
Now we construct the function near the origin by induction on the dimension of the fan . When , it is the fan of consisting of three cones -, and +. One can construct the bump function which is equal to near and supported in a sufficiently small neighborhood of . For the induction step, we consider an -dimensional fan . For any point near but not equal to , we have for some . Then we can decompose locally as . Applying the induction hypothesis to gives a bump function compactly supported in any sufficiently small neighborhood of (for the directions, we do not need the induction hypothesis to get the bump function). This produces a partition of unity outside . Finally, letting and extending it continuously to the origin gives the desired function. ∎
Lemma 3.10 produces a partition of unity for the complex of monodromy invariant differential forms on , which satisfies the requirement in Condition 4.7 below. In particular, the cohomology of computes . Given a point , we can take an element , compactly supported in an arbitrarily small neighborhood , to represent a non-zero element in the cohomology .
4. Smoothing of maximally degenerate Calabi–Yau varieties via dgBV algebras
In this section, we review and refine the results in [8] concerning smoothing of the maximally degenerate Calabi–Yau log variety over using the local smoothing models ’s specified in §2.4. In order to relate with tropical geometry on , we will choose so that it is the pre-image of an open subset in .
4.1. Good covers and local smoothing data
Given and a point , we take a sufficiently small open subset . We need to construct a local smoothing model on the Stein open subset .
- •
-
•
If , we assume that for and for other ’s. Note that may not be a small open subset in as we may contract a polytope via (Figure 5). If we write as lattices, then for each direct summand , we have a commutative diagram
so that both and are coming from pullbacks of some subsets under the projection maps and respectively. From this, we see that and while for other ’s. Now we take for and otherwise accordingly. Then we can take introduced in (2.17) and the map defined by
(4.1) Note that the third line of this formula is different from that of equation (2.19) because we do not specify a point . By shrinking if necessary, one can show that it is an embedding using an argument similar to [28, Thm. 2.6]. This is possible because we can check that the Jacobian appearing in the proof of [28, Thm. 2.6] is invertible for all point in , which is a connected compact subset by property in Assumption 3.5.
Condition 4.1.
An open cover of is said to be good if
-
(1)
for each , there exists a unique such that for some ;
-
(2)
only when or , and if this is the case, we have either or .
Given a good cover of , we have the corresponding Stein open cover of given by for each . For each , the infinitesimal local smoothing model is given as a log space over (see (2.14)). Let be the -order thickening over and be the open inclusion. As in [8, §8], we obtain coherent sheaves of BV algebras (and modules) over from these local smoothing models. But for the purpose of this paper, we would like to push forward these coherent sheaves to and work with the open subsets ’s. This leads to the following modification of [8, Def. 7.6] (see also [8, Def. 2.14 and 2.20]):
Definition 4.2.
For each , we define
-
•
the sheaf of -order polyvector fields to be (i.e. push-forward of relative log polyvector fields on );
-
•
the -order log de Rham complex to be (i.e. push-forward of log de Rham differentials) equipped with the de Rham differential which is naturally a dg module over ;
-
•
the local log volume form as a nowhere vanishing element in and the -order volume form to be .
Given , there are natural maps which induce the maps . Before taking the push-forward , each is a sheaf of flat -modules with the property that by [17, Cor. 7.4 and 7.9]. In other words, we have a short exact sequence of coherent sheaves
Applying , which is exact, we get
As a result, we see that is a sheaf of flat -modules on , so we have for each ; a similar statement holds for .
A natural filtration is given by and taking wedge product defines the natural sheaf isomorphism . We have the space of relative log de Rham differentials.
There is a natural action for and given by contracting a logarithmic holomorphic vector field with a logarithmic holomorphic form . To simplify notations, for , we often simply write , suppressing the contraction . We define the Lie derivative via the formula (or equivalently, ). By contracting with , we get a sheaf isomorphism , which defines the BV operator by . We call it the BV operator because the BV identity:
| (4.2) |
for , where we put , defines a graded Lie bracket. This gives the structure of a sheaf of BV algebras.
4.2. An explicit description of the sheaf of log de Rham forms
Let us consider and the local model near described in §4.1, with and as in (2.17), (2.18) and an embedding . We may treat as a compact subset of via the identification . For each , we denote the corresponding element by and the corresponding function by . Similar to [17, Lem. 7.14], the germs of holomorphic functions near in the space can be written as
| (4.3) |
where is a monoid morphism such that , and it is equipped with the product (but note that in general). Thus we have .
To describe the differential forms, we consider the vector space , regarded as the space of -forms on . Write for and set , as a subset of . For an element , we have the corresponding -form under the association between and . Let be the power set of and write for . A computation for sections of the sheaf from [28, Prop. 1.12] and [17, Lem. 7.14] can then be rephrased as the following lemma.
Lemma 4.3 ([28, 17]).
The space of germs of sections of near is a subspace of given by elements of the form
where and the subspace is given as follows: we consider the pullback of the product of normal fans to and take for , where is the smallest cone in containing .
Here we can treat since is a direct summand of . A similar description for is simply given by quotienting out the direct summand in the above formula for . In particular, if we restrict ourselves to the case , a general element can be written as
One can choose a nowhere vanishing element
for some nonzero element , which is well defined up to rescaling. Any element in can be written as for some .
Recall that the subset is intersecting the singular locus (as in §4.1), where is the coordinate function of with simple zeros along for . There is a change of coordinates between a neighborhood of in and that of in given by
Under the map , we have for some connected compact subset . In the coordinates , we find that can be written as near for some nowhere vanishing function .
Lemma 4.4.
When (i.e. in the above discussion), the top cohomology group is isomorphic to , which is generated by the element .
Proof.
Given a general element as above, first observe that we can write , where and . We take a basis of , and also a partition of the lattice points in such that for . Letting
we have . So we only need to consider elements of the form . If for some , we may take for some . Now this is equivalent to as forms in . This reduces the problem to .
Working in with coordinates ’s, we can write
using the fact that is multi-circular. By writing with , we can see that any element can be represented as in the quotient , for some constant . ∎
From this lemma, we conclude that the top cohomology sheaf is isomorphic to the locally constant sheaf over .
Lemma 4.5.
The volume element is non-zero in for every .
Proof.
We first consider the case when for some maximal cell . The toric stratum associated to is equipped with the natural divisorial log structure induced from its boundary divisor. Then the sheaf of log derivations for is isomorphic to . By [28, Lem. 3.12], we have in , where is nowhere vanishing and is a non-zero constant . Thus is non-zero in the cohomology as the same is true for . Next we consider a general point . If the statement is not true, we will have for some . Then there is an open neighborhood such that this relation continues to hold. As , for those maximal cells which contain the point , we can take a nearby point and conclude that in . This contradicts the previous case. ∎
Lemma 4.6.
Suppose that . For an element of the form
with satisfying , there exist and with such that
| (4.4) |
in , where we recall that .
Proof.
To simplify notations in this proof, we will drop the subscript . We prove the first statement by induction on . The initial case is trivial. Assuming that this has been done for the -order, then, by taking an arbitrary lifting of to the -order, we have
for some . By Lemmas 4.4 and 4.5, we have for some and some suitable constant . Letting and , we have
By defining in , we obtain the desired expression. ∎
4.3. A global pre-dgBV algebra from gluing
One approach for smoothing is to look for gluing morphisms between the local smoothing models which satisfy the cocycle condition, from which one obtain a -order thickening over . This was done by Kontsevich–Soibelman [36] (in 2d) and Gross–Siebert [29] (in general dimensions) using consistent scattering diagrams. If such gluing morphisms ’s are available, one can certainly glue the global -order sheaves , and the volume form .
In [8], we instead took suitable dg-resolutions ’s of the sheaves ’s (more precisely, we used the Thom–Whitney resolution in [8, §3]) to construct gluings
of sheaves which only preserve the Gerstenhaber algebra structure but not the differential. The key discovery in [8] was that, as the sheaves ’s are soft, such a gluing problem could be solved without any information from the complicated scattering diagrams. What we obtained is a pre-dgBV algebra555This was originally called an almost dgBV algebra in [8], but we later found the name pre-dgBV algebra from [16] more appropriate. , in which the differential squares to zero only modulo . Using well-known algebraic techniques [48, 33], we can solve the Maurer–Cartan equation and construct the thickening . In this subsection, we will summarize the whole procedure, incorporating the nice reformulation by Felten [16] in terms of deformations of Gerstenhaber algebras.
To begin with, we assume the following condition holds:
Condition 4.7.
There is a sheaf of unital differential graded algebras (abbrev. as dga) (over or ) over , with degrees for some , such that
-
•
the natural inclusion (or ) of the locally constant sheaf (concentrated at degree ) gives a resolution, and
-
•
for any open cover , there is a partition of unity subordinate to , i.e. we have with and such that is locally finite and .
It is easy to construct such an and there are many natural choices. For instance, if is a smooth manifold, then we can simply take the usual de Rham complex on . In §3.3.2, the sheaf of monodromy invariant differential forms we constructed using the (singular) integral affine structure on is another possible choice for (with degrees ). Yet another variant, namely the sheaf of monodromy invariant tropical differential forms, will be constructed in §5.1; this links tropical geometry on with the smoothing of the maximally degenerate Calabi–Yau variety .
Let us recall how to obtain a gluing of the dg resolutions of the sheaves and using any possible choice of such an . We fix a good cover of and the corresponding Stein open cover of , where for each .
Definition 4.8.
We define and , which gives a sheaf of dgBV algebras over . The dgBV structure is defined componentwise by
for and for each open subset .
Definition 4.9.
We define and , which gives a sheaf of dgas over equipped with the natural filtration inherited from . The structures are defined componentwise by
for and for each open subset .
There is an action of on by contraction defined by the formula
for , and for each open subset . Note that the local holomorphic volume form satisfies , and we have the identity of operators.
The next step is to consider gluing of the local sheaves ’s for higher orders . Similar constructions have been done in [8, 16] using the combinatorial Thom–Whitney resolution for the sheaves ’s. We make suitable modifications of those arguments to fit into our current setting.
First, since and are divisorial deformations (in the sense of [28, Def. 2.7]) of the intersection , we can use [28, Thm. 2.11] and the fact that is Stein to obtain an isomorphism of divisorial deformations which induces the gluing morphism that in turn gives .
Definition 4.10.
A -order Gerstenhaber deformation of is a collection of gluing morphisms of the form
for some with , such that the cocycle condition
is satisfied.
An isomorphism between two -order Gerstenhaber deformations and is a collection of automorphisms of the form
for some with , such that
A slight modification of [16, Lem. 6.6], with essentially the same proof, gives the following:
Proposition 4.11.
Given a -order Gerstenhaber deformation , the obstruction to the existence of a lifting to a -order deformation lies in the Čech cohomology (with respect to the cover )
The isomorphism classes of -order liftings are in
Fixing a -order lifting , the automorphisms fixing are in
Since satisfies Condition 4.7, we have . In particular, we always have a set of compatible Gerstenhaber deformations where and any two of them are equivalent. Fixing such a set , we obtain a set of Gerstenhaber algebras which is compatible, in the sense that there are natural identifications .
We can also glue the local sheaves ’s of dgas using . First, we can define using . For each fixed , we can write as before. Then
| (4.5) |
where we recall that , preserves the dga structure and the filtration on ’s. As a result, we obtain a set of compatible sheaves of dgas. The contraction action is also compatible with the gluing construction, so we have a natural action of on .
Next, we glue the operators ’s and ’s.
Definition 4.12.
A -order pre-differential on is a degree operator obtained from gluing the operators specified by a collection of elements such that and
Two pre-differentials and are equivalent if there is a Gerstenhaber automorphism (for the deformation ) such that .
Notice that we only have , which is why we call it a pre-differential. Using the argument in [8, Thm. 3.34] or [16, Lem. 8.1], we can always lift any -order pre-differential to a -order pre-differential. Furthermore, any two such liftings differ by a global element . We fix a set of such compatible pre-differentials. For each , the action of on is given by gluing of the action of on . On the other hand, the elements
| (4.6) |
glue to give a global element , and for different ’s, these elements are compatible. Computation shows that on and on .
To glue the operators ’s, we need to glue the local volume elements ’s to a global . We consider an element of the form , where satisfies . Given a -order global volume element , we take a lifting such that
for some element . By construction, gives a Čech -cycle in which is exact. So there exist ’s such that , and we can modify as , which gives the desired -order volume element. Inductively, we can construct compatible volume elements , . Any two such volume elements and differ by , where is some global element. Notice that unless mod .
Using the volume element (we omit the dependence on if there is no confusion), we may now define the global BV operator by
| (4.7) |
which can locally be written as . We have . The local elements
| (4.8) |
glue to give a global element which satisfies . Also, the elements and satisfy the relation by a local calculation.
In summary, we obtain pre-dgBV algebras and pre-dgas with a natural contraction action of on , and also volume elements . We set
and define a total de Rham operator by
| (4.9) |
which preserves the filtration . Using the contraction to pull back the operator, we obtain the operator acting on . Direct computation shows that , and indeed it plays the role of the de Rham differential on a smooth manifold. Readers may consult [8, §4.2] for the computations and more details.
Definition 4.13.
We call (resp. ) the sheaf of (resp. -order) smooth relative polyvector fields over , and (resp. ) the sheaf of (resp. -order) smooth forms over . We denote the corresponding total complexes by (resp. ) and (resp. ).
4.4. Smoothing by solving the Maurer–Cartan equation
With the sheaf of pre-dgBV algebras defined, we can now consider the extended Maurer–Cartan equation
| (4.10) |
for , where . Setting gives the (classical) Maurer–Cartan equation
| (4.11) |
for . To inductively solve these equations, we need two conditions, namely the holomorphic Poincaré Lemma and the Hodge-to-de Rham degeneracy.
We begin with the holomorphic Poincaré Lemma, which is a local condition on the sheaves ’s. We consider the complex , where
There is a natural exact sequence
| (4.12) |
where as elements in .
Condition 4.14.
We say that the holomorphic Poincaré Lemma holds if at every point , the complex is acyclic, where denotes the stalk of at .
The holomorphic Poincaré Lemma for our setting was proved in [28, proof of Thm. 4.1], but a gap was subsequently pointed out by Felten–Filip–Ruddat in [17], who used a different strategy to close the gap and give a correct proof in [17, Thm. 1.10]. From this condition, we can see that the cohomology sheaf is free over (cf. [34, Lem. 4.1]). We will need freeness of the cohomology over , which can be seen by the following lemma (see [34] and [8, §4.3.2] for similar arguments).
Lemma 4.15.
Proof.
First of all, applying the functor to the exact sequence
gives the following exact sequence of sheaves on :
This is true because every sheaf in the first exact sequence is a direct limit of coherent analytic sheaves, commutes with direct limits of sheaves, and as the fiber is a compact Hausdorff topological space; see e.g. [32]. By taking a Cartan–Eilenberg resolution, we have the implication:
for any open subset , where is the derived global section functor in the derived category of sheaves. In our case, and we have . Furthermore, we see that
This can be seen by taking a double complex resolving such that computes . The spectral sequence associated to the double complex has the -page given by , which is if because is a direct limit of coherent analytic sheaves. Therefore, is a quasi-isomorphism. Combining these, we obtain that for each .
Next, by Condition 4.7, is a resolution with a partition of unity, so the cohomology of the complex
computes . Through an isomorphism , we can identify the operator:
with , and hence deduce that is acyclic for any open subset .
Now, we consider the global sheaf of complexes on obtained by gluing the local sheaves . We also have obtained by gluing , and obtained by gluing . Then there is an exact sequence of complexes of sheaves
To see that the complex is acyclic, we consider the total Čech complex associated to the cover . The associated spectral sequence has zero page, thus is indeed acyclic. As a result, the map is an isomorphism. Finally, surjectivity of the map follows from the fact that the isomorphism factors through . ∎
The Hodge-to-de Rham degeneracy is a global Hodge-theoretic condition on . We consider the Hodge filtration ; the spectral sequence associated to it computes the hypercohomology of the complex of sheaves
Condition 4.16.
We say that the Hodge-to-de Rham degeneracy holds for if the spectral sequence associated to the above Hodge filtration degenerates at .
Under the assumption that is strongly simple (Definition 2.10), the Hodge-to-de Rham degeneracy for the maximally degenerate Calabi–Yau scheme was proved in [28, Thm. 3.26]. This was later generalized to the case when is only simple (instead of strongly simple)666The subtle difference between the log Hodge group and the affine Hodge group when is just simple, instead of strongly simple, was studied in details by Ruddat in his thesis [42]. and further to log toroidal spaces in Felten–Filip–Ruddat [17] using different methods.
We consider the dgBV algebra equipped with the operator .
Lemma 4.17.
Under Condition 4.16 (the Hodge-to-de Rham degeneracy), is a free -module.
Proof.
Recall that we are working with a good cover , so that the inverse image is Stein for each . We have and
If is Stein, then and hence
The hypercohomology of is computed using the Čech double complex
with respect to the Stein open cover . Similarly, the hypercohomology of the complex is computed using the Čech double complex with respect to the cover ; here, the Hodge filtration is induced from the filtration .
These two Čech complexes, as well as their corresponding Hodge filtrations, are identified as for each . Hence, under Condition 4.16, we have degeneracy also for , or equivalently, that is a free -module. In view of the isomorphisms and
we conclude that is a free -module as well. ∎
For the purpose of this paper, we restrict ourselves to the case that
where and . The extended Maurer-Cartan equation (4.10) can be decomposed, according to orders in , into the (classical) Maurer–Cartan equation (4.11) for and the equation
| (4.13) |
Theorem 4.18.
Proof.
The first statement follows from [8, Thm. 5.6] and [8, Lem. 5.12]: Starting with a -order solution for (4.10), one can always use [8, Thm. 5.6] to lift it to a general . The argument in [8, Lem. 5.12] shows that we can choose such that the component of in is zero. As a result, the component of in is again a solution to (4.10).
For the second statement, we argue that, given , there always exists such that is a solution to (4.10). We need to solve the equation (4.13) by induction on the order . The initial case is trivial by taking . Suppose the equation can be solved for . Then we take an arbitrary lifting to the -order. We can define an element by
which satisfies . Therefore, the class lies in the cohomology
where the last equivalence is from [27, Prop. 2.37]. By our assumption in §2, we have , and hence we can find an element such that . Letting proves the induction step from the -order to the -order. Now, applying the first statement, we can lift the solution to which satisfies equation (4.10), and hence solves (4.11). ∎
From Theorem 4.18, we obtain a solution to the Maurer–Cartan equation (4.11), from which we obtain the sheaves and over . These sheaves are locally isomorphic to and , so we may treat them as obtained from gluing of the local sheaves ’s and ’s. From these, we can extract consistent and compatible gluings satisfying the cocycle condition, and hence obtain a -th order thichening of over ; see [8, §5.3]. Also, , as a section of over , defines a holomorphic volume form on the -th order thickening .
4.4.1. Normalized volume form
For later purposes, we need to further normalize the holomorphic volume
by adding a suitable power series to , so that the condition that , where is a nearby -torus in the smoothing, is satisfied.
The -order Hodge bundle over is defined as the cohomology
equipped with a Gauss–Manin connection , where is the connecting homomorphism of the long exact sequence associated to
| (4.14) |
here is the -dimensional graded vector space spanned by the degree element . We denote . Restricting to the -order, we have , which is a nilpotent operator acting on , where . If we consider the top cohomoloy , which is -dimensional, we see that . So is a flat connection without log poles at . Hence, we can find a basis (order by order in ) to identify , which also trivializes the flat connection as .
Since , we can fix a non-zero generator and choose a representative . Then the element (which may simply be written as ) represents a section in . A direct computation shows that , i.e. it is a flat section to all orders. The pairing with the -order volume form gives a non-zero element in .
Definition 4.19.
The volume form is said to be normalized if is flat under .
In other words, we can write under the identification
By modifying to , this can always be achieved. Further, after the modification, still solves (4.10).
5. From smoothing of Calabi–Yau varieties to tropical geometry
5.1. Tropical differential forms
To tropicalize the pre-dgBV algebra , we need to replace the Thom–Whitney resolution used in [8] by a geometric resolution. To do so, we first need to recall some background materials from our previous works [7, §4.2.3] and [9, §3.2]. Of crucial importance is the notion of differential forms with asymptotic support (which will be called tropical differential forms in this paper) that originated from multi-valued Morse theory and Witten deformations. Such differential forms can be regarded as distribution-valued forms supported on tropical polyhedral subsets. This key notion allows us to develop tropical intersection theory via differential forms, and in particular, define the intersection pairing between possibly non-transversal tropical polyhedral subsets simply using the wedge product.
Let be an open subset of . We consider the space , where we take sections of and is a coordinate on . Let be the subset of -forms such that, for each , there exist a neighborhood , constants , and a sufficiently small real number such that for all and for ; here, the -norm is defined by for any section of the tensor bundle , where we fix a constant metric on and use the induced metric on ; denotes an operator of the form , where is a torsion-free, flat connection defining an affine structure on and is an affine coordinate system (note that is not the Gauss–Manin connection in the previous section). Similarly, let be the set of -forms such that, for each , there exist a neighborhood , a constant , and a sufficiently small real number such that for all and for .
The assignment (resp. ) defines a sheaf (resp. ) on ([7, Defs. 4.15 & 4.16]). Note that and are closed under the wedge product, and the de Rham differential . Since is a dg ideal of , the quotient is a sheaf of dgas when equipped with the de Rham differential.
Now suppose is a convex open set. By a tropical polyhedral subset of , we mean a connected convex subset of which is defined by finitely many affine equations or inequalities over of the form .
Definition 5.1 ([7], Def. 4.19).
A -form is said to have asymptotic support on a closed codimension tropical polyhedral subset with weight , denoted as , if the following conditions are satisfied:
-
(1)
For any , there is a neighborhood such that .
-
(2)
There exists a neighborhood of such that on , where is a non-zero affine -form (defined up to non-zero constant) which is normal to , and .
-
(3)
For any , there exists a convex neighborhood equipped with an affine coordinate system such that parametrizes codimension affine linear subspaces of parallel to , with corresponding to the subspace containing . With the foliation , where and is the normal bundle of , we require that, for all and multi-indices , the estimate
holds for some constant and , where and .777For , we use the convention that and also set .
Observe that and . It follows that
The weight defines a filtration of (we drop the dependence from the notation whenever it is clear from the context):888Note that is equal to the codimension of .
This filtration, which keeps track of the polynomial order of for -forms with asymptotic support on , provides a convenient tool to express and prove results in asymptotic analysis.
Definition 5.2 ([9], Def. 3.10).
A differential -form is in if there exist polyhedral subsets of codimension such that . If, moreover, , then we write . For every , let .
Example 5.3.
Let and be an affine coordinate on . Then we consider the -dependent -form
Direct calculations in [7, Lem 4.12] showed that has asymptotic support on the hyperplane defined by .
The hyperplane separates into two chambers and . If we fix a base point in and apply the integral operator in [7, Lem. 4.23], we obtain which has asymptotic support on , playing the role of a step function.
Taking finite products of elements of the above form, we obtain with asymptotic support on arbitrary tropical polyhedral subsets of . Any forms obtained from a finite number of steps of applying the differential , applying the integral operator and taking wedge product are in .
We say that two closed tropical polyhedral subsets of codimension intersect transversally if the affine subspaces of codimension and which contain and , respectively, intersect transversally. This definition applies also when or .
Lemma 5.4 ([7, Lem. 4.22]).
-
(1)
Let be closed tropical polyhedral subsets of codimension , and , respectively, such that contains and is normal to . Then if and intersect transversally with , and otherwise.
-
(2)
We have . In particular, is a dg subalgebra and is a dg ideal.
Definition 5.5.
Let be the sheafification of the presheaf defined by . We call the quotient sheaf the sheaf of tropical differential forms, which is a sheaf of dgas on with structures .
From [9, Lem. 3.6], we learn that is a resolution. Furthermore, given any point and a sufficiently small neighborhood , we can show that there exists with compact support in and satisfying near (using an argument similar to the proof of Lemma 3.10). Therefore, has a partition of unity subordinate to a given open cover. Replacing the sheaf of de Rham differential forms on by the sheaf of tropical differential forms, we can construct a particular complex on the integral tropical manifold satisfying Condition 4.7, which dictates the tropical geometry of .
Definition 5.6.
Given a point as in §3.3.2 (with a chart as in equation (3.10)), the stalk of at is defined as . This defines the complex (or simply ) of monodromy invariant tropical differential forms on . A section is a collection of elements , such that each can be represented by in a small neighborhood for some tropical differential form on , and satisfies the relation in for every .
Notice that the definition of requires the projection map in equation (3.11) to be affine, while that of in §3.3.2 does not. But like , satisfies Condition 4.7 and can be used for the purpose of gluing the sheaf of dgBV algebras in §4.3. In the rest of this section, we shall use the notations and to denote the complexes of sheaves constructed using .
5.2. The semi-flat dgBV algebra and its comparison with the pre-dgBV algebra
In this section, we define a twisting of the semi-flat dgBV algebra by the slab functions (or initial wall-crossing factors) in §2.4, and compare it with the dgBV algebra we constructed in §4.3 using gluing of local smoothing models. The key result is Lemma 5.10, which is an important step in the proof of our main result.
We start by recalling some notations from §2.4. Recall that for each vertex , we fix a representative of the strictly convex multi-valued piecewise linear function to define the cone and the monoid . The natural projection induces a surjective ring homomorphism ; we denote by the image of under the natural projection. We consider for some containing , and write for the function corresponding to . The element together with the corresponding function determine a family , whose central fiber is given by . The variety is equipped with the divisorial log structure induced by , which is log smooth. We write if we need to emphasize the log structure.
Since is orientable, we can choose a nowhere vanishing integral element . We fix a local representative for every vertex and for every maximal cell . Writing , we have the corresponding relative volume form
in . Now the relative log polyvector fields can be written as
The volume form defines a BV operator via contraction , which is given explicitly by
A Schouten–-Nijenhuis–type bracket is given by extending the following formulae skew-symmetrically:
This gives the structure of a BV algebra.
5.2.1. Construction of the semi-flat sheaves
For each , we shall define a sheaf (resp. ) of -order semi-flat log vector fields (resp. semi-flat log de Rham forms) over the open dense subset defined by
where consists of ’s such that and of ’s that intersect with . These sheaves use the natural divisorial log structure on and will not depend on the slab functions ’s. This construction is possible because we are using the much more flexible Euclidean topology on , instead of the Zariski topology on .
For , recall that we have for some . We also have , which is isomorphic to , because . The local -order thickening
is obtained by identifying as . Choosing a different vertex , we can use the parallel transport from to within and the difference between two affine functions to identify the monoids . We take
Next, we need to glue the sheaves ’s along neighborhoods of codimension one cells ’s. For each codimension one cell , we fix a primitive normal to and label the two adjacent maximal cells and so that is pointing into . There are two situations to consider.
The simpler case is when , where the monodromy is trivial. In this case, we have , with the gluing as described below Definition 2.13 using the open gluing data . We take the -order thickening given by
equipped with the divisorial log structure induced by . We extend the open gluing data
to
so that , which acts as an automorphism of . In this way we can extend the gluing to
by twisting with the ring homomorphism induced by . On a sufficiently small neighborhood of , we take
Choosing a different vertex , we may use parallel transport to identify the fans , and further use the difference to identify the monoids . One can check that the sheaf is well-defined.
The more complicated case is when , where the monodromy is non-trivial. We write , where is the unique component which contains the vertex in its closure. We fix one , the corresponding , and a sufficiently small open subset of . We assume that the neighborhood of intersects neither nor for any possible and . Then we consider the scheme-theoretic embedding
given by
We denote by the -order thickening of in and equip it with the divisorial log structure which is log smooth over (note that it is different from the local model introduced earlier in §4 because the latter depends on the slab functions , as we can see explicitly in §5.2.2, while the former doesn’t). We take
The gluing with nearby maximal cells on the overlap is given by parallel transporting through the vertex to relate the monoids and constructed from , and twisting the map with the open gluing data
using previous liftings of to . We obtain a commutative diagram of holomorphic maps
where and the vertical arrow on the right hand side respects the log structures. The induced isomorphism
of sheaves on the overlap then gives the desired gluing for defining the sheaf on . Note that the cocycle condition is trivial here as there is no triple intersection of any three open subsets from , and .
Similarly, we can define the sheaf of semi-flat log de Rham forms, together with a relative volume form obtained from gluing the local ’s specified by the element as described in the beginning of §5.2.
It would be useful to write down elements of the sheaf more explicitly. For instance, fixing a point , we may write
| (5.1) |
and use to stand for the semi-flat holomorphic vector field associated to an element .
Note that analytic continuation around the singular locus acts non-trivially on the semi-flat sheaf due to the presence of non-trivial monodromy of the affine structure. Below is a simple example.
Example 5.7.
We consider the local affine charts which appeared in Example 2.3, equipped with a strictly convex piecewise linear affine function on whose change of slopes is . Let us study the analytic continuation of a local section along the loop which starts at a point , as shown in Figure 6.
First, we can identify both and with the monoid in the cone via parallel transport through . Writing , , and , we have . Now the analytic continuation of along (going from the chart to the chart and then back to ) is given by as a sequence of elements:
via the following sequence of maps between the stalks over and :
So we see that the analytic continuation along maps to .
is equipped with the BV algebra structure inherited from (as described in the beginning of §5.2), which agrees with the one induced from the volume form . This allows us to define the sheaf of semi-flat tropical vertex Lie algebras as
| (5.2) |
Remark 5.8.
The sheaf can actually be extended over the non-essential singular locus because the monodromy around that locus acts trivially, but this is not necessary for our later discussion.
5.2.2. Explicit gluing away from codimension
When we define the sheaves ’s in §4.1, the open subset is taken to be a sufficiently small neighborhood of for some . In fact, we can choose one of these open subsets to be the large open dense subset . In this subsection, we construct the sheaves and on using an explicit gluing of the underlying complex analytic space.
Over for or over for with , we have , which was just constructed in §5.2.1. So it remains to consider such that . The log structure of is prescribed by the slab functions ’s, which restrict to functions ’s on the torus . Each of these can be pulled back via the natural projection to give a function on . In this case, we may fix the log chart given by the equation
Write for the corresponding -order thickening in , which gives a local model for smoothing (as in §4). We take
We have to specify the gluing on the overlap with the adjacent maximal cells . This is given by first using parallel transport through to relate the monoids and as in the semi-flat case, and then an embedding via the formula
| (5.3) |
where , are treated as maps as before. We observe that there is a commutative diagram of log morphisms
where . The induced isomorphism
of sheaves on the overlap then provides the gluing for defining the sheaf on . Hence, we obtain a sheaf of BV algebras, where the BV structure is inherited from the local models and . Similarly, we can define the sheaf of log de Rham forms over , together with a relative volume form by gluing the local ’s.
5.2.3. Relation between the semi-flat dgBV algebra and the log structure
The difference between and is that analytic continuation along a path in , where , induces a non-trivial action on (the semi-flat sheaf) but not on (the corrected sheaf). This is because, near a singular point of the affine structure on , there is another local model for constructed in 4.1, where restrictions of sections are invariant under analytic continuation (cf. Example 5.7). This is in line with the philosophy that monodromy is being cancelled by the slab functions ’s (which we also call initial wall-crossing factors). In view of this, we should be able to relate the sheaves and by adding back the initial wall-crossing factors ’s.
Recall that the slab function is a function on , whose zero locus is for such that . Also recall that, for containing , is the unique contractible component in such that , as defined in Assumption 3.5. Note that the inverse image under the generalized moment map is also a contractible open subset. It contains the -dimensional stratum in that corresponds to . Since , we can define in a small neighborhood of , and it can further be extended to the whole of because this subset is contractible. Restricting to the open dense torus orbit , we obtain , which can in addition be lifted to a section in for a sufficiently small .
Now we resolve the sheaves and by the complex introduced in §5.1. We let
and equip it with , and , making it a sheaf of dgBV algebras. Over the open subset , using the explicit description of , we consider the element
| (5.4) |
where is any -form with asymptotic support in and whose integral over any curve transversal to going from to is asymptotically ; such a 1-form can be constructed using a family of bump functions in the normal direction of similar to Example 5.3 (see also [7, §4]). We can further extend the section to the whole by setting it to be outside a small neighborhood of in .
Definition 5.9.
The sheaf of semi-flat polyvector fields is defined as
which is equipped with a BV operator , a wedge product (and hence a Lie bracket ) and the operator
where and . We also define the sheaf of semi-flat log de Rham forms as
equipped with , ,
and a contraction action by elements in .
It can be easily checked that , so we have a sheaf of dgBV algebras.
On the other hand, we write
which is equipped with the operators , and . The following important lemma is a comparison between the two sheaves of dgBV algebras.
Lemma 5.10.
There exists a set of compatible isomorphisms
of sheaves of dgBV algebras such that for each .
There also exists a set of compatible isomorphisms
of sheaves of dgas preserving the contraction action and such that for each . Furthermore, the relative volume form is identified via .
Proof.
Outside those ’s such that , the two sheaves are identical. So we will take a component of and compare the sheaves on a neighborhood .
We fix a point and describe the map at the stalks of the two sheaves. First, the preimage can be identified as a real -dimensional torus in . We have an identification , and we choose the unique primitive element in the ray pointing into . As analytic spaces, we write
where and , and
The germ of analytic functions can be written as
Using the embedding in §5.2.2, we can write
with the relation (here is the change of slopes for across ). For the elements and in , we have the identities (we omit the dependence on when we write elements in the stalks of sheaves):
describing the embedding . For polyvector fields, we can write
The BV operator is described by the relations , , and
| (5.5) |
Similarly, we can write down the stalk for . As a module over , we have ; the ring structure on differs from that on and is determined by the relation . The embedding is given by
The formulae for the BV operator are the same as that for , except that for the last equation in (5.5), we have instead.
We apply the argument in [7, §4], where we considered a scattering diagram consisting of only one wall, to relate these two sheaves. We can find a set of compatible elements , where for , such that and . Explicitly, is a step-function-like section of the form
For each , we also define , as an element in . Now we define the map at the stalks by writing
(and similarly for ), and extending the formulae
through the tensor product and skew-symmetrically in ’s.
To see that is the desired isomorphism, we check all the relations by computations:
-
•
Since , we have
similarly, we have . Hence, we have .
-
•
We have and
i.e. the map preserves the product structure.
-
•
From the fact that , we see that commutes with , and hence . We also have .
-
•
Again from , we have
-
•
Finally, we have
We conclude that is an isomorphism of dgBV algebras. We need to check that the map agrees with the isomorphism induced simply by the identity , where . For this purpose, we consider two nearby maximal cells such that . We have , and the gluing of over is given by parallel transporting through , and then by the formulae
| (5.6) |
The only difference for gluing of is the last equation in (5.6), which is now replaced by the formula . Since we have
on a sufficiently small neighborhood of , we see that under the gluing map of on . This shows the compatibility of with the gluing of and over . A similar argument applies for .
The proof for is similar and will be omitted. The volume form is preserved under because we have . This completes the proof of the lemma. ∎
5.2.4. A global sheaf of dgLas from gluing of the semi-flat sheaves
We shall apply the procedure described in §4.3 to the semi-flat sheaves to glue a global sheaf of dgLas. First of all, we choose an open cover satisfying the Condition 4.1, together with a decomposition such that is a cover of the semi-flat part , and is a cover of a neighborhood of .
For each , we have a compatible set of local sheaves of BV algebras, local sheaves of dgas, and relative volume elements , (as in §4.1). We can further demand that, over the semi-flat part , we have , and , and hence and for .
Using the construction in §4.3, we obtain a Gerstenhaber deformation specified by , which give rise to sets of compatible global sheaves and , . Restricting to the semi-flat part, we get two Gerstenhaber deformations and , which must be equivalent as . So we have a set of compatible isomorphisms locally given by for some , for each , and they fit into the following commutative diagram
Since the pre-differential on obtained from the construction in §4.3 is of the form for some , pulling back via gives a global element such that
Theorem 4.18 gives a Maurer–Cartan solution such that , together with a holomorphic volume form , compatible for each . We denote the pullback of under ’s to as , and that of volume form to as . We see that the equation
is satisfied, or equivalently, that is a solution to the extended Maurer–Cartan equation 4.10.
Lemma 5.11.
If the holomorphic volume form is normalized in the sense of Definition 4.19, then we can find a set of compatible , such that
As a consequence, the Maurer–Cartan solution is gauge equivalent to a solution of the form for some , via the gauge transformation .
Proof.
We should construct by induction on as in the proof of Lemma 4.6. Namely, suppose is constructed for the -order, then we shall lift it to the -order. We prove the existence of a lifting at every stalk and use partition of unity to glue a global lifting .
First of all, we can always find a gauge transformation such that
So we have , which implies that . We can write in the stalk at for some germ of holomorphic functions. Applying Lemma 4.6, we can further choose so that . In a sufficiently small neighborhood , we find an element as in Definition 4.19. The fact that the volume form is normalized forces to be constant with respect to the Gauss–Manin connection . Tracing through the exact sequence (4.14) on , we can lift to which is closed under . As a consequence, we have , and hence we conclude that .
Now we have to solve for a lifting such that up to the -order. This is equivalent to solving for a lifting satisfying for the -order once the -order is given. Take an arbitrary lifting to the -order, and making use of the formula in [8, Lem. 2.8], we have
where . From , we use induction on the order to prove that up to order . Therefore we can write
for some , by the fact that the cohomology sheaf under is free over (see the discussion right after Condition 4.14). Setting will then solve the equation. ∎
The element obtained in Lemma 5.11 can be used to conjugate the operator to get , i.e.
The volume form will be holomorphic under the operator . From the equation (4.13), we observe that . Furthermore, the image of under the isomorphism in Lemma 5.10 gives , and an operator of the form
| (5.7) |
where , that acts on .
Equipping with this operator, the semi-flat sheaf can be glued to the sheaves ’s for , preserving all the operators. More explicitly, on each overlap , we have
| (5.8) |
defined by
for , which sends the operator to .
Definition 5.12.
We call , equipped with the structure of a dgLa using and inherited from , the sheaf of semi-flat tropical vertex differential graded Lie algebras (abbrev. as sf-TVdgLa).
Note that . Also, we have since , and a direct computation shows that . Thus , and the operator preserves the sub-dgLa .
From the description of the sheaf , we can see that locally on , is supported on finitely many codimension one polyhedral subsets, called walls or slabs, which are constituents of a scattering diagram. This is why we use the subscript ‘s’ in , which stands for ‘scattering’.
5.3. Consistent scattering diagrams and Maurer–Cartan solutions
5.3.1. Scattering diagrams
In this subsection, we recall the notion of scattering diagrams introduced by Kontsevich–Soibelman [36] and Gross–Siebert [29], and make modifications to suit our needs. We begin with the notion of walls from [29, §2]. Let
be equipped with a polyhedral decomposition induced from and . For the exposition below, we will always fix and consider all these structures modulo .
Definition 5.13.
A wall consists of
-
•
a maximal cell ,
-
•
a closed -dimensional tropical polyhedral subset of such that
-
•
a choice of a primitive normal , and
-
•
a section of the tropical vertex group over a sufficiently small neighborhood of .
We call the wall-crossing factor associated to the wall . We may write a wall as for simplicity.
A wall cannot be contained in with . We define a notion of slabs for these subsets of codimension one strata intersecting . The difference is that we have an extra term coming from the slab function .
Definition 5.14.
A slab consists of
-
•
an -cell such that ,
-
•
a closed -dimensional tropical polyhedral subset of ,
-
•
a choice of a primitive normal , and
-
•
a section of over a sufficiently small neighborhood of .
The wall-crossing factor associated to the slab is given by
where is the unique vertex such that contains and
(cf. equation (5.4)). We may write a slab as for simplicity.
Remark 5.15.
In the above definition, a slab is not allowed to intersect the singular locus . This is different from the situation in [29, §2]. However, in our definition of consistent scattering diagrams, we will require consistency around each stratum of .
Example 5.16.
We consider the 3-dimensional example shown in Figure 7, from which we can see possible supports of the walls and slabs. There are two adjacent maximal cells intersecting at with colored in red. The -dimensional polyhedral subsets colored in blue can support walls and the polyhedral subset colored in green can support a slab because it is lying inside with .
Definition 5.17.
A (-order) scattering diagram is a countable collection
of walls or slabs such that the intersections of any two walls/slabs is at most an -dimensional tropical polyhedral subset, and is locally finite in .
Our notion of scattering diagrams is more flexible than the one defined in [36, 29] in two ways: First, there is no relation between the affine direction orthogonal to a wall or a slab and its wall crossing factor. As a result, we cannot allow overlapping of walls/slabs in their relative interior because in that case their associated wall crossing factors are not necessarily commuting. Second, we only require that the intersection of with is a locally finite collection of , which implies that we allow a possibly infinite number of walls/slabs approaching strata of . In the construction of the scattering diagram associated to a Maurer–Cartan solution below, all the walls/slabs will be compact subsets of . These walls will not intersect , as illustrated in Figure 7. However, there could be a union of infinitely many walls limiting to some strata of . See also Remark 1.2.
Example 5.18.
For the 2-dimensional example shown in Figure 8, we see a vertex and its adjacent cells, and the singular locus consists of the red crosses. In our version of scattering diagrams, we allow infinitely many intervals limiting to or .
Given a scattering diagram , we can define its support as . There is an induced polyhedral decomposition on such that its -cells are closed subsets of some walls or slabs, and all intersections of walls or slabs are lying in the union of the -cells. We write for the collection of all the -cells in this polyhedral decomposition. We may assume, after further subdividing the walls or slabs in if necessary, that every wall or slab is an -cell in . We call an -cell a joint, and a connected component of a chamber.
Given a wall or slab, we shall make sense of wall crossing in terms of jumping of holomorphic functions across it. Instead of formulating the definition in terms of path-ordered products of elements in the tropical vertex group as in [29], we will express it in terms of the action by the tropical vertex group on the local sections of . There is no harm in doing so since we have the inclusion , i.e. a relative vector field is determined by its action on functions.
In this regard, we would like to define the (-order) wall-crossing sheaf on the open set
which captures the jumping of holomorphic functions described by the wall-crossing factor when crossing a wall/slab. We first consider the sheaf of holomorphic functions over the subset , and let
To extend it through the walls/slabs, we will specify the analyic continuation through for each . Given a wall/slab with two adjacent chambers , and pointing into , and a point with the germ of wall-crossing factors near , we let
but with a different gluing to nearby chambers : in a sufficiently small neighborhood of , the gluing of a local section is given by
| (5.9) |
In this way, the sheaf extends to .
Now we can formulate consistency of a scattering diagram in terms of the behaviour of the sheaf over the joints ’s and -dimensional strata of . More precisely, we consider the push-forward along the embedding , and its stalk at and for strata . Similar to above, we can define the (-order) sheaf by using and considering equation (5.9) modulo . There is a natural restriction map . Taking tensor product, we have , where .
The proof of the following lemma will be given in Appendix §A.
Lemma 5.19 (Hartogs extension property).
We have
where is the inclusion. Moreover, for any scattering diagram , we have
where is the inclusion.
Lemma 5.20.
The -order sheaf is isomorphic to the sheaf .
Proof.
In view of Lemma 5.19, we only have to show that the two sheaves are isomorphic on the open subset . Since we work modulo , only the wall-crossing factor associated to a slab matters. So we take a point for some vertex , and compare with . From the proof of Lemma 5.10, we have
with the relation . The gluings with nearby maximal cells of both and are simply given by the parallel transport through and the formulae
in the proof of Lemma 5.10.
Now for the wall-crossing sheaf , the wall-crossing factor acts trivially except on the two coordinate functions because for . The gluing of to the nearby maximal cells which obeys wall crossing is given by
in a sufficiently small neighborhood of . Here, the reason that we have on is simply because we have in the gluing of . For the same reason, we see that the gluing of agrees with that of and . ∎
Definition 5.21.
A (-order) scattering diagram is said to be consistent if there is an isomorphism as sheaves of -algebras on each open subset .
The above consistency condition would imply that is surjective for any and hence is a sheaf of free -modules on . We are going to see that agrees with the push-forward of the sheaf of holomorphic functions on a (-order) thickening of the central fiber under the modified moment map .
Let us elaborate a bit on the relation between this definition of consistency and that in [29]. Assuming we have a consistent scattering diagram in the sense of [29], then we obtain a -order thickening of which is locally modeled on the thickenings ’s by [28, Cor. 2.18]. Pushing forward via the modified moment map , we obtain a sheaf of algebras over lifting , which is locally isomorphic to the ’s. This consequence is exactly what we use to formulate our definition of consistency.
Lemma 5.22.
Suppose we have such that is Stein, and an isomorphism of sheaves of -algebras which is the identity modulo . Then there is a unique isomorphism of analytic spaces inducing .
Proof.
From the description in §2.4, we can embed both families , over as closed analytic subschemes of and respectively, where projection to the second factor defines the family over . Let and be the corresponding ideal sheaves, which can be generated by finitely many elements. We can take Stein open subsets and such that their intersections with the subschemes give and respectively. By taking global sections of the sheaves over , we obtain the isomorphism . Using the fact that is Stein, we can lift ’s, where ’s are restrictions of coordinate functions to , to holomorphic functions on . In this way, can be lifted as a holomorphic map . Restricting to , we see that the image lies in , and hence we obtain the isomorphism . The uniqueness follows from the fact the is determined by . ∎
5.3.2. Constructing consistent scattering diagrams from Maurer–Cartan solutions
We are finally ready to demonstrate how to construct a consistent scattering diagram in the sense of Definition 5.21 from a Maurer–Cartan solution obtained in Theorem 4.18. As in §5.2.4, we obtain a -order Maurer–Cartan solution and define its scattered part as . From this, we want to construct a -order scattering diagram .
We take an open cover by pre-compact convex open subsets of such that, locally on , can be written as a finite sum
where has asymptotic support on a codimension one polyhedral subset , and . We take a partition of unity subordinate to the cover such that has asymptotic support on a compact subset of . As a result, we can write
| (5.10) |
where each has asymptotic support on the compact codimension one subset . The subset will be the support of our scattering diagram .
We may equip with a polyhedral decomposition such that all the boundaries and mutual intersections of ’s are contained in -dimensional strata of . So, for each -dimensional cell of , if for some , then we must have . Let , which is a finite set of indices. We will equip the -cells ’s of with the structure of walls or slabs.
We first consider the case of a wall. Take such that for all with . We let , choose a primitive normal of , and give the labels to the two adjacent chambers so that is pointing into . In a sufficiently small neighborhood of , we have and we may write
where each has asymptotic support on . Since locally on any Maurer–Cartan solution is gauge equivalent to , there exists an element such that
Such an element can be constructed inductively using the procedure in [37, §3.4.3], and can be chosen to be of the form
| (5.11) |
for some . From this we obtain the wall-crossing factor associated to the wall
| (5.12) |
Remark 5.23.
For the case where for some with , we will define a slab. We take and as above, and let the slab . The primitive normal is the one we chose earlier for each . Again we work in a small neighborhood of with two adjacent chambers . As in the proof of Lemma 5.10, we can find a step-function-like element of the form
to solve the equation on . In other words,
is an isomorphism of sheaves of dgLas. Computations using the formula in [8, Lem. 2.5] then gives the identity
Once again, we can find an element such that
and hence a corresponding element of the form (5.11). From this we get
| (5.13) |
and hence the wall-crossing factor associated to the slab .
Next we would like to argue that consistency of the scattering diagram follows from the fact that is a Maurer–Cartan solution. First of all, on the global sheaf over , we have the operator which satisfies and . This allows us to define the sheaf of -order holomorphic functions as
for each . It is a sequence of sheaves of commutative -algebras over , equipped with a natural map for that is induced from the maps for . By construction, we see that .
We claim that the maps ’s are surjective. To prove this, we fix a point and take an open chart containing in the cover of we chose at the beginning of §5.2.4. There is an isomorphism identifying the differential with by our construction. Write and notice that squares to zero, which means that is a solution to the Maurer–Cartan equation for . We apply the same trick as above to the local open subset , namely, any Maurer–Cartan solution lying in is gauge equivalent to the trivial one, so there exists such that
As a result, the map is an isomorphism of dgLas, sending isomorphically onto .
We shall now prove the consistency of the scattering diagram by identifying the associated wall-crossing sheaf with the sheaf of -order holomorphic functions.
Theorem 5.24.
There is an isomorphism of sheaves of -algebras on . Furthermore, the scattering diagram associated to the Maurer–Cartan solution is consistent in the sense of Definition 5.21.
Proof.
To prove the first statement, we first notice that there is a natural isomorphism
so we only need to consider those points where is either a wall or a slab. Since , we will work on the semi-flat locus and use the model , which is equipped with the operator . Via the isomorphism
from Lemma 5.10, we may write
We fix a point and consider the stalk at for both sheaves. In the above construction of walls and slabs from the Maurer–Cartan solution , we first take a sufficiently small open subset and then find a gauge transformation of the form in the case of a wall, and of the form in the case of a slab. We have
by construction, so this further induces an isomorphism
of -algebras.
It remains to see how the stalk is glued to nearby chambers . For this purpose, we let
as in equation (5.12) in the case of a wall, and
as in (5.13) in the case of a slab. Then, the restriction of an element to a nearby chamber is given by
in a sufficiently small neighborhood . This agrees with the description of the wall-crossing sheaf in equation (5.9). Hence we obtain an isomorphism .
To prove the second statement, we first apply pushing forward via to the first statement to get the isomorphism
Now, by the discussion right before this proof, we may identify with locally. But the sheaf , which is isomorphic to the restriction of to as sheaves of -modules, satisfies the Hartogs extension property from to by Lemma 5.19. So we have . Hence, we obtain
from which follows the consistency of the diagram . ∎
Remark 5.25.
From the proof of Theorem 5.24, we actually have a correspondence between step-function-like elements in the gauge group and elements in the tropical vertex group as follows. We fix a generic point in a joint , and consider a neighborhood of of the form , where is a neighborhood of in and is a disk in the normal direction of . We pick a compact annulus surrounding , intersecting finitely many walls/slabs. We let be the walls/slabs in anti-clockwise direction. For each , we take an open subset just containing the wall such that . The following Figure 9 below illustrates the situation.
As in the proof of Theorem 5.24, there is a gauge transformation on each of the form
where for a slab and for a wall. These are step-function-like elements in the gauge group satisfying
where is the wall crossing factor associated to .
On the overlap (where we set if ), there is a commutative diagram
allowing us to interpret the wall crossing factor as the gluing between the two sheaves and over .
Notice that the Maurer–Cartan element is global. On a small neighborhood containing , we have the sheaf on , and there is an isomorphism
Composing with the isomorphism
we have a commutative diagram of isomorphisms
This is a Čech-type cocycle condition between the sheaves ’s and , which can be understood as the original consistency condition defined using path-ordered products in [36, 29]. In particular, taking a local holomorphic function in and restricting it to , we obtain elements in that jump across the walls according to the wall crossing factors ’s.
Appendix A The Hartogs extension property
The following lemma is an application of the Hartogs extension theorem [41].
Lemma A.1.
Consider the analytic space for some and an open subset of the form , where and is a neighborhood of the origin . Let , where (i.e. is the complement of complex codimension orbits in ). Then the restriction is a ring isomorphism.
Proof.
We first consider the case where and . We can further assume that consists of just one cone , because the holomorphic functions on are those on that agree on the overlaps. So we can write
i.e. as Laurent series converging in . We may further assume that is a sufficiently small Stein open subset. Take . We have the corresponding holomorphic function on for each point , which can be extended to using the Hartogs extension theorem [41] because is a compact subset of such that is connected. Therefore, we have for for each , and hence is an element in .
For the general case, we use induction on the codimension of to show that any holomorphic function can be extended through with . Taking a point , a neighborhood of can be written as . By the induction hypothesis, we know that holomorphic functions can already be extended through . We conclude that any holomorphic function can be extended through . ∎
We will make use of the following version of the Hartogs extension theorem, which can be found in e.g. [31, p. 58], to handle extension within codimension one cells ’s and maximal cells ’s.
Theorem A.2 (Hartogs extension theorem, see e.g. [31]).
Let be a domain with , and such that is still a domain. Suppose is a non-empty open subset, and is compact for every , where is projection along one of the coordinate direction. Then the natural restriction is an isomorphism.
Proof of Lemma 5.19.
To prove the first statement, we apply Lemma A.1. So we only need to show that, for , a holomorphic function in can be extended uniquely to , where is some neighborhood of . Writing , we may simply prove that this is the case with consisting of a single ray as in the proof of Lemma A.1. Thus we can assume that and the open subset for some connected . We observe that extensions of holomorphic functions from to can be done by covering the former open subset with Hartogs’ figures.
To prove the second statement, we need to further consider extensions through for a joint . For those joints lying in some codimension one stratum , the argument is similar to the above. So we assume that is a maximal cell. We take a point and work in a sufficiently small neighborhood of . In this case, we may find a codimension one rational hyperplane containing , together with a lattice embedding which induces the projection along one of the coordinate directions. Letting and applying Theorem A.2, we obtain extensions of holomorphic functions in . ∎
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