On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces

Mirror symmetry is a duality between the symplectic geometry of a Calabi–Yau manifold $X$ and the complex geometry of its mirror $\check{X}$. With the help of mirror symmetry, one can achieve a lot of enumerative invariants of Calabi–Yau manifolds, which are a priori hard to compute.

To construct the mirror for a Calabi–Yau manifold, Strominger–Yau–Zaslow proposed the following conjectures [SYZ]: First of all, a Calabi–Yau manifold $X$ near the large complex structure limit admits a special Lagrangian fibration. This is one of the very few geometric descriptions of Calabi–Yau manifolds. Second, the mirror $\check{X}$ of $X$ can be constructed as the dual torus fibration of $X$. Third, the Ricci-flat metric on $X$ is closed to the semi-flat metric, with corrections coming from the holomorphic discs with boundaries on special Lagrangian torus fibres.

For a long time, Strominger–Yau–Zaslow conjecture serves a guiding principle for mirror symmetry. Many of its implications are proved as the building blocks for understanding mirror symmetry. For instance, it provides a geometric way of realizing the homological mirror functor [LYZ]. However, there is very few progress on the original conjecture itself. Only very few examples of special Lagrangian fibrations are known due to technical difficulties of knowing explicit form of Ricci-flat metric. From the conjecture, one need to know the Ricci-flat metric for the existence of special Lagrangian fibration. While the explicit form of the Ricci-flat metric would involve the correction from the holomorphic discs. To retrieve such information, one need to know the boundary conditions, which are provided by the special Lagrangian torus fibres. Thus, the special Lagrangian fibration, the Ricci-flat metric and the correction from holomorphic discs form an iron triangle and firmly linked to each other. Actually, all the examples in the literature are either with respect to the flat metric or the hyperKähler rotation of the holomorphic Lagrangian fibrations. Furthermore, one usually can only track the hyperKähler manifold via Torelli type theorem after hyperKähler rotation rather than writing down the explicit equation.

To get around the analytic difficulties, Kontsevich–Soibelman [KS1], Gross–Siebert [GS1] developed the algebraic alternative to construct the mirror families using rigid analytic spaces. One takes the dual intersection complex $B$ of the maximal degenerate Calabi–Yau varieties, there is a natural integral affine structures with singularities on $B$. By studying the scattering diagrams on $B$, one can reconstruct the Calabi–Yau family near the large complex structure limit. It is a folklore theorem that the affine manifold $B$ is the base for the Strominger–Yau–Zaslow conjecture, while the support of the scattering diagrams are the projection of the holomorphic discs with boundaries on special Lagrangian torus fibres. There are many success of understanding mirror symmetry via this algebraic approach.

On the other hand, one can use Lagrangian Floer theory to construct mirrors and prove homological mirror symmetry. Fukaya [F0] has proposed family Floer homology which was further developed by Tu [T4] and Abouzaid [A2, A3]. The family Floer mirror is constructed as the set of Maurer–Cartan elements for the $A_{\infty}$ structures of the Lagrangian torus fibres quotient by certain equivalences. As Lagrangian torus fibres bound Maslov index zero holomorphic discs, the Maurer–Cartan elements will jump and induces non-trivial gluing of charts. It is expected that such jumps behave the same way as the cluster transformations associate to the ones in the scattering diagram.

A symplectic realization of the SYZ mirror construction was first illustrated in some inspiring examples by Auroux [A]. Using symplectic geometry, the SYZ mirror construction was realized for toric Calabi–Yau manifolds [CLL] by Chan, Leung and the first named author. They have interesting mirror maps and Gromov-Witten theory. The mirror construction for blowing-up of toric hypersurfaces was realized by Abouzaid-Auroux-Katzarkov [AAK]. Fukaya–Oh–Ohta–Ono [FOOO-T1, FOOO-T2, FOOO-MS] developed the Floer-theoretical construction in great detail for compact toric manifolds, which generalize and strengthen the result of Cho–Oh [CO] for toric Fano manifolds.

In all these cases, the mirrors constructed in symplectic geometry coincide with the ones produced from Gross–Siebert program. The holomorphic discs can be written down explicitly and no scattering of Maslov index zero discs occur.

Singular SYZ fibers are the sources of Maslov index zero holomorphic discs and quantum corrections. In [CHL, CHL-nc], Cho, Hong and the first named author found a way to construct a localized mirror of a Lagrangian immersion by solving the Maurer–Cartan equation for the formal deformations coming from immersed sectors. Moreover, gluing between the local mirror charts based on Fukaya isomorphisms was developed in [CHL-glue]. Applying to singular fibers, it gives a canonical (partial) compactification of the SYZ mirror by gluing the local mirror charts of singular fibers with those of regular tori [HKL].

In general, it is difficult to explicitly compute the Floer theoretical mirror. Maslov index zero discs can glue to new families of Maslov index zero discs, which is analogue of scattering or wall-crossing in Gross–Siebert program. It is in general complicated to control the scattering of Maslov index zero discs.

With the assumption that the Lagrangian fibration is special, one can have extra control of the locus of torus fibres bounding holomorphic discs. They form affine lines with respect to the complex affine structure. In particular, this allows us to study a version of open Gromov–Witten invariants defined by the third author and identified them with the tropical disc counting [L1, L2, L14].

It is reasonable to expect that the Gross–Siebert mirror and the Floer-theoretical mirror are equivalent. The first step toward such statement is to identify the affine manifolds with singularities of the SYZ fibration and the one used in the Gross–Siebert program.

In this paper, we will establish first such a statement for the case of $\mathbb{P}^{2}$.

The Gross–Siebert type mirror construction of $\mathbb{P}^{2}\setminus E$ is done by Carl–Pomperla–Siebert [CPS] and the mirror is the fiberwise compactification of its Landau–Ginzburg mirror. In particular, it has the following description: First, take the toric variety $\mathbb{P}^{2}/\mathbb{Z}_{3}$, whose moment-map polytope is dual to that of $\mathbb{P}^{2}$, see Figure 1. We have the meromorphic function $W=z+w+1/zw$ on $\mathbb{P}^{2}/\mathbb{Z}_{3}$. The pole divisor of $W$ is the sum of the three toric divisors. The zero divisor of $W$ intersects with the pole divisor at three points. We blow up $\mathbb{P}^{2}/\mathbb{Z}_{3}$ at these three points, so that $W$ induces an elliptic fibration. (We can further blow up the three orbifold points of $\mathbb{P}^{2}/\mathbb{Z}_{3}$ to make the total space smooth.) Finally we delete the strict transform of the three toric divisors (which is the fiber at $\infty$) and this defines the mirror space. The Landau–Ginzburg superpotential is the elliptic fibration map induced by $W$. It is also worth noticing that the theorem is also achieved by Pierrick Beausseau with a different approach [P]. We refer the readers for the inspiring heuristic discussion there about such an expectation from a different point of view.

For the family Floer mirror, it is glued from torus charts, which are the deformation spaces of Lagrangian torus fibers. Due to scattering of Maslov zero holomorphic discs, there are infinitely many walls and chambers in this case, and each chamber corresponds to a torus chart.

On the other hand, in the Fano situation of this paper, we can use the method in [CHL-glue, HKL] to construct a $\mathbb{C}$-valued mirror. The special Lagrangian fibration on $\mathbb{P}^{2}\setminus E$ [CJL] has three singular fibers which are nodal tori. Instead of the (infinitely many) torus fibers, we take the monotone moment-map torus together with three monotone Lagrangian immersions (in place of the singular SYZ fibers), and glue their deformation spaces together to construct the mirror.

More precisely, the gluing construction has to be carried out over the Novikov field

$$\Lambda:=\left\{\sum_{i=0}^{\infty}a_{i}\mathbf{T}^{A_{i}}\mid a_{i}\in\mathbb{C},A_{i}\in\mathbb{R}\textrm{ and increases to }+\infty\right\}$$

so that the Lagrangian deformation spaces have the correct topology and dimension. See Remark 4.10. After we glue up a space over $\Lambda$ using Lagrangian Floer theory, we restrict to $\mathbb{C}$ to get a $\mathbb{C}$-valued mirror.

The authors would like to thank S.-T. Yau for constant encouragement and the Center of Mathematical Sciences and Applications for the wonderful research environment. The first author expresses his gratitude to Cheol-Hyun Cho, Hansol Hong and Yoosik Kim for the useful joint works. The third author wants to thank Peirrick Beasseau, Tristan Collins, Adam Jacob for related discussion. The first author is supported by Simons Collaboration Grant #580648. The second author is supported by the Center of Mathematical Sciences and Applications. The third author is supported by Simons Collaboration Grant #635846.

We will first review the results in [CJL]: Let $Y$ be a del Pezzo surface or a rational elliptic surface. $D\in|-K_{Y}|$ be a smooth anti-canonical divisor and $X=Y\setminus D$. There exists a meromorphic volume form $\Omega$ on with simple pole along $D$ which is unique up to a $\mathbb{C}^{\ast}$-scaling. Therefore, one can view $X$ as a log Calabi–Yau surface. Moreover, Tian–Yau proved the following theorem:

We will assume that $2\omega_{TY}^{2}=\Omega\wedge\bar{\Omega}$ after a suitable scaling of $\Omega$.

It is conjectured by Yau and also Auroux [A2] that there exists a special Lagrangian fibration on $X$. The conjecture is proved by Colllins–Jacob–Lin earlier.

Although the proof of the existence of special Lagrangian fibration in [CJL] still largely use the hyperKähler structure, an important difference from the earlier examples is that one knows which complex structure can support the special Lagrangian fibration. Moreover, one can use algebraic geometry to understand the complex structure after the hyperKähler rotation.

From the asymptotic behavior of $\check{\Omega}$, one has

In particular, the rational elliptic surface $\check{Y}$ has singular configuration $I_{9}I_{1}^{3}$ for the case $Y=\mathbb{P}^{2}$ [CJL]. The extremal rational elliptic surfaces have no deformation and thus can be identified by explicit equation. In the case of $Y=\mathbb{P}^{2}$, $\check{X}$ can actually be realized as the fibrewise compactification of the Landau–Ginzburg mirror

(2.2)

$\displaystyle\begin{split}W\colon(\mathbb{C}^{\ast})^{2}&\longrightarrow\mathbb{C}\\ (t_{1},t_{2})&\mapsto t_{1}+t_{2}+\frac{1}{t_{1}t_{2}}.\end{split}$

It is straight-forward to check that $W$ has three critical values $\lambda_{0},\lambda_{1},\lambda_{2}$ and the cross-ratio with $\infty$ is fixed. Thus, we may assume that $\lambda_{i}=3\zeta^{i}$, where $\zeta=\exp{(2\pi i/3)}$. The fibres of $W$ are three-punctured elliptic curves. By computing the global monodromy which is conjugating to

$$\begin{bmatrix}1&9\\ 0&1\end{bmatrix},$$

the Lefschetz fibration $W\colon(\mathbb{C}^{\ast})^{2}\rightarrow\mathbb{C}$ can be compactified to such an extremal rational elliptic surface by adding three sections and an $I_{9}$-fibre at infinity.

There is a $\mathbb{Z}_{3}$-action $(x,y)\mapsto(\zeta x,\zeta y)$ on $(\mathbb{C}^{\ast})^{2}$ which induces a $\mathbb{Z}_{3}$-action on the base $\mathbb{P}^{1}$ permuting the three critical values. Let $E_{0}$ be the fibre over $0\in\mathbb{P}^{1}$ which is fixed by the $\mathbb{Z}_{3}$-action.

We will describe the orientation explicitly in §3.3 (C).

Given a special Lagrangian fibration $X\rightarrow B_{\mathrm{SYZ}}$ with respect to $(\omega,\Omega)$, we will denote $L_{q}$ for the fibre over $q\in B_{\mathrm{SYZ}}$. Let $B_{0}$ be the complement of discriminant locus, then there exists an integral affine structure on $B_{0}$ [H2] which we will now explain below: Choose a reference fibre $L_{q_{0}}$ and basis $e_{i}\in\mathrm{H}_{1}(L_{q_{0}})$. For a nearby torus fibre $L_{q}$ and a path $\phi$ connecting $q$ and $q_{0}$, let $C_{i}$ be the union of the parallel transport of $e_{i}$ along $\phi$. Then the complex affine coordinate $f_{i}(q)$ of $q$ is defined to be

(2.3)

$\displaystyle f_{i}(q):=\int_{C_{i}}\mbox{Im}\Omega,$

which is well-defined since $L_{q},L_{q_{0}}$ are special Lagrangians. It is straight-forward to check that for a different choice of the basis and paths, the transition function falls in $\mathrm{GL}(n,\mathbb{Z})\rtimes\mathbb{R}^{n}$, where $n=\mbox{dim}_{\mathbb{R}}L_{q}$. Thus, $B_{0}$ is an integral affine manifold and we say $B_{\mathrm{SYZ}}$ is an integral affine manifold with singularities $\Delta=B_{\mathrm{SYZ}}\setminus B_{0}$. The above integral affine structure is usually known as the complex affine structure of the special Lagrangian fibration in the context of mirror symmetry.

From now on, we concentrate on the case $Y=\mathbb{P}^{2}$ with the Landau–Ginzburg potential function (2.2). Recall that $\mathbb{P}^{2}$ is defined by the polytope $\Delta=\mathrm{Conv}\{(-1,-1),(2,-1),(-1,2)\}$. Let $\nabla=\Delta^{\vee}$ be the dual polytope. We denote by $\mathbf{P}_{\nabla}$ the toric variety defined by $\nabla$ and by $\widetilde{\mathbf{P}}_{\nabla}\to\mathbf{P}_{\nabla}$ the maximal projective crepant partial resolution of $\mathbf{P}_{\nabla}$, which is a resolution in the present case.

We denote by $q$ the coordinate of the target space of the potential function $W$ in (2.2) and regard $(W-q\cdot 1)$ as a holomorphic section of the anti-canonical bundle over $\widetilde{\mathbf{P}}_{\nabla}$. Precisely, the monomials $t_{1}$, $t_{2}$, $t_{1}^{-1}t_{2}^{-1}$ correspond to the integral points $(1,0),(0,1),(-1,-1)$ in $\nabla$ and the monomial $t_{1}^{0}t_{2}^{0}=1$ corresponds to the integral point $(0,0)$. The subvariety $\{W-q\cdot 1=0\}\subset\widetilde{\mathbf{P}}_{\nabla}$ gives the desired compactification of our fiber $W^{-1}(q)$. The family $\{W-q\cdot 1=0\}$ is a pencil spanned by the section $1$ and $t_{1}+t_{2}+t_{1}^{-1}t_{2}^{-1}$, and can be extended to a family over $\mathbb{P}^{1}$. It is straightforward to check that the sections $1$ and $t_{1}+t_{2}+t_{1}^{-1}t_{2}^{-1}$ intersect at three points. Blowing-up the base locus gives a morphism $\check{Y}\to\mathbb{P}^{1}$. The fiber at $\infty\in\mathbb{P}^{1}$ is a union of proper transforms of toric divisors in $\widetilde{\mathbf{P}}_{\nabla}$, which is a $I_{9}$ fiber. For simplicity, the proper transform of the $I_{9}$ fiber in $\check{Y}$ is also denoted by $I_{9}$.

Let $\check{X}:=\check{Y}\setminus I_{9}$. First of all, it is clear that

$$\mathrm{H}_{4}(\check{Y},\mathbb{Z})\simeq\mathbb{Z},~{}\mathrm{H}_{2}(\check{Y},\mathbb{Z})\simeq\mathbb{Z}^{10},~{}\mbox{and}~{}\mathrm{H}_{0}(\check{Y},\mathbb{Z})\simeq\mathbb{Z}.$$

Secondly, from the Poincaré duality for orientable manifolds, we have

$$\mathrm{H}_{k}(\check{X},\mathbb{Z})\simeq\mathrm{H}^{4-k}_{\mathrm{c}}(\check{X},\mathbb{Z})\simeq\mathrm{H}^{k}(\check{X},\mathbb{Z}),~{}\forall~{}k.$$

Finally, let $U$ be the preimage of a small neighborhood around $\infty\in\mathbb{P}^{1}$ under $\check{Y}\to\mathbb{P}^{1}$. $I_{9}$ is a retract of $U$. Utilizing the Mayer–Vietoris resolution for simple normal crossing varieties, one can easily derive

$$\mathrm{H}^{2}(I_{9},\mathbb{Z})\simeq\mathbb{Z}^{9},~{}\mathrm{H}^{1}(I_{9},\mathbb{Z})\simeq\mathbb{Z},~{}\mbox{and}~{}\mathrm{H}^{0}(I_{9},\mathbb{Z})\simeq\mathbb{Z}.$$

Consider the Mayer–Vietoris sequence assicoated to the pair $(U,\check{X})$, we can show that

$$\mathrm{H}^{2}(\check{X},\mathbb{C})\simeq\mathrm{H}_{2}(\check{X},\mathbb{C})\simeq\mathbb{C}^{2}$$

We put $E_{q}=\{W-q\cdot 1=0\}\subset\check{Y}$. It follows that $\mathrm{H}_{2}(\check{X},\mathbb{Z})$ is generated by the class of $S^{1}\times S^{1}\subset(\mathbb{C}^{\ast})^{2}$ and the class of $E_{q}$.

From the construction, the standard toric form

$$\frac{\mathrm{d}t_{1}}{t_{1}}\wedge\frac{\mathrm{d}t_{2}}{t_{2}}$$

on $(\mathbb{C}^{\ast})^{2}$ extends to a meromorphic form on $\widetilde{\mathbf{P}}_{\nabla}$ with poles along the union of toric divisors. Via the pullback further to $\check{Y}$, we obtain a meromorphic $2$-form which has poles exactly along $I_{9}$.

In what follows, we set

$$\check{\Omega}=\sqrt{-1}\cdot\frac{\mathrm{d}t_{1}}{t_{1}}\wedge\frac{\mathrm{d}t_{2}}{t_{2}}.$$

The meromorphic top form $\check{\Omega}$ has the property that

$$\mathrm{Re}\left.\check{\Omega}\right|_{\mathrm{H}_{2}(\check{X},\mathbb{Z})}\equiv 0.$$

We can represent $\check{\Omega}$ in a different way, which turns out to be useful in the sequel. It follows from (2.2) that

	$\displaystyle\mathrm{d}q$	$\displaystyle=\mathrm{d}t_{1}+\mathrm{d}t_{2}-\frac{t_{1}\mathrm{d}t_{2}+t_{2}\mathrm{d}t_{1}}{(t_{1}t_{2})^{2}}$
		$\displaystyle=\left(1-\frac{1}{t_{1}^{2}t_{2}}\right)\mathrm{d}t_{1}+\left(1-\frac{1}{t_{1}t_{2}^{2}}\right)\mathrm{d}t_{2}.$

A direct calculation gives

$$\mathrm{d}q\wedge\mathrm{d}t_{1}=\frac{1-t_{1}t_{2}^{2}}{t_{2}}\frac{\check{\Omega}}{\sqrt{-1}}$$

and therefore

(3.1)

$$\check{\Omega}=\sqrt{-1}\cdot\frac{t_{2}}{1-t_{1}t_{2}^{2}}\mathrm{d}q\wedge\mathrm{d}t_{1}$$

provided that $1-t_{1}t_{2}^{2}\neq 0$.

In the previous section, we have well understood the affine structure associated to the special Lagrangian fibration on $\mathbb{P}^{2}\setminus E$, where $E$ is a smooth elliptic curve. In this section, we construct the Floer theoretical mirror of $\mathbb{P}^{2}$ relative to $E$, which is a direct application of the gluing method developed in [CHL-glue, HKL].

The strategy is the following. The special Lagrangian fibration has exactly three singular fibers. Each of these is a nodal torus pinched at one point. However, these singular fibers are located in different energy levels, in the sense that the pseudo-isomorphisms between their formal deformations involve Novikov parameter. The resulting mirror would be defined over $\Lambda$.

To simplify the situation, we take the following Lagrangians instead of the special Lagrangian fibers. We take a monotone moment-map fiber of $\mathbb{P}^{2}$, and use symplectic reduction by $\mathbb{S}^{1}$ to construct three monotone immersed Lagrangians, which play the role of the above three singular fibers. We consider the weakly unobstructed deformation spaces of these Lagrangians, and glue them together via quasi-isomorphisms in the Fukaya category.

Using these monotone Lagrangians, the gluing relations will be defined over $\mathbb{C}$, and hence we can reduce to a $\mathbb{C}$-valued mirror. Moreover, the construction of [CHL-glue] produces a mirror functor from the Fukaya category to the mirror matrix factorization category $\mathrm{Fuk}(\mathbb{P}^{2})\to\mathrm{MF}(\check{X}_{\mathbb{C}},\tilde{W})$, which induces a derived equivalence [CHL-toric].