Complexes, Residues and obstructions
for log-symplectic manifolds

The Grothendieck principal parts sheaf $P(L)$ (see EGA) is a rank-$(n+1)$ bundle on $X$ defined as

where $\Delta\subset X\times X$ is the diagonal and $p_{1},p_{2}:X\times X\to X$ are the projections. We have a short exact sequence

whose corresponding extension class in $\operatorname{Ext}^{1}(L,\Omega^{1}_{X}\otimes L)=H^{1}(X,\Omega^{1}_{X})$ coincides with $c_{1}(L)$. The sheaf

which likewise has extension class $c_{1}(L)$, is called the normalized principal parts sheaf. The map $P(L)\to L$ admits a splitting $d_{L}:L\to P(L)$ that is a derivation, i.e.

In fact, $d_{L}$ the universal derivation on $L$. Moreover $P(L)$ is generated over $\mathcal{O}_{X}$ by the image of $d_{L}$. Likewise, $P_{0}(L)$ is generated by elements of the form $\operatorname{dlog}(u):=d_{L}u\otimes u^{-1}$ where $u$ is a local generator of $L$.

It is well known that $P(L^{m+1})\simeq P(L)\otimes L^{m},m\geq 0$ which in particular yields a derivation $L^{n+1}\to P(L)\otimes L^{n},n\geq 0$. In fact, This map extends to a complex that we denote by $P_{n+1}^{\bullet}(L)$ or just $P^{\bullet}(L)$ and call the ( $(n+1)$st) principal parts complex of $L$:

The differential is given, in terms of local $\mathcal{O}_{X}$-generators $u_{1},...,u_{k},v_{1},...,v_{\ell}$ of L, by

and extending using additivity and the derivation property. There are also similar shorter complexes

Note the exact sequences

These sequences splits locally and also split globally whenever $L$ is a flat line bundle. In such cases, we get a short exact sequence

The principal parts complex $P^{\bullet}(L)$ may be tensored with $L^{j-n-1}$, for any $j>0$, yielding the $j$-th principal parts complex:

The differential is defined by setting

where $u_{1},...,u_{i},v$ are local generators for $L$, and extending by additivity and the derivation property. Thus, $P^{\bullet}(L)=P^{\bullet}_{n+1}(L)$.

An important property of principal parts complexes is the following:

The above constructions have an obvious extension to the log situation. Thus let $D$ be a divisor with normal crossings on $X$. We define $P(L)\langle\log{D}\rangle$ as the image of $P(L)$ under the inclusion $\Omega_{X}\to\Omega_{X}\langle\log{D}\rangle$, and likewise for $P_{0}(L)\langle\log{D}\rangle$. Then as above we get complexes

Let $F\subset\Omega_{X}\langle\log{D}\rangle$ be an integrable subbundle of rank $m$. Then $F$ gives rise to a foliated De Rham complex $\wedge^{\bullet}(\Omega_{X}\langle\log{D}\rangle/F)$, we well as a foliated principal parts sheaf $P^{1}_{F}(L)\langle\log{D}\rangle=P^{1}(L)\langle\log{D}\rangle/F\otimes L$. Putting these together, we obtain the foliated principal parts complexes (where $P_{0,F}(L)\langle\log{D}\rangle:=P_{0}(L)\langle\log{D}\rangle/F$):

Note that the proof of Proposition 1 made no use of of the acyclicity of the De Rham complex. Hence the same proof applies verbatim to yield

Let $f_{i}:X_{i}\to X$ be the normalization of the $i$-fold locus of $D$. A point on $X_{i}$ consists of a point on $D$ together with a choice of $i$ distinct local branches of $D$ at it. There is a canonical induced normal-crossing divisor $D_{i}$ on $X_{i}$: at a point where $x_{1}...x_{m}$ is an equation for $D$ and $x_{1},...,x_{i}$ are the chosen branches, the equation of $D_{i}$ is $x_{i+1}...x_{m}$. Note the exact sequence

where $N_{f_{1}}$ is the normal bundle to $f_{1}$ which fits in an exact sequence

Locally, $N_{f_{1}}$ coincides with $x_{1}^{-1}\mathcal{O}_{X}/\mathcal{O}_{X}$ where $x_{1}$ is a ’branch equation’: to be precise, if $K$ denotes the kernel of the natural surjection $f_{1}^{-1}\mathcal{O}_{X}\to\mathcal{O}_{X_{1}}$, then $J=K/K^{2}=K\otimes_{f^{-1}\mathcal{O}_{X}}\mathcal{O}_{X_{1}}$ is an invertible $\mathcal{O}_{X_{1}}$-module locally generated by $x_{1}$ and $N_{f_{1}}=J^{-1}$. Note that

Interestingly, even though the differential on the pullback De Rham complex $f_{1}^{-1}\Omega_{X}^{\bullet}$ does not extend to $f^{-1}\Omega_{X}^{\bullet}\otimes\mathcal{O}_{X_{1}}$, the analogous assertion for the log complex does hold: the differential on $f_{1}^{-1}\Omega_{X}^{\bullet}\langle\log{D}\rangle$ extends to what might be called the restricted log complex:

This is due to the identity (where $x_{1}$ denotes a branch equation)

Note that the residue map yields an exact sequence

Note that the rsidue map commutes with exterior derivative. Therefore this sequence induces a short exact sequence of complexes

Furthermore, a twisted form of the restricted log complex, called the normal log complex, also exists:

this is thanks to the identity, where $\omega$ is any log form,

Now recall the exact sequence coming from the residue map

In fact, it is easy to check that this exact sequence has extension class $c_{1}(N_{f_{1}})$ hence identifies $f_{1}^{*}\Omega_{X}\langle\log{D}\rangle$ with $P_{0}(N_{f_{1}})$ so that the normal log complex (9) may be identified with the principal parts complex $P^{\bullet}(N_{f_{1}})$:

Similarly, a pull back log complex $f_{k}^{*}\Omega^{\bullet}_{X}\langle\log{D}\rangle=f_{k}^{-1}\Omega_{X}^{\bullet}\langle\log{D}\rangle\otimes\mathcal{O}_{X_{k}}$ exists for all $k\geq 1$. A similar twisted log complex also exists the determinant of the normal bundle $N_{f_{k}}$:

This comes from (where $x_{1},...,x_{k}$ are the branch equations at a given point of $X_{k}$):

We have a short exact sequence of vector bundles on $X_{k}$:

where $\nu_{k}$ is the local system of branches of $D$ along $X_{k}$ and the right map is multiple residue. Taking exterior powers, we get various exact Eagon-Northcott complexes. In particular, we get surjections, called iterated Poincaé residue:

${\det}_{\mathbb{C}}(\nu_{k})$ is a rank-1 local system on $X_{k}$ which may be called the ’normal orientation sheaf’. The maps for $i\geq k$ together yield a surjection

We will use $\Omega_{X}^{+\bullet}$ to denote $\bigoplus\limits_{i>0}\Omega^{i}_{X}$ and similarly for the log versions. This to match with the Lichnerowicz-Poisson complex $T^{\bullet}_{X}$ and $T^{\bullet}_{X}\langle-\log{D}\rangle$. Thus, interior multiplication by $\Phi$ induces and isomorphism $T^{\bullet}_{X}\langle-\log{D}\rangle\to\Omega^{\bullet}_{X}\langle\log{D}\rangle$. Equivalently, $\Phi$ itself is a form im $\Omega^{2}_{X}\langle\log{D}\rangle$ inducing a nondegenerate pairing on $T_{X}\langle-\log{D}\rangle$. In terms of local coordinates, at a point of multiplicity $m$ on $D$, we have a basis for $\Omega_{X}\langle\log{D}\rangle$ of the form

and then

We have an inclusion of complexes

where, for $X$ compact Kähler, the first complex controls ’locally trivial’ deformations of $(X,\Pi)$, i.e. deformations of $(X,\Pi)$ inducing a locally trivial deformation of $D=[\Pi^{n}]$, and the second complex controls all deformations of $(X,\Pi)$. It is known (see e.g. [14]) that locally trivial deformations of $(X,\Pi)$ are always unobstructed and have an essentially Hodge-theoretic (hence topological) character, so one is interested in conditions to ensure that the above inclusion induces an isomorphism on deformation spaces; as is well known, the latter would follow if one can show that the cokernel of this inclusion has vanishing $\mathbb{H}^{1}$.

Our approach to this question starts with the above ’multiplication by $\Phi$’ isomorphism

This isomorphism extends to an isomorphism to $T^{\bullet}_{X}$ with a certain subcomplex of $\Omega^{+\bullet}_{X}(*D)$, the meromorphic forms regular off $D$, that we call the log plus complex and denote by $\Omega_{X}^{+\bullet}\langle\log{{}^{+}D}\rangle$.

Our goal then becomes that of comparing the log and log-plus complexes. To this end we introduce a filtration on $\Omega^{+\bullet}_{X}\langle\log{{}^{+}D}\rangle$, essentially the filtration induced by the exact sequence

and its isomorphic copy

where $f_{1}:X_{1}\to D\subset X$ is the normalization of $D$ and $N_{f_{1}}$ is the associated normal bundle (’branch normal bundle’). We will show that the first graded piece is always an exact complex. The second graded piece is much more subtle. We will show that it is locally exact in degree 0 unless the log-symplectic form $\Phi$, i.e. the matrix $(b_{ij})$ above satisfies some special relations with integer coefficients.

The computations of this section are all local in character, though the applications are global.

Let $f_{i}:X_{i}\to X$ be the normalization of the $i$-fold locus of $D$, $D_{i}$ the induced normal-crossing divisor on $X_{i}$. Thus a point of $X_{i}$ consists of a point $p$ of $D$ together with a choice of an unordered set $S$ of $i$ branches of $D$ through $p$ and $D_{i}$ is the union of the branches of $D$ not in $S$. We consider first the codimension-1 situation. As above, we have a residue exact sequence

(the right-hand map given by residue is locally evaluation on $x_{1}\operatorname{\partial}_{x_{1}}$ where $x_{1}$ is a local equation for the branch of $D$ through the given point of $X_{1}$ ). Note that if $\eta$ comes from a closed form on $X$ near $D$ then $\mathrm{Res}(\eta)$ is a constant.

Dualizing (15), we get

where the left-hand map, the ’co-residue’, is locally multiplication by $x_{1}\operatorname{\partial}_{x_{1}}$ where $x_{1}$ is a branch equation). Set

Then $v_{1}$ is canonical as section of $f_{1}^{*}T_{X}\langle-\log{D}\rangle$ , independent of the choice of local equation $x_{1}$. By contrast, $\operatorname{\partial}_{x_{1}}$ as section of $f_{1}^{*}T_{X}$ is canonical only up to a tangential field to $X_{1}$, and generates $f_{1}^{*}T_{X}$ modulo $T_{X_{1}}\langle-\log{D}\rangle$.

Now $f_{1}^{*}\Omega^{1}_{X}\langle\log{D}\rangle$ and $f^{*}_{1}T_{X}\langle-\log{D}\rangle$ admit mutually inverse isomorphisms

The composite

has a rank-1 kernel that is the kernel of the Poisson vector on $X_{1}$ induced by $\Pi$, aka the conormal to the symplectic foliation on $X_{1}$. Now set

Then $\psi_{1}$ is locally the form in $\Omega_{X_{1}}\langle\log{D_{1}}\rangle$ denoted by $x_{1}\phi_{1}$ in [14]. Again $\psi_{1}$ is canonically defined, independent of choices and corresponds to the first column of the $B=(b_{ij})$ matrix for a local coordinate system $x_{1},x_{2},...$ compatible with the normal-crossing divisor $D$. By contrast, $\phi_{1}$, which depends on the choice of local equation $x_{1}$, is canonical up to a log form in $\Omega_{X_{1}}\langle\log{D_{1}}\rangle$ and generates $\Omega_{X_{1}}\langle\log{{}^{+}D_{1}}\rangle$ modulo the latter.

In $X_{1}\setminus D_{1}$, $\Phi$ is locally of the form $\operatorname{dlog}(x_{1})\wedge dx_{2}+$(symplectic), so there $\psi_{1}=dx_{2}$. Note that by skew-symmetry we have

Thus, locally $\psi_{1}\in\Omega_{X_{1}}\langle\log{D_{1}}\rangle$. In terms of the matrix $B$ above, $\psi_{1}=\sum\limits_{j>1}b_{1j}\operatorname{dlog}(x_{j})$. Note that $\psi_{1}$ which corresponds to the Hamiltonian vector field $v_{1}$, is a closed form. Consequently, $\psi_{1}$ defines a foliation on $X_{1}$. Let $Q_{1}^{\bullet}=\psi_{1}\Omega_{X_{1}}^{\bullet}$ be the associated foliated De Rham complex $\psi_{1}\Omega_{X_{1}}^{\bullet}$:

endowed with the foliated differential.

Note that the residue exact sequence (15) induces the Poincaré residue sequence

Again the Poincaré residue of a closed form is closed. Now the exact sequence

yields

and this sequence induces the $\mathcal{F}_{\bullet}$ filtration on the log-plus complex $\Omega_{X}^{\bullet}\langle\log{{}^{+}D}\rangle$.

Now consider first the first graded $\mathcal{G}^{\bullet}_{1}=(\mathcal{F}^{\bullet}_{1}/\mathcal{F}^{\bullet}_{0})[1]$ which is supported in codimension 1. (the shift is so that $\mathcal{G}^{\bullet}$ starts in degree 0). Then $\mathcal{G}_{1}^{\bullet}$ is a (finite) direct image of a complex of $X_{1}$ modules:

Using Lemma 3, we can easily show:

Next we study $\mathcal{G}_{2}$, which is supported on $X_{2}$. We consider a connected, nonempty open subset $W\subset X_{2}$, for example an entire component, over which the ’normal orientation sheaf’ $\nu_{2}:X_{2,1}\to X_{2}$, i.e. the local $\mathbb{Z}_{2}$-system of branches of $X_{1}$ along $X_{2}$, is trivial (we can take $W=X_{2}$ if, e.g. $D$ has global normal crossings). Such a subset $W$ of $X_{2}$ is said to be a normally split subset of $X_{2}$ and a normal splitting of $W$ is an ordering of the branches is specified. Obviously $X_{2}$ is covered by such subsets $W$. Likewise, for a subset $Z\subset X_{k}$.