Complexes, residues and obstructions for log-symplectic manifolds

We consider compact K\"ahlerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic structure $\Phi$, a generically nondegenerate closed 2-form with simple poles on a divisor $D$ with local normal crossings. A simple linear inequality involving the iterated Poincar\'e residues of $\Phi$ at components of the double locus of $D$ ensures that the pair $(X, \Phi)$ has unobstructed deformations and that $D$ deforms locally trivially.

Understanding log-symplectic manifolds unavoidably involves understanding their deformations. In the very special case of symplectic manifolds, where D = 0, the classical theorem of Bogomolov [2] shows that the pair (X, Φ) has unobstructed deformations. In [14] we obtained a generalization of this result which holds when Φ satisfied a certain 'very general position' condition with respect to D (the original statement is corrected in the subsequent erratum/corrigendum). Namely, we showed in this case that (X, Φ) has albeit from a more global viewpoint. In fact as in [14] it turns out to be more convenient to transport the situation over to the De Rham side where it becomes an inclusion where the latter 'log-plus' complex is a certain complex of meromorphic forms with poles on D. We study a filtration, introduced in [14], interpolating between the two complexes, especially its first two graded pieces. As we show, the first piece is automatically exact, while 0-acyclicity for the second piece leads to the above cocycle condition. See §3 for details. We begin the paper with a couple of auxiliary, independent sections. In §1 we construct a 'principal parts complex' associated to an invertible sheaf L on a smooth variety, extending the principal parts sheaf P(L) together with the universal derivation L → P(L). We show this complex is always exact. In §2 we show that, for any normalcrossing divisor D ⊂ X on any smooth variety, the log complex Ω • X log D -unlike Ω • X itself-can be pulled back to a complex of vector bundles on the normalization of D.
These complexes play a role in our analysis of the aforementioned inclusion map. I am grateful to Brent Pym for helpful communications, in particular for communicating Example 8.

PRINCIPAL PARTS COMPLEX
In this section X denotes an arbitrary n-dimensional smooth complex variety and L denotes an invertible sheaf on X.
1.1. Principal parts. The Grothendieck principal parts sheaf P(L) (see EGA) is a rank-(n + 1) bundle on X defined as , which likewise has extension class c 1 (L), is called the normalized principal parts sheaf. The map P(L) → L admits a splitting d L : L → P(L) that is a derivation, i.e.
In fact, d L the universal derivation on L. Moreover P(L) is generated over O X by the image of d L . Likewise, P 0 (L) is generated by elements of the form dlog(u) := d L u ⊗ u −1 where u is a local generator of L.

Complex.
It is well known that P(L m+1 ) ≃ P(L) ⊗ L m , m ≥ 0 which in particular yields a derivation L n+1 → P(L) ⊗ L n , n ≥ 0. In fact, This map extends to a complex that we denote by P • n+1 (L) or just P • (L) and call the ( (n + 1)st) principal parts complex of L: The differential is given, in terms of local O X -generators u 1 , ..., u k , v 1 , ..., v ℓ of L, by and extending using additivity and the derivation property. There are also similar shorter complexes L m → P(L)L m−1 → ... → ∧ m P(L). 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 • (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 • (L) = P • n+1 (L). An important property of principal parts complexes is the following: Proposition 1. For any local system S, the complexes P • j (L) ⊗ S are null-homotopic and exact for all j > 0. Proof. The assertion being local, we may assume L is trivial and S = C so the i-th term of Then a homotopy is given by 0 id 0 0 . Thus, P • j (L) is null-homotopic, hence exact.
1.3. Log version. 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) log D as the image of P(L) under the inclusion Ω X → Ω X log D , and likewise for P 0 (L) log D . Then as above we get complexes 1.4. Foliated version. Let F ⊂ Ω X log D be an integrable subbundle of rank m. Then F gives rise to a foliated De Rham complex ∧ • (Ω X log D /F), we well as a foliated principal parts sheaf P 1 Putting these together, we obtain the foliated principal parts complexes (where P 0,F (L) log D := P 0 (L) log D /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 Proposition 2. For any local system S, the complexes P • j,F (L) log D ⊗ S are null-homotopic and exact for all j > 0.

CALCULUS ON NORMAL CROSSING DIVISORS
In this section X denotes a smooth variety or complex manifold and D denotes a locally normal-crossing divisor on X. Our aim is to show that the log complex on X, unlike its De Rham analogue, can be pulled back to the normalization of D.
where N f 1 is the normal bundle to f 1 which fits in an exact sequence ).

2.2.
Pulling back log complexes. Interestingly, even though the differential on the pullback De Rham complex f −1 1 Ω • X does not extend to f −1 Ω • X ⊗ O X 1 , the analogous assertion for the log complex does hold: the differential on f −1 1 Ω • X log D 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 ω is any log form, Now recall the exact sequence coming from the residue map so that the normal log complex (9) may be identified with the principal parts complex P • (N f 1 ): This comes from (where x 1 , ..., x k are the branch equations at a given point of X k ):

Iterated residue.
We have a short exact sequence of vector bundles on X k : where ν 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 C (ν k ) is a rank-1 local system on X k which may be called the 'normal orientation sheaf'. The maps for i ≥ k together yield a surjection

COMPARING LOG AND LOG PLUS COMPLEXES
In this section X denotes a log-symplectic smooth variety with log-symplectic form Φ and corresponding Poisson vector Π = Φ −1 , and D denotes the degeneracy divisor of Π or polar divisor of Φ. Our aim is to prove Theorem 6 which shows that deformations of (X, Φ) coincide with locally trivial deformations of (X, Φ, D) and are unobstructed.
3.1. Setting up. We will use Ω +• X to denote i>0 Ω i X and similarly for the log versions. This to match with the Lichnerowicz-Poisson complex T • X and T • X − log D . Thus, interior multiplication by Φ induces and isomorphism T • X log D inducing a nondegenerate pairing on T X − log D . In terms of local coordinates, at a point of multiplicity m on D, we have a basis for Ω X log D of the form We have an inclusion of complexes where, for X compact Kähler, the first complex controls 'locally trivial' deformations of (X, Π), i.e. deformations of (X, Π) inducing a locally trivial deformation of D = [Π n ], and the second complex controls all deformations of (X, Π). It is known (see e.g. [14]) that locally trivial deformations of (X, Π) 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 H 1 .
Our approach to this question starts with the above 'multiplication by Φ' isomor- . This isomorphism extends to an isomorphism to T • X with a certain subcomplex of Ω +• X ( * D), the meromorphic forms regular off D, that we call the log plus complex and denote by Our goal then becomes that of comparing the log and log-plus complexes. To this end we introduce a filtration on Ω +• X log + D , essentially the filtration induced by the exact sequence 0 → T X − log D → T X → f 1 * N f 1 → 0 7 and its isomorphic copy where f 1 : X 1 → D ⊂ 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 Φ, 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.
3.2. Residues and duality. Let f i : X i → 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 ∂ x 1 where x 1 is a local equation for the branch of D through the given point of X 1 ). Note that if η comes from a closed form on X near D then Res(η) is a constant.
Dualizing (15), we get where the left-hand map, the 'co-residue', is locally multiplication by Then v 1 is canonical as section of f * 1 T X − log D , independent of the choice of local equation x 1 . By contrast, ∂ x 1 as section of f * 1 T X is canonical only up to a tangential field to X 1 , and generates f * has a rank-1 kernel that is the kernel of the Poisson vector on X 1 induced by Π, aka the conormal to the symplectic foliation on X 1 . Now set Then ψ 1 is locally the form in Ω X 1 log D 1 denoted by x 1 φ 1 in [14]. Again ψ 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, φ 1 , which depends on the choice of local equation x 1 , is canonical up to a log form in Ω X 1 log D 1 and generates Ω X 1 log + D 1 modulo the latter.
In X 1 \ D 1 , Φ is locally of the form dlog(x 1 ) ∧ dx 2 +(symplectic), so there ψ 1 = dx 2 . Note that by skew-symmetry we have Thus, locally ψ 1 ∈ Ω X 1 log D 1 . In terms of the matrix B above, Note that ψ 1 which corresponds to the Hamiltonian vector field v 1 , is a closed form. Consequently, ψ 1 defines a foliation on X 1 . Let Q • 1 = ψ 1 Ω • X 1 be the associated foliated De Rham complex ψ 1 Ω • X 1 : . 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 and this sequence induces the F • filtration on the log-plus complex Ω • X log + D .

First graded piece. Now consider first the first graded
which is supported in codimension 1. (the shift is so that G • starts in degree 0). Then G • 1 is a (finite) direct image of a complex of X 1 modules: .. Using Lemma 3, we can easily show: , hence is null-homotopic and exact, hence G • 1 is exact.
3.4. Second graded piece. Next we study G 2 , which is supported on X 2 . We consider a connected, nonempty open subset W ⊂ X 2 , for example an entire component, over which the 'normal orientation sheaf' ν 2 : X 2,1 → X 2 , i.e. the local 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 ⊂ X k .
3.4.1. Iterated residue. Over a normally split subset W, we have a diagram where Res i denote the (Poincaré) residues along the branches of X 1 over X 2 . Set This is essentially what is called the biresidue by Matviichuk et al., see [9]. Thus, when c W 0, we have a basis for the log forms .., η 2 n = dx 2n m = multiplicity of D, m ≥ 2, and then If W may be not be normally orientable (e.g. an entire component of X 2 ) then c W is defined only up to sign; if c W = 0 we say that W is non-residual, otherwise it is residual.

Non-residual case.
Here we consider the case c W = 0. Note that in that case we may express Φ along W in the form where the gammas are closed log forms in the coordinates on W, i.e. x 3 , ..., x 2n . Moreover, γ 3 ∧ γ 4 0 because Φ n is divisible by dlog(x 1 ) dlog(x 2 ). Also, unless γ 3 , γ 4 are both holomorphic (pole-free), there is another component W ′ of X 2 such that c W ′ 0 (in particular, W ∩ D 2 ∅). Hence if no such W ′ exists, we may by suitably modifying coordinates, assume locally that γ 3 = dx 3 , γ 4 = dx 4 . A similar argument, or induction, applies to γ 5 . This means we are essentially in the P-normal case considered in [13]. This we conclude: 2O W → f * 2 Ω X log D | W , whose image we denote by M 2W . It has a local basis (ψ 11 = x 1 φ 1 , ψ 12 = x 2 φ 2 ) corresponding to the basis (e 1 , e 2 ) of 2O W . In term of the B-matrix, we have As ψ 11 , ψ 12 are closed, M 2 is integrable. LetΩ denote the quotient f * 2 Ω X log D | W /M 2W . Then we have an isomorphismΩ given explicitly byω → ω − Res 1 (ω)ψ 12 /c W − Res 2 (ω)ψ 11 /c W (because Res 2 (ψ 11 ) = Res 1 (ψ 12 ) = c W , residues with respect to the two branches of D). Now set N 2 = det N f 2 , an invertible sheaf on X 2 .
is the direct image of a complex on X 2 : where a local generator of N 2 has the form 1/x 1 x 2 and the differential has the form 3.4.4. Zeroth differential. Using the identification (19), the zeroth differential has the form 11 The form ψ 2 = − dlog(x 1 x 2 ) + (ψ 11 + ψ 12 )/c W has zero residues with respect to x 1 , x 2 , hence yields a form in Ω X 2 log D 2 . Changing the local equations x 1 , x 2 changes ψ by adding a holomorphic (pole-free) form on X 2 .
For g nonzero (21) can be rewritteñ When does this operator have a nontrivial kernel? First, if g is constant then ψ 2 = 0 on W which is im[possible if W meets D 2 . Next, locally at a point x ∈ W \ D 2 ∩ W, clearly g/x 1 x 2 holomorphic and nonzero in the kernel exists locally since ψ 2 is closed and holomorphic so ψ 2 = dh for a holomorphic function h and we can take g = e h . Moreover nonzero solutions to d(g/x 1 x 2 ) = 0 differ by a multiplicative constant. The condition that the local solutions patch is clearly that 1 2πi γ ψ 2 be an integer for any loop γ in W \ D 2 ∩ W. Now ψ 2 is defined only modulo a holomorphic form on X 2 while H 1 (W \ D 2 ∩ W) is generated modulo H 1 (W) by small loops normal to components of D 2 , So the relevant condition is just integrality over such loops γ. At a simple point of D 2 ∩ W, the condition that g exist locally as a holomorphic function with no pole on D 2 is clearly that for γ as above, oriented positively, the integer 1 2πi γ ψ 2 is nonnegative, so that g has no pole on D 2 . In other words, that the sum of the first 2 columns of the B matrix, normalized so that b 12 = −b 21 = 1, should be a nonnegative integer vector. Finally by Hartogs, if g is holomorphic off the singular locus of D 2 ∩ W, it extends holomorphically to W.
3.4.5. Special components. Now let Z be a component of D 2 ∩ W and assume W and Z are both normally split so that the branches of D along W may be labelled 12 while those along Z may be labelled 123. Thus branches of X 2 over Z are labelled 12, 23, 31 and the preceding discussion shows that the zeroth differential has nontrivial kernel along Z only if the iterated residues of Φ along these branches, denoted c 21 , c 23 , c 31 , assuming c 12 0, satisfy