2.1 Logarithmic differential forms

The notion of a differential form with logarithmic singularities along a hypersurface was introduced by Deligne [Reference DeligneDel70] in the normal crossings case, and extended to general hypersurfaces by Saito [Reference SaitoSai80]. In this section, we briefly recall the notions and results from Saito’s paper that will be relevant in our study of log symplectic structures.

Here and throughout, $\mathsf{X}$ is a complex manifold and $\mathsf{D}\subset \mathsf{X}$ is a reduced hypersurface; thus, $\mathsf{D}$ may have several irreducible components, but each component is taken with multiplicity one. We denote by $\mathsf{D}_{\text{sing}}$ the singular locus of $\mathsf{D}$ , i.e. the closed analytic subspace defined by the vanishing of the one-jet of a local defining equation for $\mathsf{D}$ . We recall that $\mathsf{D}$ is normal if and only if $\dim \mathsf{D}_{\text{sing}}\leqslant \dim \mathsf{D}-2=\dim \mathsf{X}-3$ . This condition should not be confused with the normal crossings condition, which means that $\mathsf{D}$ is locally isomorphic to the union of a collection of coordinate hyperplanes in $\mathbb{C}^{n}$ , and therefore has a singular locus of dimension $\dim \mathsf{X}-2$ .

We denote by $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\mathsf{D})$ the sheaf of meromorphic differential forms that are holomorphic on $\mathsf{X}\setminus \mathsf{D}$ and have, at worst, poles of order one on every irreducible component of $\mathsf{D}$ .

Just as the holomorphic forms $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }$ are dual to the exterior powers $\mathscr{X}_{\mathsf{X}}^{\bullet }=\unicode[STIX]{x039B}^{\bullet }{\mathcal{T}}_{\mathsf{X}}$ of the tangent sheaf, the dual of $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D})$ is the sheaf $\mathscr{X}_{\mathsf{X}}^{\bullet }(-\text{log}\,\mathsf{D})\subset \mathscr{X}_{\mathsf{X}}^{\bullet }$ of logarithmic multiderivations, i.e. the multiderivations that are tangent to $\mathsf{D}$ in the sense that their action preserves the defining ideal ${\mathcal{O}}_{\mathsf{X}}(-\mathsf{D})\subset {\mathcal{O}}_{\mathsf{X}}$ .

By definition, the de Rham differential maps logarithmic forms to logarithmic forms. Hence we have a complex of sheaves $(\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D}),d)$ , called the logarithmic de Rham complex. The cohomology of this complex is, in general, quite sensitive to the singularities of $\mathsf{D}$ . In [Reference SaitoSai80], Saito explains that every logarithmic form $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{k}(\log \mathsf{D})$ has a residue $\text{Res}\,\unicode[STIX]{x1D714}$ , which is a meromorphic form on $\mathsf{D}$ that is holomorphic on the smooth locus. In this way, we obtain an exact sequence of complexes

where $\unicode[STIX]{x1D6FA}_{\mathsf{D}}^{\bullet ,\text{reg}}$ is the image of the residue map. We recall that $\unicode[STIX]{x1D6FA}_{\mathsf{D}}^{\bullet ,\text{reg}}$ may be identified [Reference AleksandrovAle90] with the ‘regular’ differential forms on $\mathsf{D}$ in the sense of [Reference BarletBar78]. In particular, when $\mathsf{D}$ is normal, we have that $\unicode[STIX]{x1D6FA}_{\mathsf{D}}^{\bullet ,\text{reg}}=(\unicode[STIX]{x1D6FA}_{\mathsf{D}}^{\bullet })^{\vee \vee }$ is the double dual of the usual differential forms on $\mathsf{D}$ .

The sheaf $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D})$ is a coherent ${\mathcal{O}}_{\mathsf{X}}$ -module, but in general it will not be locally free, i.e. there will be no holomorphic vector bundle on $\mathsf{X}$ whose sheaf of sections is $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D})$ . However, $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D})$ is a reflexive ${\mathcal{O}}_{\mathsf{X}}$ -module, meaning that the canonical map $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D})\rightarrow \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D})^{\vee \vee }$ to the double dual is an isomorphism. As a result, elements of $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D})$ exhibit the Hartogs phenomenon [Reference HartshorneHar80]: sections defined away from an analytic subspace of codimension at least two extend uniquely to all of  $\mathsf{X}$ .

Amongst all the singular hypersurfaces $\mathsf{D}\subset \mathsf{X}$ , the ones for which the sheaf $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})$ is locally free are rather special. They are called free divisors. For example, if $\mathsf{D}$ is smooth or has only normal crossings singularities, then $\mathsf{D}$ is a free divisor. In this case, we simply have $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{k}(\log \mathsf{D})\cong \unicode[STIX]{x039B}^{k}\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})$ . In general, the freeness of a divisor is determined by the structure of its singular locus.

This theorem implies, in particular, that any plane curve is a free divisor, but isolated hypersurface singularities in $\mathbb{C}^{n}$ for $n\geqslant 3$ are not free.

2.2 Log symplectic manifolds

Let $\mathsf{X}$ be a complex manifold of dimension $2n$ with $n\geqslant 1$ , and let $\mathsf{D}\subset \mathsf{X}$ be a reduced divisor. We say that a global logarithmic two-form $\unicode[STIX]{x1D714}\in \mathsf{H}^{0}(\mathsf{X},\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2}(\log \mathsf{D}))$ is nondegenerate if its top power $\unicode[STIX]{x1D714}^{n}$ is a nonvanishing section of the line bundle ${\mathcal{K}}_{\mathsf{X}}(\mathsf{D})$ , where ${\mathcal{K}}_{\mathsf{X}}=\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2n}$ is the canonical bundle of $\mathsf{X}$ . Equivalently, a logarithmic two-form is nondegenerate if and only if the induced map

(1) $$\begin{eqnarray}\begin{array}{@{}ccccc@{}}\unicode[STIX]{x1D714}^{\flat } & : & \mathscr{X}_{\mathsf{X}}^{1}(-\text{log}\,\mathsf{D}) & \rightarrow & \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})\\ & & Z & \mapsto & i_{Z}\unicode[STIX]{x1D714}\end{array}\end{eqnarray}$$

is an isomorphism of ${\mathcal{O}}_{\mathsf{X}}$ -modules.

The existence of a nonvanishing section of $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2n}(\mathsf{D})$ is a logarithmic analogue of the usual Calabi–Yau condition (triviality of the canonical class). It is equivalent to the divisor $\mathsf{D}$ being an anticanonical divisor, so we make the following definition.

A log symplectic form on a log Calabi–Yau manifold $(\mathsf{X},\mathsf{D})$ is a nondegenerate global logarithmic two-form $\unicode[STIX]{x1D714}\in \mathsf{H}^{0}(\mathsf{X},\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2}(\log \mathsf{D}))$ that is closed, i.e.  $d\unicode[STIX]{x1D714}=0$ . We remark that the hypersurface $\mathsf{D}$ is completely determined by the meromorphic two-form $\unicode[STIX]{x1D714}$ , since it is the polar divisor of $\unicode[STIX]{x1D714}^{n}$ . For this reason, we will often refer to the pair $(\mathsf{X},\unicode[STIX]{x1D714})$ as a log symplectic manifold, the hypersurface $\mathsf{D}$ being implicit. We will always assume that $\mathsf{D}$ is nonempty, so symplectic manifolds do not count as log symplectic manifolds in our terminology.

Example 2.4 (Log symplectic surfaces).

Let $(\mathsf{X},\mathsf{D})$ be a log Calabi–Yau surface (e.g. a Del Pezzo surface containing an elliptic curve). Since $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{3}(\log \mathsf{D})=0$ , every logarithmic two-form is closed, so that a log symplectic structure on $(\mathsf{X},\mathsf{D})$ is determined by an arbitrary nonvanishing section of the line bundle ${\mathcal{K}}_{\mathsf{X}}(\mathsf{D})$ .

When $\mathsf{X}=\mathbb{C}^{2}$ with coordinates $(x,y)$ , every log symplectic structure can be written as

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\frac{dx\wedge dy}{f}\end{eqnarray}$$

where $f\in {\mathcal{O}}_{\mathbb{C}^{2}}$ is square-free and $\mathsf{D}$ is its zero locus. Hence $\mathsf{D}$ can be an arbitrary plane curve, with no constraints on the singularities.

When $(\mathsf{X},\mathsf{D})$ is compact, the log symplectic form $\unicode[STIX]{x1D714}$ is unique up to rescaling by a constant. The birational classification of projective log symplectic surfaces can be extracted from the full birational classification of projective surfaces containing an effective anticanonical divisor [Reference Bartocci and MacríBM05, Reference IngallsIng98]; it turns out that the only possible singularities for $\mathsf{D}$ are of type $A_{1}$ , $A_{2}$ , $A_{3}$ or $D_{4}$ .

Example 2.5 (Toric examples).

Let $\mathsf{X}=\mathbb{C}^{2n}$ with coordinates $(x_{1},\ldots ,x_{2n})$ , and let $\mathsf{D}\subset \mathsf{X}$ be the union of the coordinate axes, so that $\mathsf{X}\setminus \mathsf{D}$ is the complex torus $\mathsf{T}=(\mathbb{C}^{\times })^{2n}$ . Let $\unicode[STIX]{x1D706}=(\unicode[STIX]{x1D706}_{ij})_{1\leqslant i,j\leqslant 2n}$ be a skew-symmetric matrix of complex numbers. Then the two-form

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\mathop{\sum }_{1\leqslant i<j\leqslant 2n}\unicode[STIX]{x1D706}_{ij}\frac{dx_{i}}{x_{i}}\wedge \frac{dx_{j}}{x_{j}}\end{eqnarray}$$

is closed and logarithmic, and is invariant under the action of $\mathsf{T}$ . It will be nondegenerate, defining a log symplectic form on $(\mathbb{C}^{2n},\mathsf{D})$ , if and only if the matrix $\unicode[STIX]{x1D706}$ is nonsingular. If we compactify $\mathbb{C}^{2n}$ to a projective space by adding a hyperplane at infinity, we obtain the log symplectic structures characterized in Theorem 1.2. Similar structures can also be obtained by choosing other toric compactifications of $\mathsf{T}$ .

If $(\mathsf{X},\unicode[STIX]{x1D714})$ is a log symplectic manifold and $p\in \mathsf{D}$ is a smooth point of the polar divisor, then a version of the Darboux theorem holds in a neighbourhood of $p$ . Namely, there exist coordinates $(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})$ centred at $p$ in which $\mathsf{D}$ is the zero locus of $x_{1}$ and

(2) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\frac{dx_{1}}{x_{1}}\wedge dy_{1}+\mathop{\sum }_{i=2}^{n}dx_{i}\wedge dy_{i}.\end{eqnarray}$$

For a discussion of the proof, see [Reference GotoGot02, Lemma 1.2] or [Reference Guillemin, Miranda and PiresGMP14, Proposition 19].

Every log symplectic form $\unicode[STIX]{x1D714}$ has an inverse, which is a global logarithmic biderivation

$$\begin{eqnarray}\unicode[STIX]{x1D70E}\in \mathsf{H}^{0}(\mathsf{X},\mathscr{X}_{\mathsf{X}}^{2}(-\text{log}\,\mathsf{D}))\subset \mathsf{H}^{0}(\mathsf{X},\mathscr{X}_{\mathsf{X}}^{2})\end{eqnarray}$$

with Schouten bracket $[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}]=0$ . Thus $\unicode[STIX]{x1D70E}$ defines a Poisson bracket on ${\mathcal{O}}_{\mathsf{X}}$ by the usual formula

$$\begin{eqnarray}\{f,g\}=\langle df\wedge dg,\unicode[STIX]{x1D70E}\rangle\end{eqnarray}$$

for $f,g\in {\mathcal{O}}_{\mathsf{X}}$ . Moreover, the Pfaffian $\unicode[STIX]{x1D70E}^{n}\in \mathsf{H}^{0}(\mathsf{X},\mathscr{X}_{\mathsf{X}}^{2n})$ , which is a section of the anticanonical line bundle, gives a reduced defining equation for $\mathsf{D}$ . From this point of view, $\mathsf{D}$ is the degeneracy divisor of $\unicode[STIX]{x1D70E}$ , the locus where its rank drops. We will say that a Poisson structure $\unicode[STIX]{x1D70E}$ is log symplectic if it is induced by a log symplectic form in this way.

Conversely, given a generically nondegenerate Poisson structure $\unicode[STIX]{x1D70E}$ with a reduced degeneracy divisor $\mathsf{D}$ , it is always the case that $\unicode[STIX]{x1D70E}\in \mathsf{H}^{0}(\mathsf{X},\mathscr{X}_{\mathsf{X}}^{2}(-\text{log}\,\mathsf{D}))$ . Indeed, $\mathsf{D}\subset \mathsf{X}$ is a Poisson subspace (see [Reference PolishchukPol97, Corollary 2.3] or [Reference Gualtieri and PymGP13, Proposition 6]), which implies that $\unicode[STIX]{x1D70E}$ is tangent to $\mathsf{D}$ . The anchor map

$$\begin{eqnarray}\begin{array}{@{}ccccc@{}}\unicode[STIX]{x1D70E}^{\sharp } & : & \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1} & \rightarrow & \mathscr{X}_{\mathsf{X}}^{1}\\ & & \unicode[STIX]{x1D6FC} & \mapsto & i_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D70E}\end{array}\end{eqnarray}$$

then extends to an isomorphism $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})\rightarrow \mathscr{X}_{\mathsf{X}}^{1}(-\text{log}\,\mathsf{D})$ , so that $\unicode[STIX]{x1D70E}^{\sharp }$ may be inverted to obtain a log symplectic form $\unicode[STIX]{x1D714}$ . Notice that this argument works no matter how singular $\mathsf{D}$ is, thanks to the reflexivity of the sheaves involved. We therefore have the following basic result.

Recall that for any Poisson bivector field $\unicode[STIX]{x1D70E}$ , the image of the anchor map $\unicode[STIX]{x1D70E}^{\sharp }:\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}\rightarrow \mathscr{X}_{\mathsf{X}}^{1}$ is an involutive subsheaf of $\mathscr{X}_{\mathsf{X}}^{1}$ and gives rise to a foliation of $\mathsf{X}$ by even-dimensional symplectic leaves. For a log symplectic structure on a connected complex manifold $\mathsf{X}$ of dimension $2n$ , the open complement $\mathsf{X}\setminus \mathsf{D}$ is a symplectic leaf of dimension $2n$ . Meanwhile, the smooth locus of $\mathsf{D}$ is foliated by leaves of dimension $2n-2$ ; this foliation is defined by the kernel of the residue one-form $\text{Res}\,\unicode[STIX]{x1D714}$ , which is closed. Then, the singular locus $\mathsf{D}_{\text{sing}}$ is foliated by leaves of dimension $2n-2$ or less. Notice that having an open symplectic leaf is not enough to guarantee that $\unicode[STIX]{x1D70E}$ is log symplectic, since the latter is possible even when the Pfaffian is not reduced.

We close this section with a description of the symplectic leaves of one of the elliptic log symplectic structures that will be the focus of § 4.

Example 2.7 (An elliptic log symplectic form).

Let $(w,x,y,z)$ be coordinates on $\mathbb{C}^{4}$ and let $\unicode[STIX]{x1D702},\unicode[STIX]{x1D708}\in \mathbb{C}$ be constants. Then the Poisson brackets

$$\begin{eqnarray}\displaystyle \{w,x\} & = & \displaystyle x,\quad \{y,z\}=\unicode[STIX]{x1D702}\,x^{2}+\unicode[STIX]{x1D708}\,yz,\nonumber\\ \displaystyle \{w,y\} & = & \displaystyle y,\quad \{z,x\}=\unicode[STIX]{x1D702}\,y^{2}+\unicode[STIX]{x1D708}\,zx,\nonumber\\ \displaystyle \{w,z\} & = & \displaystyle z,\quad \{x,y\}=\unicode[STIX]{x1D702}\,z^{2}+\unicode[STIX]{x1D708}\,xy\nonumber\end{eqnarray}$$

define a log symplectic structure whose degeneracy divisor $\mathsf{D}\subset \mathbb{C}^{4}$ is given by the zeros of the cubic function

$$\begin{eqnarray}f={\textstyle \frac{1}{3}}\unicode[STIX]{x1D702}(x^{3}+y^{3}+z^{3})+\unicode[STIX]{x1D708}\,xyz.\end{eqnarray}$$

Thus, for generic values of $\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D708}$ , the hypersurface is a product $\mathsf{D}=\mathbb{C}\times \mathsf{D}_{0}$ , where $\mathsf{D}_{0}\subset \mathbb{C}_{x,y,z}^{3}$ is the cone over an elliptic curve $\mathsf{E}\subset \mathbb{P}^{2}$ . The singular locus $\mathsf{D}_{\text{sing}}\subset \mathsf{D}$ is a subscheme of multiplicity 8 supported on the line $x=y=z=0$ .

The complement $\mathbb{C}^{4}\setminus \mathsf{D}$ is a symplectic leaf of dimension four, and the individual points of $\mathsf{D}_{\text{sing}}$ are zero-dimensional leaves. The two-dimensional symplectic leaves give a regular foliation of the smooth locus $\mathsf{D}_{\text{reg}}=\mathsf{D}\setminus \mathsf{D}_{\text{sing}}$ , which has the following description. Since $\mathsf{D}_{0}$ is the cone over $\mathsf{E}$ and $\mathsf{D}=\mathbb{C}\times \mathsf{D}_{0}$ , we have a natural projection $\unicode[STIX]{x1D70B}:\mathsf{D}_{\text{reg}}\rightarrow \mathbb{C}\times \mathsf{E}$ whose fibres are copies of $\mathbb{C}^{\times }$ . There is then a unique nonvanishing one-form $\unicode[STIX]{x1D6FC}\in \mathsf{H}^{0}(\mathsf{E},\unicode[STIX]{x1D6FA}_{\mathsf{E}}^{1})$ such that

$$\begin{eqnarray}\text{Res}\,\unicode[STIX]{x1D714}=\unicode[STIX]{x1D70B}^{\ast }(p_{1}^{\ast }\,dw-p_{2}^{\ast }\unicode[STIX]{x1D6FC})\in \unicode[STIX]{x1D6FA}_{\mathsf{D}_{\text{reg}}}^{1},\end{eqnarray}$$

where $w$ is the coordinate on $\mathbb{C}$ as above, and the maps $p_{1}:\mathbb{C}\times \mathsf{E}\rightarrow \mathbb{C}$ and $p_{2}:\mathbb{C}\times \mathsf{E}\rightarrow \mathsf{E}$ are the projections.

Let $Z=\unicode[STIX]{x1D6FC}^{-1}\in \mathsf{H}^{0}(\mathsf{E},\mathscr{X}_{\mathsf{E}}^{1})$ be the vector field dual to $\unicode[STIX]{x1D6FC}$ . Then $Z$ generates an action of the additive group $(\mathbb{C},+)$ on $\mathsf{E}$ by translations in the group law of the elliptic curve. Combining this action with the standard translation action on $\mathbb{C}$ , we obtain the diagonal action of $(\mathbb{C},+)$ on the product $\mathbb{C}\times \mathsf{E}$ . The symplectic leaves of $\mathsf{D}_{\text{reg}}$ are precisely the preimages of the orbits of this action under the projection $\unicode[STIX]{x1D70B}:\mathsf{D}_{\text{reg}}\rightarrow \mathbb{C}\times \mathsf{E}$ .

2.3 Stability under deformations

For a compact complex manifold $\mathsf{X}$ , we denote by $\mathsf{Pois}(\mathsf{X})\subset \mathsf{H}^{0}(\mathsf{X},\mathscr{X}_{\mathsf{X}}^{2})$ the space of Poisson structures on $\mathsf{X}$ . It is the algebraic subvariety consisting of those sections $\unicode[STIX]{x1D70E}$ that satisfy the integrability condition $[\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}]=0\in \mathsf{H}^{0}(\mathsf{X},\mathscr{X}_{\mathsf{X}}^{3})$ . This condition amounts to a system of homogeneous quadratic equations on the finite-dimensional vector space $\mathsf{H}^{0}(\mathsf{X},\mathscr{X}_{\mathsf{X}}^{2})$ . The description of the irreducible components of $\mathsf{Pois}(\mathsf{X})$ is a difficult problem and has important implications for noncommutative geometry. We observe that some components of this variety are defined by log symplectic structures.

3.1 Gorenstein singular loci

The first step is to understand the simpler question of which hypersurfaces $\mathsf{D}\subset \mathsf{X}$ admit a nondegenerate logarithmic two-form $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2}(\log \mathsf{D})$ or, equivalently, a nondegenerate logarithmic biderivation $\unicode[STIX]{x1D70E}\in \mathscr{X}_{\mathsf{X}}^{2}(-\text{log}\,\mathsf{D})$ ), without worrying about the integrability condition $d\unicode[STIX]{x1D714}=0$ . Clearly, if $\mathsf{D}$ is a free divisor and $\dim \mathsf{X}=2n$ is even, then such a logarithmic two-form exists locally. Indeed, in a sufficiently small open set, we may simply take an ${\mathcal{O}}_{\mathsf{X}}$ -module basis $\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n},\unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{n}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})$ and set

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FC}_{1}\wedge \unicode[STIX]{x1D6FD}_{1}+\cdots +\unicode[STIX]{x1D6FC}_{n}\wedge \unicode[STIX]{x1D6FD}_{n}.\end{eqnarray}$$

Of course, such a form will not, in general, be closed, and may not extend to a global logarithmic two-form.

Meanwhile, the log symplectic structure in Example 2.7 has a degeneracy hypersurface whose singular locus has codimension three in the ambient space. In light of the Jacobian criterion for freeness (Theorem 2.2), such a hypersurface cannot be a free divisor. Nevertheless, this criterion has an analogue in the skew-symmetric setting, which we now explain.

Let $(\mathsf{X},\mathsf{D})$ be a connected log Calabi–Yau manifold of dimension $2n$ , and consider the anticanonical line bundle ${\mathcal{K}}_{\mathsf{X}}^{\vee }=\mathscr{X}_{\mathsf{X}}^{2n}\cong {\mathcal{O}}_{\mathsf{X}}(\mathsf{D})$ . Let ${\mathcal{A}}$ be its Atiyah algebroid, i.e. the sheaf of first-order differential operators on ${\mathcal{K}}_{\mathsf{X}}^{\vee }$ . Thus ${\mathcal{A}}$ fits into an exact sequence

(3)

and splittings of this sequence of Lie algebroids are in bijective correspondence with flat connections on ${\mathcal{K}}_{\mathsf{X}}^{\vee }$ . Sections of $\unicode[STIX]{x039B}^{2}{\mathcal{A}}$ can then be interpreted as skew-symmetric bidifferential operators ${\mathcal{K}}_{\mathsf{X}}^{\vee }\times {\mathcal{K}}_{\mathsf{X}}^{\vee }\rightarrow ({\mathcal{K}}_{\mathsf{X}}^{\vee })^{\otimes 2}$ , and the second exterior power of (3) gives a symbol map $\unicode[STIX]{x039B}^{2}{\mathcal{A}}\rightarrow \mathscr{X}_{\mathsf{X}}^{2}$ .

Given a nondegenerate logarithmic biderivation $\unicode[STIX]{x1D70E}\in \mathsf{H}^{0}(\mathsf{X},\mathscr{X}_{\mathsf{X}}^{2}(-\text{log}\,\mathsf{D}))$ , there is a canonical section

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{{\mathcal{A}}}\in \mathsf{H}^{0}(\mathsf{X},\unicode[STIX]{x039B}^{2}{\mathcal{A}})\end{eqnarray}$$

whose symbol is $\unicode[STIX]{x1D70E}$ . This section can be described as follows: choose a local trivialization $\unicode[STIX]{x1D707}\in {\mathcal{K}}_{\mathsf{X}}^{\vee }$ and write $\unicode[STIX]{x1D70E}^{n}=h\unicode[STIX]{x1D707}$ for a function $h\in {\mathcal{O}}_{\mathsf{X}}(-\mathsf{D})$ . Consider the vector field

(4) $$\begin{eqnarray}Z=h^{-1}i_{dh}\unicode[STIX]{x1D70E}\in \mathscr{X}_{\mathsf{X}}^{1}(-\text{log}\,\mathsf{D}).\end{eqnarray}$$

Then the bidifferential operator $\unicode[STIX]{x1D70E}_{{\mathcal{A}}}$ may be defined by the formula

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{{\mathcal{A}}}(f\unicode[STIX]{x1D707},g\unicode[STIX]{x1D707})=(\langle df\wedge dg,\unicode[STIX]{x1D70E}\rangle +gZ(f)-fZ(g))\unicode[STIX]{x1D707}^{2}\in ({\mathcal{K}}_{\mathsf{X}}^{\vee })^{2}\end{eqnarray}$$

for $f,g\in {\mathcal{O}}_{\mathsf{X}}$ . One can check that this definition is independent of the choice of local trivialization.

Proof. This theorem was proved in [Reference Gualtieri and PymGP13] under the assumption that $\unicode[STIX]{x1D70E}$ is a Poisson structure, but the integrability condition was not actually used in the proof. Therefore, we shall omit most of the proof here, and simply recall why the Pfaffians define the singular locus.

A local trivialization $\unicode[STIX]{x1D707}\in {\mathcal{K}}_{\mathsf{X}}^{\vee }$ gives a splitting ${\mathcal{A}}\cong \mathscr{X}_{\mathsf{X}}^{1}\oplus {\mathcal{O}}_{\mathsf{X}}$ of (3) and hence a decomposition $\unicode[STIX]{x039B}^{2n}{\mathcal{A}}\cong \mathscr{X}_{\mathsf{X}}^{2n}\oplus \mathscr{X}_{\mathsf{X}}^{2n-1}$ . With respect to this decomposition, we may write $\unicode[STIX]{x1D70E}_{{\mathcal{A}}}^{n}=(\unicode[STIX]{x1D70E}^{n},nZ\wedge \unicode[STIX]{x1D70E}^{n-1})$ with $Z$ as defined in (4). However,

$$\begin{eqnarray}\displaystyle Z\wedge \unicode[STIX]{x1D70E}^{n-1} & = & \displaystyle (h^{-1}i_{dh}\unicode[STIX]{x1D70E})\wedge \unicode[STIX]{x1D70E}^{n-1}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{n}h^{-1}i_{dh}\unicode[STIX]{x1D70E}^{n}\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{n}i_{dh}\unicode[STIX]{x1D707}\nonumber\end{eqnarray}$$

and hence $\unicode[STIX]{x1D70E}_{{\mathcal{A}}}^{n}=(h\unicode[STIX]{x1D707},i_{dh}\unicode[STIX]{x1D707})$ . Since $\unicode[STIX]{x1D707}$ is nonvanishing, the zero locus of $\unicode[STIX]{x1D70E}_{{\mathcal{A}}}^{n}$ coincides with the simultaneous vanishing locus of $h$ and $dh$ , which is precisely the singular locus of $\mathsf{D}$ , as claimed.◻

Locally, this theorem has a partial converse.

Proof. Let ${\mathcal{O}}_{}={\mathcal{O}}_{\mathbb{C}^{2n,0}}$ be the ring of germs of analytic functions and denote by $\mathscr{X}_{}^{1}=\mathscr{X}_{\mathbb{C}^{2n},0}^{1}$ the ${\mathcal{O}}_{}$ -module of vector fields. Let $h\in {\mathcal{O}}_{}$ be a defining equation for $\mathsf{D}$ , and let ${\mathcal{O}}_{\mathsf{D}_{\text{sing}}}$ be the ring of functions on the singular locus. Setting ${\mathcal{A}}={\mathcal{O}}_{}\oplus \mathscr{X}_{}^{1}$ , the map $(h,dh):{\mathcal{A}}\rightarrow {\mathcal{O}}_{}$ has cokernel ${\mathcal{O}}_{\mathsf{D}_{\text{sing}}}$ , giving the beginning of a free resolution of ${\mathcal{O}}_{\mathsf{D}_{\text{sing}}}$ . Its kernel is identified with $\mathscr{X}_{}^{1}(-\text{log}\,\mathsf{D})$ by the projection ${\mathcal{A}}\rightarrow \mathscr{X}_{}^{1}$ .

On the other hand, by the Buchsbaum–Eisenbud structure theorem [Reference Buchsbaum and EisenbudBE77] for codimension-three Gorenstein ideals, there exist an integer $k>0$ , a free module ${\mathcal{E}}$ of rank $2k+1$ and a skew-symmetric form $\unicode[STIX]{x1D70C}\in \unicode[STIX]{x039B}^{2}{\mathcal{E}}$ such that the minimal free resolution of ${\mathcal{O}}_{\mathsf{D}_{\text{sing}}}$ has the form

where the outer maps are the Pfaffians $\unicode[STIX]{x1D70C}^{k}\in \unicode[STIX]{x039B}^{2k}{\mathcal{E}}\cong {\mathcal{E}}^{\vee }\otimes \det {\mathcal{E}}$ . Therefore, by minimality, there exists a free module ${\mathcal{F}}$ equipped with an isomorphism ${\mathcal{A}}\cong ({\mathcal{E}}\otimes \det {\mathcal{E}}^{\vee })\oplus {\mathcal{F}}$ that identifies the submodule $\mathscr{X}_{}^{1}(-\text{log}\,\mathsf{D})\subset {\mathcal{A}}$ with $\mathsf{img}(\unicode[STIX]{x1D70C})\oplus {\mathcal{F}}$ .

Since the ranks of ${\mathcal{A}}$ and ${\mathcal{E}}$ are odd, the rank of ${\mathcal{F}}$ must be even. Hence we may choose a nondegenerate skew form $\unicode[STIX]{x1D70C}^{\prime }\in \unicode[STIX]{x039B}^{2}{\mathcal{F}}$ . Under the isomorphism ${\mathcal{A}}\cong {\mathcal{E}}\oplus {\mathcal{F}}$ , the sum $\unicode[STIX]{x1D70E}_{{\mathcal{A}}}=\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D70C}^{\prime }$ defines an element of $\unicode[STIX]{x039B}^{2}{\mathcal{A}}$ , and we obtain a free resolution

of ${\mathcal{O}}_{\mathsf{D}_{\text{sing}}}$ . Let $\unicode[STIX]{x1D70E}$ be the image of $\unicode[STIX]{x1D70E}_{{\mathcal{A}}}$ under the projection $\unicode[STIX]{x039B}^{2}{\mathcal{A}}\rightarrow \mathscr{X}_{}^{2}$ . The fact that the sequence above is a complex implies that $\unicode[STIX]{x1D70E}$ lies in $\mathscr{X}_{\mathsf{X}}^{2}(-\text{log}\,\mathsf{D})$ . Moreover, the map $(h,dh)$ is identified with the Pfaffian $\unicode[STIX]{x1D70E}_{{\mathcal{A}}}^{n}$ under the isomorphism $\unicode[STIX]{x039B}^{2n}{\mathcal{A}}\cong \mathscr{X}_{}^{2n}\oplus \mathscr{X}_{}^{2n-1}\cong {\mathcal{O}}_{}\oplus \unicode[STIX]{x1D6FA}_{}^{1}$ given by an appropriate choice of volume form. Hence $h^{-1}\unicode[STIX]{x1D70E}^{n}$ trivializes $\mathscr{X}_{}^{2n}$ , so $\unicode[STIX]{x1D70E}$ is nondegenerate.◻

Example 3.3 (Quasi-homogeneous surfaces).

Let $f\in \mathbb{C}[x,y,z]$ be a quasi-homogeneous polynomial with weights $(a,b,c)\in \mathbb{Z}_{{>}0}^{3}$ , and suppose that $0$ is the only critical point of $f$ . Let $\mathsf{D}_{0}\subset \mathbb{C}^{3}$ be the zero locus of $f$ . Because of the quasi-homogeneity, the singular locus of the product $\mathsf{D}=\mathbb{C}\times \mathsf{D}_{0}\subset \mathbb{C}^{4}$ is the complete intersection defined by the equations $\unicode[STIX]{x2202}_{x}f=\unicode[STIX]{x2202}_{y}f=\unicode[STIX]{x2202}_{z}f=0$ , and hence it is Gorenstein.

Correspondingly, $\mathsf{D}$ supports a nondegenerate logarithmic biderivation, an example of which may be constructed as follows. Let $E=ax\unicode[STIX]{x2202}_{x}+by\unicode[STIX]{x2202}_{y}+cz\unicode[STIX]{x2202}_{z}$ be the weighted Euler vector field, and extend the coordinates $(x,y,z)$ on $\mathbb{C}^{3}$ to a coordinate system $(w,x,y,z)$ on $\mathbb{C}^{4}$ . Then one can readily check that

$$\begin{eqnarray}\unicode[STIX]{x1D70E}=E\wedge \unicode[STIX]{x2202}_{w}+i_{df}(\unicode[STIX]{x2202}_{x}\wedge \unicode[STIX]{x2202}_{y}\wedge \unicode[STIX]{x2202}_{z})\in \mathscr{X}_{\mathbb{C}^{4}}^{2}(-\text{log}\,\mathsf{D})\end{eqnarray}$$

is a nondegenerate logarithmic biderivation. However, it will not in general be integrable. In fact, we will see in § 4 that $\mathsf{D}$ supports a log symplectic form if and only if the degree of $f$ is equal to $a+b+c$ .◻

3.2 The local Moser trick

We now return to the integrable case, in which the nondegenerate logarithmic form is closed. Clearly, if $(\mathsf{X},\unicode[STIX]{x1D714})$ and $(\mathsf{X}^{\prime },\unicode[STIX]{x1D714}^{\prime })$ are log symplectic structures that are isomorphic in neighbourhoods of points $p\in \mathsf{X}$ and $p^{\prime }\in \mathsf{X}^{\prime }$ , then their degeneracy hypersurfaces $\mathsf{D}$ and $\mathsf{D}^{\prime }$ are also isomorphic in the same neighbourhoods. In other words, the singularity type of $\mathsf{D}$ at $p$ is a local invariant of the log symplectic structure. In this section, we study log symplectic structures for which the underlying singularity type is fixed, and find local analogues of several standard cohomological properties of compact symplectic manifolds and log symplectic manifolds with smooth degeneracy hypersurfaces [Reference Guillemin, Miranda and PiresGMP14].

For a reduced hypersurface germ $\mathsf{D}\subset (\mathsf{X},p)$ , let $\mathsf{D}=\mathsf{D}_{1}\cup \cdots \cup \mathsf{D}_{k}$ be a decomposition into irreducible components, and let $h_{1},\ldots ,h_{k}$ be defining equations for the components. If the ${\mathcal{O}}_{\mathsf{X},p}$ -module $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})_{p}$ is generated by the meromorphic forms $h_{1}^{-1}dh_{1},\ldots ,h_{k}^{-1}h_{k}$ and the holomorphic forms $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}$ , we will say that $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})_{p}$ is generated by logarithmic differentials. Saito has shown that $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})_{p}$ is generated by logarithmic differentials if and only if each component $\mathsf{D}_{i}$ is normal and the intersections of the components are sufficiently transverse; see [Reference SaitoSai80, Theorem 2.9] for the exact formulation. These conditions are satisfied, in particular, if $\mathsf{D}$ is smooth, normal, or has normal crossings singularities.

Our results pertain to the local logarithmic de Rham cohomology at a point $p\in \mathsf{D}_{\text{sing}}$ , which is the stalk cohomology $\mathsf{H}_{\text{dR},p}^{\bullet }(\mathsf{X},\log \mathsf{D})$ of the complex of sheaves $(\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{\bullet }(\log \mathsf{D}),d)$ at $p$ .

Proposition 3.4. Let $(\mathsf{X},\mathsf{D},\unicode[STIX]{x1D714})$ be a log symplectic manifold of dimension $2n$ , and let $p\in \mathsf{D}_{\text{sing}}$ be a singular point at which $\mathsf{D}$ is normal. (Hence $\dim \mathsf{D}_{\text{sing}}=2n-3$ at $p$ by Theorem 3.1.) Then the local logarithmic cohomology class

$$\begin{eqnarray}[\unicode[STIX]{x1D714}]\in \mathsf{H}_{\text{dR},p}^{2}(\mathsf{X},\log \mathsf{D})\end{eqnarray}$$

is nonzero.

Proof. Suppose, to the contrary, that $[\unicode[STIX]{x1D714}]=0$ . Then

$$\begin{eqnarray}\unicode[STIX]{x1D714}=d\unicode[STIX]{x1D6FC}\end{eqnarray}$$

for some $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})_{p}$ . Let $f$ be a reduced defining equation for $\mathsf{D}$ near $p$ . Since $\mathsf{D}$ is normal, it is irreducible, so that $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})_{p}$ is generated by the logarithmic differential $f^{-1}df$ . We may therefore write

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}=gf^{-1}df+\unicode[STIX]{x1D6FD}\end{eqnarray}$$

with $g\in {\mathcal{O}}_{\mathsf{X},p}$ and $\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{\mathsf{X},p}^{1}$ , so that

$$\begin{eqnarray}\unicode[STIX]{x1D714}=f^{-1}dg\wedge df+d\unicode[STIX]{x1D6FD}.\end{eqnarray}$$

Therefore

$$\begin{eqnarray}\unicode[STIX]{x1D714}^{n}=nf^{-1}\,dg\wedge df\wedge d\unicode[STIX]{x1D6FD}^{n-1}+(d\unicode[STIX]{x1D6FD})^{n}.\end{eqnarray}$$

Viewed as a section of $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2n}(\mathsf{D})$ , the $2n$ -form $(d\unicode[STIX]{x1D6FD})^{n}$ vanishes at $p$ . Nondegeneracy of $\unicode[STIX]{x1D714}$ then implies that the section

$$\begin{eqnarray}f^{-1}\,dg\wedge df\wedge d\unicode[STIX]{x1D6FD}^{n-1}\end{eqnarray}$$

is a local trivialization of $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2n}(\mathsf{D})$ near $p$ , which in turn implies that $df(p)\neq 0$ . But $p$ is a singular point of $\mathsf{D}$ , and hence $df(p)=0$ , a contradiction.◻

Lemma 3.6 (Local Moser trick).

Let $\mathsf{D}\subset \mathsf{X}$ be a reduced hypersurface and let $p\in \mathsf{D}_{\text{sing}}$ be a point such that $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})_{p}$ is generated by logarithmic differentials. Suppose that $\unicode[STIX]{x1D714}_{t}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2}(\log \mathsf{D})_{p}$ , $t\in [0,1]$ , is a smooth family of germs of log symplectic forms such that the local logarithmic cohomology class

$$\begin{eqnarray}[\unicode[STIX]{x1D714}_{t}]\in \mathsf{H}_{\text{dR},p}^{2}(\mathsf{X},\log \mathsf{D})\end{eqnarray}$$

is independent of $t$ . Then there exists a family of germs of automorphisms $\unicode[STIX]{x1D719}_{t}\in \mathsf{Aut}(\mathsf{X},\mathsf{D})_{p}$ such that $\unicode[STIX]{x1D719}_{t}^{\ast }\unicode[STIX]{x1D714}_{t}=\unicode[STIX]{x1D714}_{0}$ .

Proof. By assumption, the logarithmic two-form

$$\begin{eqnarray}\dot{\unicode[STIX]{x1D714}}_{t}=\frac{d\unicode[STIX]{x1D714}_{t}}{dt}\end{eqnarray}$$

is exact. We may therefore choose a smooth family $\unicode[STIX]{x1D6FC}_{t}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\mathsf{D})_{p}$ of logarithmic one-forms such that $d\unicode[STIX]{x1D6FC}_{t}=\dot{\unicode[STIX]{x1D714}}_{t}$ . Let $f_{1},\ldots ,f_{k}$ be defining equations for the irreducible components of $\mathsf{D}$ . Because $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})_{p}$ is generated by logarithmic differentials, we may write

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{t}=\mathop{\sum }_{i=1}^{k}g_{i,t}f_{i}^{-1}df_{i}+\unicode[STIX]{x1D6FD}_{t}\end{eqnarray}$$

where $g_{i,t}\in {\mathcal{O}}_{\mathsf{X},p}$ and $\unicode[STIX]{x1D6FD}_{t}\in \unicode[STIX]{x1D6FA}_{\mathsf{X},p}^{1}$ . Replacing $g_{i,t}$ with $g_{i,t}-g_{i,t}(p)$ and changing $\unicode[STIX]{x1D6FD}_{t}$ by an exact form depending on $t$ , we may assume that $g_{i,t}(p)=0$ and $\unicode[STIX]{x1D6FD}_{i,t}(p)=0$ for all $t$ . Hence $\unicode[STIX]{x1D6FC}_{t}$ vanishes at $p$ when viewed as a section of $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})$ .

By nondegeneracy, there is a unique time-dependent vector field $Z_{t}\in \mathscr{X}_{\mathsf{X}}^{1}(-\text{log}\,\mathsf{D})_{p}$ such that

$$\begin{eqnarray}i_{Z_{t}}\unicode[STIX]{x1D714}_{t}=\unicode[STIX]{x1D6FC}_{t},\end{eqnarray}$$

and since $\unicode[STIX]{x1D6FC}_{t}$ vanishes at $p$ , the holomorphic vector field $Z_{t}$ also vanishes at $p$ for all $t$ . Therefore $Z_{t}$ integrates to a family of germs of automorphisms $\unicode[STIX]{x1D719}_{t}\in \mathsf{Aut}(\mathsf{X},p)$ , and since $Z_{t}$ is logarithmic, these automorphism preserve the subgerm $(\mathsf{D},p)\subset (\mathsf{X},p)$ .

We now apply the standard calculation

$$\begin{eqnarray}\displaystyle \frac{d\unicode[STIX]{x1D719}_{t}^{\ast }\unicode[STIX]{x1D714}_{t}}{dt}=\mathscr{L}_{Z_{t}}\unicode[STIX]{x1D714}_{t}=i_{Z_{t}}d\unicode[STIX]{x1D714}_{t}+di_{Z_{t}}\unicode[STIX]{x1D714}_{t}=d\unicode[STIX]{x1D6FC}_{t}=\dot{\unicode[STIX]{x1D714}}_{t}, & & \displaystyle \nonumber\end{eqnarray}$$

which implies that $\unicode[STIX]{x1D719}_{1}^{\ast }\unicode[STIX]{x1D714}_{1}=\unicode[STIX]{x1D714}_{0}$ , as required.◻

This result implies that the local logarithmic cohomology class determines the germ of the log symplectic form up to isomorphism.

Theorem 3.7. Let $\mathsf{D}\subset \mathsf{X}$ be a reduced hypersurface and let $p\in \mathsf{D}$ be a point such that $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}(\log \mathsf{D})_{p}$ is generated by logarithmic differentials. Suppose that $\unicode[STIX]{x1D714}_{0},\unicode[STIX]{x1D714}_{1}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2}(\log \mathsf{D})_{p}$ are two germs of log symplectic structures with the same local logarithmic cohomology classes

$$\begin{eqnarray}[\unicode[STIX]{x1D714}_{0}]=[\unicode[STIX]{x1D714}_{1}]\in \mathsf{H}_{\text{dR},p}^{2}(\mathsf{X},\log \mathsf{D}).\end{eqnarray}$$

Then $\unicode[STIX]{x1D714}_{0}$ and $\unicode[STIX]{x1D714}_{1}$ are isotopic.

Proof. For $\unicode[STIX]{x1D706}\in \mathbb{C}$ , we define the logarithmic two-form

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D706}}=(1-\unicode[STIX]{x1D706})\unicode[STIX]{x1D714}_{0}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D714}_{1}.\end{eqnarray}$$

Clearly the class $[\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D706}}]\in \mathsf{H}_{p}^{2}(\mathsf{X},\log \mathsf{D})$ is independent of $\unicode[STIX]{x1D706}$ .

Let $\mathsf{L}=\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{2n}(\mathsf{D})|_{p}$ be the fibre of the logarithmic canonical line bundle at $p$ , and consider the map

$$\begin{eqnarray}\begin{array}{@{}ccccc@{}}\text{Pf} & : & \mathbb{C} & \rightarrow & \mathsf{L}\\ & & \unicode[STIX]{x1D706} & \mapsto & (\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D706}})^{n}(p).\end{array}\end{eqnarray}$$

Then $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D706}}$ is nondegenerate in a neighbourhood of $p$ if and only if $\text{Pf}(\unicode[STIX]{x1D706})\neq 0$ . In particular, $\text{Pf}(0)$ and $\text{Pf}(1)$ are both nonzero. Since $\text{Pf}(\unicode[STIX]{x1D706})$ depends polynomially on $\unicode[STIX]{x1D706}$ , we may choose a smooth path $\unicode[STIX]{x1D6FE}:[0,1]\rightarrow \mathbb{C}$ such that $\unicode[STIX]{x1D6FE}(0)=0$ , $\unicode[STIX]{x1D6FE}(1)=1$ and $\text{Pf}(\unicode[STIX]{x1D6FE}(t))\neq 0$ for all $t$ . But then $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6FE}(t)}$ for $t\in [0,1]$ gives a smoothly varying family of log symplectic forms whose local cohomology classes are all the same. We are therefore in the situation of Lemma 3.6 and conclude that the germs of $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D714}^{\prime }$ are isomorphic.◻

The main application of this result is to give streamlined proofs of local normal forms for log symplectic structures. We give now two examples; the approach will be used again when we discuss elliptic structures in § 4.

Example 3.9 (Normal crossings).

Let $p\in \mathsf{D}$ be a point at which $\mathsf{D}$ may be decomposed locally into $k$ components with normal crossings. Choose coordinates $(x_{1},\ldots ,x_{k},y_{1},\ldots ,y_{m})$ so that $x_{1}x_{2}\cdots x_{k}$ is a local defining equation for $\mathsf{D}$ . Then the first logarithmic cohomology is a $k$ -dimensional vector space

$$\begin{eqnarray}\mathsf{V}=\mathsf{H}_{\text{dR},p}^{1}(\mathsf{X},\log \mathsf{D})=\mathbb{C}\cdot \langle [x_{1}^{-1}dx],\ldots ,[x_{k}^{-1}dx_{k}]\rangle ,\end{eqnarray}$$

and the wedge product identifies $\mathsf{H}_{\text{dR},p}^{\bullet }(\mathsf{X},\log \mathsf{D})\cong \unicode[STIX]{x039B}^{\bullet }\mathsf{V}$ as rings.

In particular, for $k\geqslant 2$ , every logarithmic two-form is cohomologous to one of the form

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{0}=\mathop{\sum }_{1\leqslant i<j\leqslant k}a_{ij}\frac{dx_{i}}{x_{i}}\wedge \frac{dx_{j}}{x_{j}}\end{eqnarray}$$

where $A=(a_{ij})$ is a skew-symmetric matrix of constants. The $(i,j)$ entry can be interpreted as a period: it is the integral of $\unicode[STIX]{x1D714}$ over a small real torus $\mathsf{T}_{ij}\cong \mathsf{S}^{1}\times \mathsf{S}^{1}$ that encircles the intersection $\mathsf{D}_{i}\cap \mathsf{D}_{j}$ in a neighbourhood of $p$ .

If $k=\dim \mathsf{X}$ , then the above calculation of the cohomology shows that any nonvanishing logarithmic volume form is nontrivial in cohomology. Now $\unicode[STIX]{x1D714}_{0}^{n}=\text{Pf}(A)(dx_{1}/x_{1})\wedge \cdots \wedge (dx_{n}/x_{n})$ , and hence a form cohomologous to $\unicode[STIX]{x1D714}_{0}^{n}$ can only be log symplectic if the Pfaffian $\text{Pf}(A)$ is nonzero, i.e. the matrix $A$ is nonsingular. In this case $\unicode[STIX]{x1D714}_{0}^{n}$ is already log symplectic. Hence every log symplectic structure on $\mathsf{D}$ is locally isomorphic to one of this form, where the nonsingular matrix $A$ is determined by the periods of the log symplectic form.

On the other hand, if $k<\dim \mathsf{X}$ , then $\unicode[STIX]{x1D714}_{0}$ is degenerate. In order to obtain a log symplectic form, we must extend $\unicode[STIX]{x1D714}_{0}$ by adding terms involving the $y$ coordinates, to obtain a cohomologous form

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\mathop{\sum }_{1\leqslant i<j\leqslant k}a_{ij}\frac{dx_{i}}{x_{i}}\wedge \frac{dx_{j}}{x_{j}}+\mathop{\sum }_{i=1}^{k}\mathop{\sum }_{j=1}^{m}b_{ij}\frac{dx_{i}}{x_{i}}\wedge dy_{j}+\mathop{\sum }_{1\leqslant i<j\leqslant m}c_{ij}dy_{i}\wedge dy_{j}\end{eqnarray}$$

for a $k\times m$ matrix of constants $B=(b_{ij})$ and an $m\times m$ skew-symmetric matrix $C=(c_{ij})$ . Then $\unicode[STIX]{x1D714}$ will be log symplectic if and only if the block matrix

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}A & B\\ -B & C\end{array}\right)\end{eqnarray}$$

of coefficients is invertible. Two such log symplectic forms will be isotopic near $p$ if and only if they share the same matrix $A$ of periods. The matrices $B$ and $C$ do not affect the isomorphism class and so can be further simplified, but the optimal simplification depends on $k$ and  $A$ .

4.1 The elliptic normal form theorem

Let $(\mathsf{X},\mathsf{D},\unicode[STIX]{x1D714})$ be a log symplectic manifold and let $\unicode[STIX]{x1D70E}$ be the corresponding Poisson bivector. Given a volume form $\unicode[STIX]{x1D707}\in {\mathcal{K}}_{\mathsf{X}}$ , we have $\unicode[STIX]{x1D70E}^{n}=f\unicode[STIX]{x1D707}^{-1}$ , where $f\in {\mathcal{O}}_{\mathsf{X}}(-\mathsf{D})$ is a local defining equation for $\mathsf{D}$ . We may then consider the log Hamiltonian vector field

$$\begin{eqnarray}Z_{\unicode[STIX]{x1D707}}=f^{-1}i_{df}\unicode[STIX]{x1D70E}\in \mathscr{X}_{\mathsf{X}}^{1}(-\text{log}\,\mathsf{D}).\end{eqnarray}$$

This vector field is nothing but the modular vector field of $\unicode[STIX]{x1D70E}$ with respect to the volume form $\unicode[STIX]{x1D707}$ , as introduced by Weinstein [Reference WeinsteinWei97] for general Poisson manifolds. One can easily verify that $Z_{\unicode[STIX]{x1D707}}$ is $\unicode[STIX]{x1D707}$ -divergence free, i.e.  $\mathscr{L}_{Z_{\unicode[STIX]{x1D707}}}\unicode[STIX]{x1D707}=0$ .

If we change the volume form, then $Z_{\unicode[STIX]{x1D707}}$ changes by a Hamiltonian vector field. Hence, if $p\in \mathsf{X}$ is a point where the Poisson structure vanishes, the value of $Z_{\unicode[STIX]{x1D707}}$ at $p$ is independent of the equation $f$ . We therefore have a natural section

$$\begin{eqnarray}Z\in \mathsf{H}^{0}(\mathsf{Y},\mathscr{X}_{\mathsf{X}}^{1}|_{\mathsf{Y}})\end{eqnarray}$$

where $\mathsf{Y}\subset \mathsf{X}$ is the vanishing locus of $\unicode[STIX]{x1D70E}$ . According to [Reference Gualtieri and PymGP13, Theorem 19], this section may alternatively be computed as follows: by restricting the one-jet of $\unicode[STIX]{x1D70E}$ to $\mathsf{Y}$ , we obtain a natural section $j^{1}\unicode[STIX]{x1D70E}|_{\mathsf{Y}}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}\otimes \mathscr{X}_{\mathsf{X}}^{2}|_{\mathsf{Y}}$ , and $Z$ is the image of this section under the interior contraction $\unicode[STIX]{x1D6FA}_{\mathsf{X}}^{1}\otimes \mathscr{X}_{\mathsf{X}}^{2}\rightarrow \mathscr{X}_{\mathsf{X}}^{1}$ . Hence, $Z$ is a linear combination of first derivatives of the Poisson structure along its zero locus.

Suppose now that $p\in \mathsf{D}_{\text{sing}}$ is a singular point at which $\mathsf{D}$ is normal. By Theorem 3.1, the codimension of $\mathsf{D}_{\text{sing}}$ in $\mathsf{X}$ is equal to three. Since $\mathsf{D}_{\text{sing}}$ is a Poisson subspace (see [Reference PolishchukPol97, Corollary 2.4] or [Reference Gualtieri and PymGP13, Lemma 2.3]), $\mathsf{D}_{\text{sing}}$ contains the symplectic leaf $\mathsf{L}\subset \mathsf{X}$ through $p$ , and hence $\mathsf{L}$ , being even-dimensional, must have codimension at least four. The rest of the paper is concerned with singularities for which the following transversality condition is satisfied.

Using Weinstein’s splitting theorem [Reference WeinsteinWei83], we see that in a sufficiently small neighbourhood of an elliptic point, the log symplectic structure decomposes as a product of a symplectic manifold of dimension $\dim \mathsf{X}-4$ and a four-dimensional purely elliptic structure. As a result, the local structure near an elliptic point is completely determined by the following.

Theorem 4.5. Let $(\mathsf{X},\mathsf{D},\unicode[STIX]{x1D714})$ be a log symplectic manifold of dimension four, and let $p\in \mathsf{D}_{\text{sing}}$ be an elliptic point. Then there exist coordinates $(w,x,y,z)$ on $\mathsf{X}$ centred at $p$ in which the Poisson brackets have the form

(5) $$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\{w,x\}=ax,\quad \{x,y\}=\unicode[STIX]{x1D706}\unicode[STIX]{x2202}_{z}f,\\ \{w,y\}=by,\quad \{y,z\}=\unicode[STIX]{x1D706}\unicode[STIX]{x2202}_{x}f,\\ \{w,z\}=cz,\quad \{z,x\}=\unicode[STIX]{x1D706}\unicode[STIX]{x2202}_{y}f\end{array} & & \displaystyle\end{eqnarray}$$

for some positive integers $(a,b,c)\in \mathbb{Z}_{{>}0}^{3}$ and polynomial $f$ appearing in Table 1, together with a constant $\unicode[STIX]{x1D706}\in \mathbb{C}^{\times }$ .

In particular, the zero locus of the Poisson structure, defined by the equations $x=y=z=0$ , is smooth at $p$ ; also, the degeneracy hypersurface, defined by the equation $f=0$ , is locally the product of a smooth curve and a simple elliptic surface singularity.

Table 1. The simple elliptic surface singularities, parametrized by $\unicode[STIX]{x1D70F}\in \mathbb{C}^{\times }$ .

The rest of this section is devoted to the proof of Theorem 4.5. The first step is given by the following lemma.

Proof. Let $Z$ be the modular vector field with respect to some volume form $\unicode[STIX]{x1D707}\in \unicode[STIX]{x1D6FA}_{\mathsf{X}}^{4}$ defined near $p$ . Since $Z(p)\neq 0$ and $\mathscr{L}_{Z}\unicode[STIX]{x1D707}=0$ , we may choose a coordinate system $(w,x,y,z)$ near $p$ such that $Z=\unicode[STIX]{x2202}_{w}$ and $\unicode[STIX]{x1D707}=dw\wedge dx\wedge dy\wedge dz$ . Since $\mathscr{L}_{Z}\unicode[STIX]{x1D70E}=0$ , one can easily compute that the germ of the Poisson tensor $\unicode[STIX]{x1D70E}$ has the form

$$\begin{eqnarray}\unicode[STIX]{x1D70E}=E\wedge \unicode[STIX]{x2202}_{w}+\unicode[STIX]{x1D70E}_{0},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{0}$ is the germ of a Poisson structure on $\mathbb{C}^{3}$ , with coordinates $(x,y,z)$ , and $E$ is the germ of a vector field on $\mathbb{C}^{3}$ satisfying $\mathscr{L}_{E}\unicode[STIX]{x1D70E}_{0}=0$ . In particular,

$$\begin{eqnarray}\unicode[STIX]{x1D70E}\wedge \unicode[STIX]{x1D70E}=f\unicode[STIX]{x1D707}^{-1},\end{eqnarray}$$

where $f$ is a function depending only on $(x,y,z)$ , so that the degeneracy hypersurface $\mathsf{D}$ is a product of a smooth curve and the surface $\mathsf{D}_{0}=f^{-1}(0)\subset \mathbb{C}_{x,y,z}^{3}$ . Since we are assuming that $\dim _{p}\mathsf{D}_{\text{sing}}=1$ , the surface $\mathsf{D}_{0}$ must have an isolated singularity at the origin.

Using the decomposition, we compute the modular vector field

$$\begin{eqnarray}Z=f^{-1}i_{df}\unicode[STIX]{x1D70E}=f^{-1}(\mathscr{L}_{E}f)\unicode[STIX]{x2202}_{w}+f^{-1}i_{df}\unicode[STIX]{x1D70E}_{0}.\end{eqnarray}$$

Since $Z=\unicode[STIX]{x2202}_{w}$ and $\unicode[STIX]{x1D70E}_{0}$ has no components involving $\unicode[STIX]{x2202}_{w}$ , this equation implies that $\mathscr{L}_{E}f=f$ . The lemma now follows from a theorem of Saito [Reference SaitoSai71].◻

Using the previous lemma and the cohomological parametrization of local normal forms (Theorem 3.7), the proof of Theorem 4.5 reduces to the following.

Proof. Let $\mathsf{D}_{0}\subset \mathbb{C}^{3}$ be the intersection of $\mathsf{D}$ with the hyperplane $w=0$ , so that $\mathsf{D}=\mathbb{C}\times \mathsf{D}_{0}$ . By the Künneth theorem for logarithmic de Rham cohomology [Reference Castro-Jiménez, Narváez-Macarro and MondCJNMM96, Lemma 2.2], the pullback along the projection $(\mathbb{C}^{4},\mathsf{D})\rightarrow (\mathbb{C}^{3},\mathsf{D}_{0})$ provides an isomorphism

$$\begin{eqnarray}\mathsf{H}_{\text{dR},0}^{2}(\mathbb{C}^{4},\log \mathsf{D})\cong \mathsf{H}_{\text{dR},0}^{2}(\mathbb{C}^{3},\log \mathsf{D}_{0}).\end{eqnarray}$$

Because of the holomorphic Poincaré lemma, the complex $\unicode[STIX]{x1D6FA}_{\mathbb{C}^{3}}^{\bullet }$ is exact in positive degrees. Hence the residue exact sequence (1) gives an isomorphism

$$\begin{eqnarray}\mathsf{H}_{\text{dR},0}^{2}(\mathbb{C}^{3},\log \mathsf{D}_{0})\cong \mathsf{H}_{0}^{1}(\unicode[STIX]{x1D6FA}_{\mathsf{D}_{0}}^{\bullet ,\text{reg}})\end{eqnarray}$$

with the cohomology of the regular differential forms on $\mathsf{D}_{0}$ . But since $\mathsf{D}_{0}$ is normal and $\unicode[STIX]{x1D6FA}_{\mathsf{D}_{0}}^{\bullet ,\text{reg}}$ is reflexive, we have $\unicode[STIX]{x1D6FA}_{\mathsf{D}_{0}}^{\bullet ,\text{reg}}\cong (\unicode[STIX]{x1D6FA}_{\mathsf{D}_{0}}^{\bullet })^{\vee \vee }={\mathcal{H}}om(\mathscr{X}_{\mathsf{D}_{0}}^{\bullet },{\mathcal{O}}_{\mathsf{D}_{0}})$ . The cohomology of this complex was computed in [Reference Eriksen and GustavsenEG09, Theorem 5.2]. From that result, we see that every cohomology class in $\mathsf{H}_{0}^{2}(\mathbb{C}^{3},\log \mathsf{D}_{0})$ is represented by a unique element of the form

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{h}=f^{-1}h\,i_{E}(dx\wedge dy\wedge dz)\end{eqnarray}$$

where $h\in \mathbb{C}[x,y,z]$ is quasi-homogeneous of degree $k-a-b-c$ .

Hence every closed logarithmic form $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{4}}^{2}(\log \mathsf{D})$ may be written as

$$\begin{eqnarray}\unicode[STIX]{x1D714}=f^{-1}h\,i_{E}(dx\wedge dy\wedge dz)+d\unicode[STIX]{x1D6FC}\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{4}}^{1}(\log \mathsf{D})$ . Since $\mathsf{D}$ is normal, $\unicode[STIX]{x1D6FA}_{\mathbb{C}^{4}}^{1}(\log \mathsf{D})$ is generated by logarithmic differentials, so $\unicode[STIX]{x1D6FC}=gf^{-1}df+\unicode[STIX]{x1D6FD}$ with $g\in {\mathcal{O}}_{\mathbb{C}^{4}}$ and $\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{4}}^{1}$ holomorphic.

We must determine when such an $\unicode[STIX]{x1D714}$ is nondegenerate. Using the identity $\mathscr{L}_{E}f=kf$ and the fact that $f$ and $df$ vanish at the origin, one easily computes that

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{2}=2khf^{-1}\,dx\wedge \,dy\wedge \,dz\wedge dg\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~(x,y,z)\cdot \unicode[STIX]{x1D6FA}_{\mathbb{C}^{4}}^{4}(\mathsf{D}). & & \displaystyle \nonumber\end{eqnarray}$$

Hence $\unicode[STIX]{x1D714}$ can only be nondegenerate if the top degree form $h\,dx\wedge dy\wedge dz\wedge dg$ is nonvanishing near the origin. This condition forces the quasi-homogeneous polynomial $h$ to be a nonzero constant. Since its degree is equal to $k-a-b-c$ , we must have $k=a+b+c$ . The first statement now follows from Saito’s classification of such quasi-homogeneous polynomials [Reference SaitoSai71]; see also the formulae in [Reference Etingof and GinzburgEG10, Proposition 2.3.2].

For the second statement, we note that since the choices of $g$ and $\unicode[STIX]{x1D6FD}$ do not affect the cohomology class of $\unicode[STIX]{x1D714}$ , we may as well take $g=k^{-1}w$ and $\unicode[STIX]{x1D6FD}=0$ . Thus every log symplectic form on $(\mathbb{C}^{4},\mathsf{D})$ is cohomologous at $0$ to the form

$$\begin{eqnarray}\unicode[STIX]{x1D714}=(k\unicode[STIX]{x1D706}f)^{-1}\,i_{E}(dx\wedge \,dy\wedge \,dz)+k^{-1}f^{-1}\,df\wedge dw\end{eqnarray}$$

for a unique constant $\unicode[STIX]{x1D706}\in \mathbb{C}^{\times }$ . Inverting $\unicode[STIX]{x1D714}$ , we obtain the Poisson brackets of (5), as required.◻

4.2 Basic properties of purely elliptic structures

4.2.3 Topological constraints

According to the local normal form, we may decompose $\mathsf{Y}$ as a disjoint union

$$\begin{eqnarray}\mathsf{Y}=\mathsf{Y}_{6}\sqcup \mathsf{Y}_{7}\sqcup \mathsf{Y}_{8}\end{eqnarray}$$

of open submanifolds, where $\mathsf{Y}_{i}$ denotes the set of points at which $\mathsf{D}$ has singularities of type  $\widetilde{E}_{i}$ .

Considering the Milnor numbers of the elliptic singularities in Table 1, we see that the length of $\mathsf{D}_{\text{sing}}$ along $\mathsf{Y}_{i}$ is equal to $i+2$ . Hence we have the following formula for the fundamental class in singular homology:

$$\begin{eqnarray}[\mathsf{D}_{\text{sing}}]=8[\mathsf{Y}_{6}]+9[\mathsf{Y}_{7}]+10[\mathsf{Y}_{8}]\in \mathsf{H}_{\dim \mathsf{X}-6}(\mathsf{X},\mathbb{Z}).\end{eqnarray}$$

Combining this result with Theorem 1.1, we obtain the proposition below.

Proposition 4.10. For a purely elliptic log symplectic manifold $(\mathsf{X},\mathsf{D},\unicode[STIX]{x1D714})$ , we have the equality

$$\begin{eqnarray}8[\mathsf{Y}_{6}]+9[\mathsf{Y}_{7}]+10[\mathsf{Y}_{8}]=(c_{1}c_{2}-c_{3})\cap [\mathsf{X}]\in \mathsf{H}_{\dim \mathsf{X}-6}(\mathsf{X},\mathbb{Z}),\end{eqnarray}$$

where $\mathsf{Y}_{i}\subset \mathsf{X}$ is the locus where $\mathsf{D}$ has singularities of type $\widetilde{E}_{i}$ .

5.1 Numerical constraints

Consider a given integral cohomology class $H\in \mathsf{H}^{2}(\mathsf{X},\mathbb{Z})$ . We define integers

$$\begin{eqnarray}a_{i}=\deg _{H}\mathsf{Y}_{i}=\int _{\mathsf{Y}_{i}}H\end{eqnarray}$$

for $i=6,7,8$ . Thus each $a_{i}$ is the total degree of a collection of elliptic curves in $\mathsf{X}$ , and from Proposition 4.10 we obtain the equation

(6) $$\begin{eqnarray}\displaystyle 8a_{6}+9a_{7}+10a_{8}=H(c_{1}c_{2}-c_{3})\cap [\mathsf{X}]\in \mathbb{Z}. & & \displaystyle\end{eqnarray}$$

If $H$ is a nef class, then each of the integers $a_{6},a_{7}$ and $a_{8}$ is nonnegative. We now specialize to the manifolds in Theorem 1.3.

Hypersurface case

Let $\mathsf{X}\subset \mathbb{P}^{5}$ be a smooth hypersurface of degree $d\leqslant 3$ , and denote by $H\in \mathsf{H}^{2}(\mathsf{X},\mathbb{Z})$ the hyperplane class. Using the exact sequence for the normal bundle, we have the total Chern class $c(\mathsf{X})=(1+H)^{6}(1+dH)^{-1}$ , from which one readily computes that

$$\begin{eqnarray}c_{1}c_{2}-c_{3}=(6d^{2}-36d+70)H^{3}.\end{eqnarray}$$

Meanwhile $H^{4}\cap [\mathsf{X}]=d$ , and hence (6) reads

(7) $$\begin{eqnarray}\displaystyle 8a_{6}+9a_{7}+10a_{8}=6d^{3}-36d^{2}+70d. & & \displaystyle\end{eqnarray}$$

The nonnegative integer solutions of this equation are displayed in Table 2. Since an elliptic curve in projective space has degree at least three, the solutions for $(a_{6},a_{7},a_{8})$ with $d=2$ cannot be degrees of collections of elliptic curves. Therefore a smooth quadric fourfold admits no purely elliptic structures.

Table 2. Nonnegative integer solutions of (7) with $d\leqslant 3$ .

Product case

Now suppose that $\mathsf{X}=\mathbb{P}^{1}\times \mathsf{W}$ , where $\mathsf{W}\subset \mathbb{P}^{4}$ is a smooth hypersurface of degree $d<5$ , i.e. a hypersurface that is Fano. Let $A,B\in \mathsf{H}^{2}(\mathsf{X},\mathbb{Z})$ be the pullbacks of the hyperplane classes on $\mathbb{P}^{1}$ and $\mathbb{P}^{4}$ , respectively. Thus $A^{2}=0$ , $B^{4}=0$ and $AB^{3}\cap [\mathsf{X}]=d$ . The total Chern class of $\mathsf{X}$ is given by the formula $c(\mathsf{X})=(1+A)^{2}(1+B)^{5}(1+dB)^{-1}$ . We conclude that

$$\begin{eqnarray}c_{1}c_{2}-c_{3}=2(d-5)^{2}AB^{2}+(5d^{2}-25d+40)B^{3}.\end{eqnarray}$$

Considering (6) with the nef classes $A$ and $B$ , we obtain the equations

(8) $$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}8a_{6}+9a_{7}+10a_{8}=2d(d-5)^{2},\\ 8b_{6}+9b_{7}+10b_{8}=d(5d^{2}-25d+40)\end{array} & & \displaystyle\end{eqnarray}$$

for the bidegrees $(a_{i},b_{i})$ of the hypothetical collections $\mathsf{Y}_{6},\mathsf{Y}_{7}$ and $\mathsf{Y}_{8}$ of elliptic curves. The solutions of these equations for which $a_{i}$ and $b_{i}$ are nonnegative integers are shown in Table 3.

Table 3. Nonnegative integer solutions of (8).

We claim that none of these solutions can represent the bidegrees of collections of elliptic curves, and hence there can be no purely elliptic log symplectic structures on $\mathsf{X}$ .

Indeed, if $d=1,2$ or $3$ , then at least one of the loci $\mathsf{Y}_{i}$ has degree zero with respect to $B$ but positive degree with respect to $A$ . Such a curve is necessarily a (nonempty) union of fibres of the projection $\mathsf{X}\rightarrow \mathsf{W}$ , which are copies of $\mathbb{P}^{1}$ . Hence they are not elliptic curves.

Likewise, if $d=4$ , then $\mathsf{Y}_{6}$ has degree one with respect to $A$ . Since $A$ is nonnegative on every connected component of $\mathsf{Y}_{6}$ , there must be a connected component $\mathsf{C}\subset \mathsf{Y}_{6}$ that has degree one with respect to $A$ . But then $\mathsf{C}$ would map isomorphically onto $\mathbb{P}^{1}$ under the projection $\mathsf{X}\rightarrow \mathbb{P}^{1}$ , and hence it is not elliptic.

5.2 Projective space

We now consider purely elliptic log symplectic structures on $\mathbb{P}^{4}$ . Let $\mathsf{D}$ be the degeneracy hypersurface and $\mathsf{Y}$ the locus where the Poisson structure vanishes. Consulting Table 2, we see that $\mathsf{Y}$ has degree at most five. Since each connected component is an elliptic curve, it must have degree at least three. Hence $\mathsf{Y}$ must be connected, and there are two possibilities: either $\mathsf{Y}$ has degree five, in which case $\mathsf{D}$ has $\widetilde{E}_{6}$ singularities, or $\mathsf{Y}$ has degree four, in which case $\mathsf{D}$ has $\widetilde{E}_{8}$ singularities. We now prove that the case of a curve of degree four is impossible, while in the degree-five case the only possibilities are the Feigin–Odesskii Poisson structures of type  $q_{5,1}$ .

5.2.2 Curve of degree four

The possibility of any other purely elliptic log symplectic structure on $\mathbb{P}^{4}$ is ruled out by the following result.

Theorem 5.4. There does not exist a purely elliptic log symplectic structure on $\mathbb{P}^{4}$ whose singular locus $\mathsf{Y}$ is a curve of degree four.

Proof. Suppose to the contrary that such a structure exists, and let $\mathsf{D}$ be its degeneracy divisor. Since $\mathsf{D}$ is normal, it must be irreducible.

Notice that the elliptic curve $\mathsf{Y}$ is necessarily the complete intersection of a hyperplane $\mathsf{H}\subset \mathbb{P}^{4}$ and two quadrics $\mathsf{Q}_{1},\mathsf{Q}_{2}\subset \mathbb{P}^{4}$ . Let $h$ be a defining equation for the hyperplane, and extend $h$ to a system $(h,x_{0},x_{1},x_{2},x_{3})$ of homogeneous coordinates for $\mathbb{P}^{4}$ . Without loss of generality, we may assume that the quadratic forms $q_{1}$ and $q_{2}$ defining $\mathsf{Q}_{1}$ and $\mathsf{Q}_{2}$ depend only on the coordinates $x_{0},\ldots ,x_{3}$ . Then, as is well known, every quadric in the pencil spanned by $q_{1}$ and $q_{2}$ has rank equal to three or four.

A theorem of Bondal [Reference BondalBon93] asserts that the Poisson bracket on $\mathbb{P}^{4}$ has a canonical lift to a Poisson bracket on the homogeneous coordinate ring $\mathbb{C}[h,x_{0},\ldots ,x_{3}]$ , such that the elementary brackets $\{h,x_{i}\}$ and $\{x_{i},x_{j}\}$ are given by homogeneous quadratic forms. To derive a contradiction, we will show by direct calculation that all Poisson brackets of the form $\{h,x_{i}\}$ must be multiples of $h$ , i.e. that the ideal $(h)$ is a Poisson ideal. This implies that the hyperplane $\mathsf{H}\subset \mathbb{P}^{4}$ is a Poisson subspace. Since $\dim \mathsf{H}=3$ , the Poisson tensor must have rank at most three on $\mathsf{H}$ , so that $\mathsf{H}\subset \mathsf{D}$ , which contradicts the irreducibility of $\mathsf{D}$ .

In our calculation, we will require the following two basic properties of the bracket.

1. (i) The ideal ${\mathcal{I}}=(h,q_{1},q_{2})$ cutting out $\mathsf{Y}$ is a Poisson ideal, i.e. $\{{\mathcal{I}},{\mathcal{O}}_{\mathbb{C}^{5}}\}\subset {\mathcal{I}}$ .

2. (ii) The homogeneous quintic polynomial $f$ defining $\mathsf{D}$ is a Casimir function, i.e.  $\{f,{\mathcal{O}}_{\mathbb{C}^{5}}\}=0$ .

These properties can easily be established using the definition of the quadratic Poisson bracket via the canonical Poisson module structure on ${\mathcal{K}}_{\mathbb{P}^{4}}^{\vee }\cong {\mathcal{O}}_{\mathbb{P}^{4}}(5)$ in [Reference PolishchukPol97, § 12]; the first property follows from the fact that $\mathsf{Y}\subset \mathbb{P}^{4}$ is a strong Poisson subspace in the sense of [Reference Gualtieri and PymGP13], and the second follows from the fact that the anticanonical section cutting out $\mathsf{D}\subset \mathbb{P}^{4}$ is a Poisson flat section [Reference PolishchukPol97, Proposition 7.4].

Since ${\mathcal{I}}$ is a Poisson ideal, the elementary brackets must have the form

(9) $$\begin{eqnarray}\{h,x_{i}\}=s_{i}q_{1}+t_{i}q_{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h\end{eqnarray}$$

for constants $s_{i},t_{i}\in \mathbb{C}$ . We therefore need to show that $s_{i}=t_{i}=0$ for all $i$ . It will be useful to define linear forms $A,B,C$ and $D$ by the formulae

(10) $$\begin{eqnarray}\begin{array}{@{}c@{}}\{h,q_{1}\}=Aq_{1}+Bq_{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h,\\ \{h,q_{2}\}=Cq_{1}+Dq_{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h.\end{array}\end{eqnarray}$$

Now we observe that, since $\mathsf{D}$ is singular along the complete intersection $\mathsf{Y}$ , the defining equation for $\mathsf{D}$ may be written as

$$\begin{eqnarray}f={\textstyle \frac{1}{2}}a_{11}q_{1}^{2}+a_{12}q_{1}q_{2}+{\textstyle \frac{1}{2}}a_{22}q_{2}^{2}+(b_{1}q_{1}+b_{2}q_{2})h\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h^{2}\end{eqnarray}$$

for quadratic forms $b_{1},b_{2}$ and linear forms $a_{11},a_{12},a_{22}$ in the variables $x_{0},\ldots ,x_{3}$ . Since $\mathsf{D}$ has $\widetilde{E}_{8}$ singularities along $\mathsf{Y}$ , the Hessian of $f$ must have rank one on the cone over $\mathsf{Y}$ in $\mathbb{C}^{5}$ . Hence the two-by-two minors of the Hessian must lie in ${\mathcal{I}}$ . In particular, the expressions

$$\begin{eqnarray}M=a_{11}a_{22}-a_{12}^{2},\quad N=b_{1}a_{12}-b_{2}a_{11}\end{eqnarray}$$

must lie in ${\mathcal{I}}$ . Therefore $M$ is a quadratic form in the pencil spanned by $q_{1}$ and $q_{2}$ . Evidently, $M$ has rank at most three, so there are exactly two possibilities: either $M$ is identically zero, or it has rank equal to three. We now consider these two cases separately.

Case 1: $M=0$ . In this case, the linear forms $a_{11}$ , $a_{12}$ and $a_{22}$ must all be constant multiples of a single form $a$ . Using the equation $M=0$ , we may factor the first three terms of $f$ and write

$$\begin{eqnarray}f={\textstyle \frac{1}{2}}aq_{1}^{2}+(b_{1}q_{1}+b_{2}q_{2})h\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h^{2}\end{eqnarray}$$

without loss of generality. Since $\mathsf{D}$ is irreducible, $f$ is not divisible by $h$ , and hence $a\neq 0$ .

The minor $N\in {\mathcal{I}}$ defined above is now given by $N=ab_{2}$ . Since $a$ is not a zero divisor modulo ${\mathcal{I}}$ , this forces $b_{2}$ to be a linear combination of $q_{1}$ and $q_{2}$ . If $b_{2}$ were a multiple of $q_{1}$ , we would have $f\in (h,q_{1})^{2}$ , which would imply that $\mathsf{D}$ is singular along the quadric surface defined by $h$ and $q_{1}$ , contradicting the normality of $\mathsf{D}$ . Therefore $b_{2}$ is linearly independent of $q_{1}$ , and so we may assume without loss of generality that

(11) $$\begin{eqnarray}f={\textstyle \frac{1}{2}}aq_{1}^{2}+bq_{1}h+{\textstyle \frac{1}{2}}q_{2}^{2}h\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h^{2}\end{eqnarray}$$

for some quadratic form $b$ .

We now compute the Poisson bracket $\{h,f\}=0$ modulo $h$ , giving the equation

$$\begin{eqnarray}{\textstyle \frac{1}{2}}\{h,a\}q_{1}^{2}=-a(Aq_{1}+Bq_{2})q_{1}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h.\end{eqnarray}$$

Since ${\mathcal{I}}$ is a Poisson ideal, the left-hand side of this equation lies in ${\mathcal{I}}^{3}$ . Since ${\mathcal{I}}$ is a complete intersection and $a$ is not a zero divisor modulo ${\mathcal{I}}$ , this forces $A=B=0$ . Therefore the brackets $\{h,q_{1}\}$ and $\{h,a\}$ are multiples of $h$ . In particular, we may write

$$\begin{eqnarray}\{h,q_{1}\}=hQ\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h^{2}\end{eqnarray}$$

for some $Q$ depending only on $x_{0},\ldots ,x_{3}$ .

If the quadratic form $q_{1}$ has rank four, we may choose the coordinates in which it takes the form $q_{1}={\textstyle \frac{1}{2}}(x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2})$ . Using (9), we compute the bracket

$$\begin{eqnarray}\{h,q_{1}\}=\biggl(\,\mathop{\sum }_{i=0}^{3}s_{i}x_{i}\biggr)q_{1}+\biggl(\,\mathop{\sum }_{i=0}^{3}t_{i}x_{i}\biggr)q_{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h.\end{eqnarray}$$

Since ${\mathcal{I}}$ is a complete intersection and $\{h,q_{1}\}$ is divisible by $h$ , we must have $s_{i}=t_{i}=0$ for all $i$ , and hence the ideal $(h)$ is Poisson.

If, on the other hand, the form $q_{1}$ has rank three, we may choose the coordinates in such a way that $q_{1}={\textstyle \frac{1}{2}}(x_{0}^{2}+x_{1}^{2}+x_{2}^{2})$ and $q_{2}={\textstyle \frac{1}{2}}x_{2}^{2}\hspace{0.2em}{\rm mod}\hspace{0.2em}(x_{0},x_{1},x_{2})^{2}$ . Calculating as above, we find that $s_{i}=t_{i}=0$ for $0\leqslant i\leqslant 2$ . Hence the Hamiltonian vector field of $h$ is given by

$$\begin{eqnarray}\{h,\cdot \}=(sq_{1}+tq_{2})\unicode[STIX]{x2202}_{x_{3}}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h\cdot \mathscr{X}_{\mathbb{C}^{5}}^{1}\end{eqnarray}$$

where $s=s_{3}$ and $t=t_{3}$ . It remains to show that $s$ and $t$ are equal to zero. Using (11), we compute

$$\begin{eqnarray}0=h^{-1}\{h,f\}=aQq_{1}+(t\unicode[STIX]{x2202}_{x_{3}}b+sx_{3})q_{1}q_{2}+(tx_{3})q_{2}^{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~(h,q_{1}^{2}).\end{eqnarray}$$

Since $x_{3}$ is not a zero divisor modulo $(h,q_{1})$ , we must have $t=0$ . Moreover, since all terms but the first lie in $(q_{1},q_{2})^{2}$ , we must have that $Q$ is a linear combination of $q_{1}$ and $q_{2}$ , say $Q=\unicode[STIX]{x1D706}_{1}q_{1}+\unicode[STIX]{x1D706}_{2}q_{2}$ . Therefore the equation above reduces to

$$\begin{eqnarray}(sx_{3}+\unicode[STIX]{x1D706}_{2}a)q_{1}q_{2}=0\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~(h,q_{1}^{2}),\end{eqnarray}$$

which implies that $sx_{3}=-\unicode[STIX]{x1D706}_{2}a$ . If $s$ were nonzero, we would have

$$\begin{eqnarray}\{h,a\}=\{h,-s\unicode[STIX]{x1D706}_{2}^{-1}x_{3}\}=-s^{2}\unicode[STIX]{x1D706}_{2}^{-1}q_{1}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h,\end{eqnarray}$$

which contradicts the fact that $\{h,a\}=0\hspace{0.2em}{\rm mod}\hspace{0.2em}h$ . Hence $s$ must be zero, completing the proof that $(h)$ is a Poisson ideal when $M=0$ .

Case 2: $M$ has rank three. In this case, the linear forms $x=a_{11}$ , $y=a_{22}$ and $z=a_{12}$ must be linearly independent, and hence we may use them as the homogeneous coordinates $x_{0}$ , $x_{1}$ and  $x_{2}$ .

The quadric $M=xy-z^{2}$ lies in ${\mathcal{I}}$ and hence is a linear combination of $q_{1}$ and $q_{2}$ . After a change of basis in the pencil spanned by $q_{1}$ and $q_{2}$ , we may assume that $M=q_{1}$ . We then choose our final homogeneous coordinate $w=x_{3}$ so that

$$\begin{eqnarray}q_{2}=w^{2}+Q(x,y,z)\end{eqnarray}$$

for a quadratic form $Q$ in three variables.

Computing the bracket $\{h,f\}=0$ , we obtain the equation

(12) $$\begin{eqnarray}\displaystyle {\textstyle \frac{1}{2}}\{h,x\}q_{1}^{2}+\{h,z\}q_{1}q_{2}+{\textstyle \frac{1}{2}}\{h,y\}q_{2}^{2} & = & \displaystyle -(xB+z(A+D)+yC)q_{1}q_{2}\nonumber\\ \displaystyle & & \displaystyle -\,(Ax+zC)q_{1}^{2}\nonumber\\ \displaystyle & & \displaystyle -\,(yD+zB)q_{2}^{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h.\end{eqnarray}$$

Since ${\mathcal{I}}$ is a Poisson ideal, the left-hand side of this equation lies in ${\mathcal{I}}^{3}$ . Thus the coefficients of $q_{1}^{2}$ , $q_{1}q_{2}$ and $q_{2}^{2}$ on the right-hand side must be linear combinations of $q_{1}$ and $q_{2}$ modulo $h$ . But these coefficients are quadratic forms that lie in the ideal $(x,y,z)$ defining the critical locus of $q_{1}=xy-z^{2}$ . Since the critical locus is not in the base locus of the pencil, these forms must all be multiples of $q_{1}$ . This puts strong constraints on $A,B,C$ and $D$ ; one can easily compute that they must be written as

$$\begin{eqnarray}A=Fy+Gz,\quad B=-Uy-Vz,\quad C=-Gx-Fz,\quad D=Vx+Uz\end{eqnarray}$$

for some constants $F,G,U,V\in \mathbb{C}$ . Then, using the form (9) of the elementary brackets, the definition (10) of $A$ and $B$ , and the formula $q_{1}=xy-z^{2}$ , one finds that

(13) $$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\{h,x\}=Fq_{1}-Uq_{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h,\\ \{h,y\}=0\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h,\\ \{h,z\}=-{\textstyle \frac{1}{2}}Gq_{1}+{\textstyle \frac{1}{2}}Vq_{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h.\end{array} & & \displaystyle\end{eqnarray}$$

Equation (12) now gives

$$\begin{eqnarray}Fq_{1}^{3}+(U+G)q_{1}^{2}q_{2}+Vq_{1}q_{2}^{2}=0\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h,\end{eqnarray}$$

which implies that $F=V=0$ and $U=-G$ . Hence $C=-Gx$ and $D=-Gz$ . Using the definition (10) of $C$ and $D$ together with the brackets (13), we find that

$$\begin{eqnarray}Gq_{2}\unicode[STIX]{x2202}_{x}Q-{\textstyle \frac{1}{2}}Gq_{1}\unicode[STIX]{x2202}_{z}Q+2\{h,w\}w=-Gxq_{1}-Gzq_{2}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~h,\end{eqnarray}$$

which is easily seen to imply that $G=0$ and that $\{h,w\}=0$ modulo $h$ . It follows that $h$ generates a Poisson ideal, as required.◻

5.3 Cubic fourfolds

In order to complete the proof of Theorem 1.3, it remains to show that a smooth cubic fourfold $\mathsf{X}$ does not admit any purely elliptic log symplectic forms. From Table 2, we know that the degeneracy divisor $\mathsf{D}\subset \mathsf{X}$ of such a structure would have $\widetilde{E}_{6}$ singularities along a locus $\mathsf{Y}\subset \mathsf{X}$ that is a union of elliptic curves of total degree six. Hence $\mathsf{Y}$ must be either a single curve of degree six or a pair of plane cubics. In Propositions 5.6 and 5.7 below, we will show that such a hypersurface cannot exist. We begin by reducing the problem to the study of singular cubic fourfolds.

Proof. Let ${\mathcal{J}}\subset {\mathcal{O}}_{\mathsf{X}}$ and ${\mathcal{I}}\subset {\mathcal{O}}_{\mathbb{P}^{5}}$ be the ideals defining $\mathsf{C}$ as a subspace of $\mathsf{X}$ and $\mathbb{P}^{5}$ , respectively, and let ${\mathcal{O}}_{\mathbb{P}^{5}}(-\mathsf{X})\subset {\mathcal{O}}_{\mathbb{P}^{5}}$ be the ideal defining $\mathsf{X}$ . By the adjunction formula, we have ${\mathcal{K}}_{\mathsf{X}}^{\vee }\cong {\mathcal{O}}_{\mathsf{X}}(3)$ , and so the anticanonical divisors with multiplicity three along $\mathsf{C}$ are cut out by sections of ${\mathcal{J}}^{3}(3)$ . To prove the existence and uniqueness of $\widetilde{\mathsf{D}}$ , we must show that the map

induced by the restriction ${\mathcal{O}}_{\mathbb{P}^{5}}\rightarrow {\mathcal{O}}_{\mathsf{X}}$ is an isomorphism.

Since $\mathsf{X}$ and $\mathsf{C}$ are smooth, the restriction gives a surjection ${\mathcal{I}}^{3}\rightarrow {\mathcal{J}}^{3}$ with kernel ${\mathcal{I}}^{3}\cap {\mathcal{O}}_{\mathbb{P}^{5}}(-\mathsf{X})$ , and this kernel is the precisely the image of the multiplication map ${\mathcal{I}}^{2}\otimes {\mathcal{O}}_{\mathbb{P}^{5}}(-\mathsf{X})\rightarrow {\mathcal{I}}^{3}$ . Twisting by ${\mathcal{O}}_{\mathbb{P}^{5}}(3)$ , we obtain an exact sequence

and the relevant part of the long exact sequence reads

Now $h^{0}({\mathcal{I}}^{2})=0$ since $\mathbb{P}^{5}$ has no nonconstant global functions. Hence the lemma will follow if we can show that $h^{1}({\mathcal{I}}^{2})=0$ as well. For this, we consider the exact sequence

defining the conormal sheaf ${\mathcal{N}}^{\vee }$ of $\mathsf{C}$ in $\mathbb{P}^{5}$ . Since ${\mathcal{N}}^{\vee }\subset \unicode[STIX]{x1D6FA}_{\mathbb{P}^{5}}^{1}|_{\mathsf{C}}\subset {\mathcal{O}}_{\mathsf{C}}(-1)^{\oplus 6}$ , we have $h^{0}({\mathcal{N}}^{\vee })=0$ . Hence the vanishing of $h^{1}({\mathcal{I}}^{2})$ follows from the vanishing of $h^{1}({\mathcal{I}})$ , which in turn follows from the exact sequence

and the vanishing of $h^{1}({\mathcal{O}}_{\mathbb{P}^{5}})$ . This completes the proof of the existence and uniqueness of $\widetilde{\mathsf{D}}$ .

To see that $\mathsf{Sec}_{2}(\mathsf{C})\subset \widetilde{\mathsf{D}}$ , let $\mathsf{W}\subset \mathbb{P}^{5}$ be a plane that hits $\mathsf{C}$ at three points $p,q,r\in \mathsf{C}$ that are not collinear. Then either $\mathsf{W}\subset \mathsf{Y}$ or the intersection $\mathsf{W}\cap \mathsf{Y}$ is a cubic curve with multiplicity three at $p,q$ and $r$ . But the latter is impossible because the only cubic curve with three non-collinear singular points is a triangle, for which the singularities are nodes.◻

Proof. Let $\mathsf{Y}\subset \mathsf{X}\subset \mathbb{P}^{5}$ be a degree-six elliptic curve. In light of Lemma 5.5, it is enough to show that a cubic fourfold $\widetilde{\mathsf{D}}$ with multiplicity three along $\mathsf{Y}$ cannot be normal.

To this end, let $\mathsf{W}\subset \mathbb{P}^{5}$ be the smallest linear subspace containing $\mathsf{Y}$ . We have $h^{0}({\mathcal{O}}_{\mathsf{Y}}(1))=6$ by Riemann–Roch, and hence $\mathsf{Y}$ is the linear projection of an elliptic normal sextic $\mathsf{Y}^{\prime }\subset \mathbb{P}^{5}$ to $\mathsf{W}$ . Since the secant planes of an elliptic normal sextic fill out all of $\mathbb{P}^{5}$ , we must have that $\mathsf{W}=\mathsf{Sec}_{2}(\mathsf{Y})$ . Hence $\mathsf{W}\subset \widetilde{\mathsf{D}}$ by Lemma 5.5. It follows that the dimension of $\mathsf{W}$ is at most four. On the other hand, the dimension of $\mathsf{W}$ is at least three because $\mathbb{P}^{2}$ does not contain an elliptic sextic.

If $\mathsf{W}$ has dimension four, then $\mathsf{W}$ is an irreducible component of $\widetilde{\mathsf{D}}$ . Hence $\widetilde{\mathsf{D}}$ is not normal.

Therefore, suppose that $\mathsf{W}$ has dimension three. Choose homogeneous coordinates $x_{0},x_{1},y_{0},\ldots ,y_{3}$ for $\mathbb{P}^{5}$ such that $\mathsf{W}$ is cut out by the equations $x_{0}=x_{1}=0$ . Since $\widetilde{\mathsf{D}}$ contains $\mathsf{W}$ , it is cut out by a cubic polynomial of the form

$$\begin{eqnarray}f=x_{0}q_{0}+x_{1}q_{1}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~(x_{0},x_{1})^{2}\end{eqnarray}$$

where $q_{0}$ and $q_{1}$ are quadratic forms in the variables $y_{0},\ldots ,y_{3}$ . Now $f$ vanishes to order three on $\mathsf{Y}$ . Hence the derivative

$$\begin{eqnarray}df=q_{0}\,dx_{0}+q_{1}\,dx_{1}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~(x_{0},x_{1})\end{eqnarray}$$

and the Hessian

$$\begin{eqnarray}\text{Hess}(f)=dq_{0}\cdot dx_{0}+dq_{1}\cdot dx_{1}\quad \hspace{0.2em}{\rm mod}\hspace{0.2em}~(x_{0},x_{1},(dx_{0})^{2},dx_{0}\cdot dx_{1},(dx_{1})^{2})\end{eqnarray}$$

must vanish on $\mathsf{Y}$ . This implies that the quadrics defined by $q_{0}$ and $q_{1}$ are singular along $\mathsf{Y}$ . But the singular locus of a quadric is a linear subspace, and $\mathsf{W}$ is the smallest linear subspace containing $\mathsf{Y}$ . Hence $q_{0}=q_{1}=0$ identically, which implies that $f\in (x_{0},x_{1})^{2}$ . Hence $\widetilde{\mathsf{D}}$ is singular along $\mathsf{W}$ , which implies that $\widetilde{\mathsf{D}}$ is not normal.◻

Proof. Because of the degrees, the curves $\mathsf{Y}_{1}$ and $\mathsf{Y}_{2}$ must be contained in planes $\mathsf{W}_{1}$ and $\mathsf{W}_{2}$ . Let us denote by $\mathsf{W}_{i}^{\bot }\subset \mathsf{H}^{0}({\mathcal{O}}_{\mathbb{P}^{5}}(1))$ the space of linear forms cutting out $\mathsf{W}_{i}$ . Because $\mathsf{Y}_{i}=\mathsf{X}\cap \mathsf{W}_{i}$ is a complete intersection, the cubic fourfolds with multiplicity three along $\mathsf{Y}_{i}$ are given by the subspace

$$\begin{eqnarray}\mathsf{Sym}^{3}\mathsf{W}_{i}^{\bot }\subset \mathsf{H}^{0}({\mathcal{O}}_{\mathbb{ P}^{5}}(3)).\end{eqnarray}$$

Hence, if $\widetilde{\mathsf{D}}\subset \mathbb{P}^{5}$ is the singular cubic fourfold provided by Lemma 5.5, its defining equation lies in the intersection

$$\begin{eqnarray}\mathsf{Sym}^{3}\mathsf{W}_{1}^{\bot }\cap \mathsf{Sym}^{3}\mathsf{W}_{2}^{\bot }=\mathsf{Sym}^{3}(\mathsf{W}_{1}^{\bot }\cap \mathsf{W}_{2}^{\bot }).\end{eqnarray}$$

Therefore $\widetilde{\mathsf{D}}$ has multiplicity three along the linear span of $\mathsf{W}_{1}$ and $\mathsf{W}_{2}$ in $\mathbb{P}^{5}$ , which implies that the singular locus of $\widetilde{\mathsf{D}}$ has codimension at most $\dim (\mathsf{W}_{1}\cap \mathsf{W}_{2})$ in $\widetilde{\mathsf{D}}$ . Thus $\widetilde{\mathsf{D}}$ can only be normal if $\mathsf{W}_{1}=\mathsf{W}_{2}$ , but then the cubic curves $\mathsf{Y}_{1}$ and $\mathsf{Y}_{2}$ lie in the same plane, and hence they cannot be disjoint.◻