The Looijenga–Lunts–Verbitsky Algebra for Primitive Symplectic Varieties with Isolated Singularities

Constructing the LLV algebra for the ordinary cohomology of a primitive symplectic variety is difficult since $H^{*}(X,\mathbb Q)$, a priori, neither carries a pure Hodge structure nor satisfies the hard Lefschetz theorem. Instead, we work with the intersection cohomology groups.

Intersection cohomology was invented by Goresky–MacPherson [GM80] as a way of generalizing Poincaré duality to singular topological spaces. Beilinson–Bernstein–Deligne [BBD82] observed, using characteristic $p$ methods, that the intersection cohomology groups of a projective variety admit a decomposition theorem with respect to projective morphisms. As a consequence, the intersection cohomology groups carry pure Hodge structures and satisfy the hard Lefschetz theorem. This was also observed by Saito [Sai88] in greater generality using the theory of mixed Hodge modules, as well as work of de Cataldo–Migliorini [dCM05] using purely Hodge theoretic techniques. The goal of this paper is to understand the total Lie algebra with respect to intersection cohomology, which we define analogously as the Lie algebra generated by the $\mathfrak{sl}_{2}$-operators corresponding to any HL class, i.e., those classes $\omega\in\operatorname{IH}^{2}(X,\mathbb Q)$ such that

$$L_{\omega}^{k}\colon\operatorname{IH}^{\dim X-k}(X,\mathbb Q)\overset{\vbox to0.0pt{\vss\hbox{$\sim$}\vskip-3.0pt}}{\longrightarrow}\operatorname{IH}^{\dim X+k}(X,\mathbb Q).$$

To this end, we define a Beauville–Bogomolov–Fujiki (BBF) form $Q_{X}$ on the intersection cohomology $\operatorname{IH}^{2}(X,\mathbb Q)$ of a primitive symplectic variety $X$; see Section 5.2.1. It is compatible with the standard BBF form $q_{X}$ on $H^{2}(X,\mathbb Q)$ (see Definition 2.5) and satisfies

$$Q_{X}|_{H^{2}(X,\mathbb Q)}=q_{X}$$

corresponding to the natural inclusion $H^{2}(X,\mathbb Q)\subset\operatorname{IH}^{2}(X,\mathbb Q)$ (see Remark 2.9).

The assumption on $b_{2}$ is due to our use of the global moduli theory of Bakker–Lehn [BL22]. We note that the case $b_{2}\leq 4$ holds assuming the surjectivity of the period map and other special cases (see Section 5.4). We emphasize the fact that our proof of Theorem 1.1 gives an algebraic proof of (1.1). We expect our methods to generalize to any primitive symplectic variety.

One of the key features of the cohomology of a hyperkähler manifold $X$ is its structure as an irreducible holomorphic symplectic manifold. The holomorphic symplectic form $\sigma$ on $X$ induces isomorphisms ${}_{X}^{n-p}\xrightarrow{\vbox to0.0pt{\vss\hbox{$\sim$}\vskip-3.0pt}}{}_{X}^{n+p}$ by wedging, where $2n$ is the (complex) dimension of $X$. Passing to cohomology, we get the symplectic hard Lefschetz theorem

$$L_{\sigma}^{p}\colon H^{n-p,q}(X)\overset{\vbox to0.0pt{\vss\hbox{$\sim$}\vskip-3.0pt}}{\longrightarrow}H^{n+p,q}(X),$$

which induces the extra symmetry on the Hodge diamond of $X$. An interesting observation is that, by deforming a compact hyperkähler manifold $X$, we can see that the hard Lefschetz theory of $H^{*}(X,\mathbb Q)$ is related to its symplectic hard Lefschetz theory due to Verbitsky’s global Torelli theorem. With this in mind, we first show that the intersection cohomology groups of primitive symplectic varieties with isolated singularities also admit this symplectic symmetry.

To prove Theorem 1.2, we study the Hodge theory of the (compactly supported) cohomology of the regular locus $U:=X_{\mathrm{reg}}$. We prove the following useful theorem, giving an analog of [Ara90, Theorem 2]

The LLV structure theorem follows from the symplectic hard Lefschetz theory. Theorem 1.2 shows that there are operators $L_{\sigma},{}_{\sigma}$ which complete to an $\mathfrak{sl}_{2}$-triple

$$\mathfrak s_{\sigma}=\langle L_{\sigma},{}_{\sigma},H_{\sigma}\rangle,$$

where $H_{\sigma}$ acts as the holomorphic weight operator $H_{\sigma}(\alpha)=(p-n)\alpha$ for an intersection $(p,q)$-class. Similarly, by conjugation we get a second $\mathfrak{sl}_{2}$-triple

$$s_{\overline{\sigma}}=\langle L_{\overline{\sigma}},{}_{\overline{\sigma}},H_{\overline{\sigma}}\rangle$$

corresponding to the antiholomorphic symplectic form $\overline{\sigma}$, where $H_{\overline{\sigma}}(\alpha)=(q-n)\alpha$. This generates an $\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$-structure on the total intersection cohomology $\operatorname{IH}^{*}(X)$. The key observation is that the Lefschetz operators for $\sigma$ and $\overline{\sigma}$ commute:

$$[L_{\sigma},L_{\overline{\sigma}}]=[{}_{\sigma},{}_{\overline{\sigma}}]=0.$$

The representation theory of this $\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$-action, along with the monodromy representation of $H^{2}(X,\mathbb C)$, describes the LLV algebra $\mathfrak g$ of intersection cohomology completely. This will lead to the proof of Theorem 1.1.

Extending a result of Verbitsky [Ver96], we show that the module generated by $\operatorname{IH}^{2}(X,\mathbb C)$ in $\operatorname{IH}^{*}(X,\mathbb C)$ is a $\mathfrak g$-module, called the Verbitsky component $V_{(n)}$. In [GKLR22], the Verbitsky component and the LLV decomposition were studied for the known examples of compact hyperkähler manifolds. In general, it is expected that $V_{(n)}$ (along with the LLV decomposition) puts restrictive conditions on the cohomology of a compact hyperkähler manifold, and we expect the same to be true for primitive symplectic varieties.

Recall that the classical Kuga–Satake construction, see [KS67], associates to a K3 surface $S$ an abelian variety $A$ and an embedding $H^{2}(S)\hookrightarrow H^{1}(A)\otimes H^{1}(A)^{*}$ of polarized weight 2 Hodge structures (see for example [Huy16, Section 2.6]). In [KSV19], this construction was extended to compact hyperkähler manifolds and the LLV algebra: If $X$ is compact hyperkähler, there exist an abelian variety $A$ and an embedding $\mathfrak g_{X}\hookrightarrow\mathfrak g_{A}$, where we note that $\mathfrak g_{A}\cong\mathfrak{so}(H^{1}(A)\otimes H^{1}(A)^{*})$ by [LL97, Proposition (3.3)]. We outline how the same construction associates to a primitive symplectic variety with isolated singularities a complex torus and an embedding of the total Lie algebras.

The Mumford–Tate algebra of a pure Hodge structure $H$ is the smallest $\mathbb Q$-algebraic subalgebra $\mathfrak m$ of $\mathfrak{gl}(H)$ for which $\mathfrak m_{\mathbb R}$ contains the Weil operator. There is a relationship between the Mumford–Tate algebra and the LLV algebra, which was studied thoroughly in [GKLR22]. We make similar observations in the isolated singularities case and show that the Mumford–Tate algebra sits naturally inside the LLV algebra for intersection cohomology. Moreover, we show that the degree of transcendence of the total intersection cohomology $\operatorname{IH}^{*}(X,\mathbb C)$ over the special Mumford–Tate group is equal to that of $\operatorname{IH}^{2}(X,\mathbb C)$, which is the second statement of Theorem 1.1.

Let $X$ be a normal complex variety. A log-resolution of singularities is a projective birational morphism $\pi\colon\widetilde{X}\to X$ from a smooth complex variety $\widetilde{X}$ which is an isomorphism over the regular locus $U$ and for which $\pi^{-1}(\Sigma)=E=\sumop\displaylimits E_{i}$ is a simple normal crossing divisor, where is the singular locus of $X$ and the $E_{i}$ are the smooth components of $E$.

We say $X$ is $\mathbb Q$-Gorenstein if there is an integer $m$ such that $mK_{X}$ is Cartier, where $K_{X}$ is the canonical divisor; the smallest such integer is called the index of $X$. If $X$ has index 1, we say that $X$ is Gorenstein.

If $X$ is $\mathbb Q$-Gorenstein of index $m$ and $\pi\colon\widetilde{X}\to X$ a log-resolution of singularities, there are integers $a_{i}$ such that

$$mK_{\widetilde{X}}=\pi^{*}(mK_{X})+\sumop\displaylimits a_{i}E_{i}.$$

We say that $X$ has canonical (resp. terminal ) singularities if we have $a_{i}\geq 0$ (resp. $a_{i}>0$) for every $i$. We call the $a_{i}$ the discrepancies of the exceptional divisors $E_{i}$.

If $X$ is just a normal variety, we say that $X$ has rational singularities if for some resolution of singularities $\pi\colon\widetilde{X}\to X$, the higher direct image sheaves satisfy $R^{i}\pi_{*}\mathscr{O}_{\widetilde{X}}=0$ for every $i>0$.

Canonical singularities are rational; see [KM98, Theorem 5.22]. Conversely, a normal variety with Gorenstein rational singularities has at worst canonical singularities. One way to see this is to consider the holomorphic extension problem for differentials: For which $p$ is the inclusion

(2.1)

$$\pi_{*}{}_{\widetilde{X}}^{p}\lhook\joinrel\longrightarrow j_{*}{}_{U}^{p}$$

is an isomorphism, where $j\colon U\hookrightarrow X$ is the inclusion of the regular locus $U:=X_{\mathrm{reg}}$? Kebekus–Schnell showed if $X$ has rational singularities, then $\pi_{*}{}_{\widetilde{X}}^{p}\hookrightarrow j_{*}{}_{U}^{p}$ is an isomorphism for each $p$, see [KS21, Corollary 1.8], using the fact that the canonical sheaf $\omega_{X}$ is reflexive by Kempf’s criterion. In particular, the discrepancies $a_{i}$ are non-negative, as holomorphic $n$-forms on $U$ extend with at worst zeros.

Since all our varieties are assumed normal, the sheaf $j_{*}{}_{U}^{p}$ is reflexive and isomorphic to the sheaf of reflexive $p$-forms

$${}_{X}^{[p]}:=({}_{X}^{p})^{**}\cong j_{*}{}_{U}^{p}.$$

Here ${}_{X}^{1}$ is the sheaf of Kähler differentials and ${}_{X}^{p}$ is the $p^{\mathrm{th}}$ exterior power.

One application of the work of Kebekus–Schnell is the existence of a functorial pullback morphism for reflexive differentials; see [KS21, Theorem 1.11]. Given a morphism $f\colon Y\to X$ of reduced complex spaces with rational singularities, there is a pullback

(2.2)

$$df\colon f^{*}{}_{X}^{[p]}\longrightarrow{}_{Y}^{[p]}$$

which satisfies natural universal properties; see [KS21, Section 14]. Specifically, it agrees with the pullback of Kähler differentials on smooth varieties, whenever this makes sense.

Originally studied by Beauville [Bea00], symplectic varieties were defined as an attempt to extend results on $K$-trivial manifolds to the singular setting. Symplectic varieties have well-behaved singularities: They are rational Gorenstein, see [Bea00, Proposition 1.3], and therefore have at worst canonical singularities. In fact, the strictly canonical locus is contained entirely in codimension 2 by [Nam01c]. Said differently, we have the following.

One of the key features of symplectic varieties is that they are stratified by varieties which, up to normalization, are again symplectic varieties. We will use the following structure theorem throughout the paper.

In [Kal09], a symplectic variety $Y_{x}$ is a formal scheme rather than the completion of some symplectic variety, as its existence is predicted by solutions to differential equations derived from the Poisson structure (see Section 3 of op. cit.). By considering [KS24, Proposition 2.3, Appendix A], we can assume $Y_{x}$ is defined (and symplectic) in an analytic neighborhood of $x$.

It would be interesting to understand how the holomorphic symplectic geometry of the regular strata $X_{i}^{\circ}$ determines the geometry of a symplectic variety $X$. For instance, if $\sigma\in H^{0}(X,{}_{X}^{[2]})$ is the class of the symplectic form on the regular locus of $X$, there is class $j_{i}^{*}\sigma\in H^{0}(X_{i}^{\mathrm{nm}},{}_{X_{i}}^{[2]})$, where $j_{i}\colon X_{i}^{\mathrm{nm}}\to X$ is the natural map from the normalization of $X_{i}$, defined by reflexive pullback for rational singularities; see [KS21, Theorem 14.1]. We claim this class is non-zero whenever $\dim X_{i}>0$. Since the problem is local, we can consider the product decomposition $\widehat{X}_{x}\cong Y_{x}\times\widehat{X_{i}^{\circ}}_{x}$ by Proposition 2.3. If $j_{i}^{*}\sigma=0$, then $\sigma=p_{1}^{*}\sigma_{Y_{x}}$, where $p_{1}\colon X\to Y_{x}$ is the projection morphism and $\sigma_{Y_{x}}$ is the symplectic form on $Y_{x}$. If $\dim Y_{x}\neq\dim X$, then $\sigma^{\dim X}:=\wedge^{\dim X}\sigma_{Y_{x}}=0$. This is absurd if $X$ is a symplectic variety, and so we see that $j_{i}^{*}\sigma$ defines a non-zero global section of ${}_{X_{i}}^{[2]}$. Since $X_{i}^{\circ}$ is symplectic, $j_{i}^{*}\sigma$ is a symplectic form.

We now transition to global properties of symplectic varieties. The Hodge theory of singular symplectic varieties has been studied in [Nam01a, Nam06, Mat01, Mat15, Sch20, BL21, BL22] at varying levels of generality; primitive symplectic varieties, which were studied in [BL22], is the most general framework for studying the global properties of symplectic singularities.

The general framework of the global moduli theory of primitive symplectic varieties works in the category of complex Kähler varieties. A Kähler form on a reduced complex analytic space $X$ is given by an open covering $X=\bigcupop\displaylimits_{i}U_{i}$ and smooth strictly plurisubharmonic functions $f_{i}\colon U_{i}\to\mathbb R$ such that on each intersection $U_{ij}$, the function $f_{ij}=f_{i}|_{U_{ij}}-f_{j}|_{U_{ij}}$ is locally the real part of a harmonic function. If $X$ admits a Kähler form, we say that $X$ is a Kähler variety. The most important property for this paper is that if $X$ is a Kähler variety, then $X$ admits a resolution by a Kähler manifold. If $X$ is a compact Kähler variety with at worst rational singularities and $\pi\colon\widetilde{X}\to X$ is a resolution of singularities, then there is an injection $H^{k}(X,\mathbb Z)\hookrightarrow H^{k}(\widetilde{X},\mathbb Z)$ for $k\leq 2$, see [BL21, Lemma 2.1], and $H^{k}(X,\mathbb Z)$ inherits a pure Hodge structure from $H^{k}(\widetilde{X},\mathbb Z)$, which is described as follows. There is a spectral sequence

$$E_{1}^{p,q}:=H^{q}(X,j_{*}{}_{U}^{p})\Longrightarrow\mathbb H^{p+q}(X,j_{*}{}_{U}^{\bullet})$$

which degenerates at $E_{1}$ for $p+q\leq 2$; see [BL21, Lemma 2.2]. For $k\leq 2$, we have isomorphisms

(2.3)

$$H^{k}(X,\mathbb C)\overset{\vbox to0.0pt{\vss\hbox{$\sim$}\vskip-3.0pt}}{\longrightarrow}\mathbb H^{k}(X,\pi_{*}{}_{\widetilde{X}}^{\bullet})\cong\mathbb H^{k}(X,j_{*}{}_{U}^{\bullet}),$$

and the Hodge filtration is obtained by these isomorphisms and the degeneration of the reflexive Hodge-to-de Rham spectral sequence.

The second cohomology of a primitive symplectic variety is therefore a pure Hodge structure. As in the case of irreducible holomorphic symplectic manifolds, the geometry is controlled by the Hodge theory on $H^{2}$. To see this, we need the following quadratic form.

This form $q_{X,\sigma}$ defines a quadratic form over $H^{2}(X,\mathbb R)$ by definition. It is non-degenerate, and the signature of $q_{X,\sigma}$ is $(3,b_{2}-3)$, where $b_{2}=\dim H^{2}(X,\mathbb R)$ is the second Betti number; see [Sch20, Theorem 2]. If we rescale $\sigma$ so that $\intop\nolimits_{X}(\sigma\overline{\sigma})^{n}=1$, then $q_{X,\sigma}$ is independent of $\sigma$. In this case, we call $q_{X}:=q_{X,\sigma}$ the Beauville–Bogomolov–Fujiki form.

We define the period domain of $X$ to be

$$\Omega:=\left\{[\sigma]\in\mathbb P(H^{2}(X,\mathbb C))\mid q_{X}(\sigma)=0,\leavevmode\nobreak\ q_{X}(\sigma,\overline{\sigma})>0\right\}.$$

There is an associated period map

$$\rho\colon\operatorname{Def}^{\,\mathrm{lt}}(X)\longrightarrow\Omega$$

from the universal family of locally trivial deformations of $X$ which sends $t$ to $H^{2,0}(X_{t})$, where $X_{t}$ is a locally trivial deformation of $X$ corresponding to a point $t\in\operatorname{Def}^{\,\mathrm{lt}}(X)$. The period map $\rho$ is a local isomorphism by the local Torelli theorem, see [BL22, Proposition 5.5], which leads to two immediate consequences. First, $q_{X}$ satisfies the Fujiki relation: There is a positive constant $c$ such that

(2.4)

$$q_{X}(\alpha)^{n}=c\intop\nolimits_{X}\alpha^{2n};$$

see [BL22, Proposition 5.15]. Second, we may rescale the BBF form to get an integral quadratic form on $H^{2}(X,\mathbb Z)$; see [BL22, Lemma 5.7]. We denote the corresponding integral lattice by $\Gamma:=(H^{2}(X,\mathbb Z),q_{X})$.

The period map also satisfies many global properties. We can view the period domain as the moduli space of all weight 2 Hodge structures on which admit a quadratic form $q$ of signature $(3,b_{2}-3)$ which is positive-definite on the real space underlying $H^{2,0}\otimes H^{0,2}$. We then write for the period domain. If $X^{\prime}$ is a locally trivial deformation of $X$ with $(H^{2}(X^{\prime},\mathbb Z),q_{X^{\prime}})\cong\Gamma$, then we get a period map $p\colon\mathscr{M}^{\prime}\to$ from the moduli space of -marked locally trivial deformations of $X^{\prime}$. The orthogonal group $O(\Gamma)$ acts on $\mathscr{M}^{\prime}$ and by changing the marking. For any connected component $\mathscr{M}$ of $\mathscr{M}^{\prime}$, we denote by $\operatorname{Mon}(\mathscr{M})\subset O(\Gamma)$ the image of the monodromy representation on second cohomology.

The intersection cohomology is defined as the hypercohomology groups with respect to the intersection complex. The intersection cohomology complex is the perverse sheaf underlying the unique Hodge module determined by the constant variation of pure Hodge structures on $\mathbb Q_{X_{\mathrm{reg}}}[\dim X]$ over the regular locus of $X$; see [Sai88, Section 5.3]. Writing $\mathcal{IC}_{X}$ for this complex, we have

$$\operatorname{IH}^{k}(X,\mathbb Q):=\mathbb H^{k-\dim X}(X,\mathcal{IC}_{X}).$$

We will use the following description of intersection cohomology throughout the paper.

Since the analytic topology of a singularity of a complex variety is preserved under locally trivial deformations, Lemma 2.10 holds for any primitive symplectic variety, as a general locally trivial deformation is projective; see [BL22, Corollary 6.10].

Let $U$ be a smooth Kähler variety, and let $\widetilde{X}$ be a smooth compactification of $U$ such that the complement $\widetilde{X}\setminus U\cong E$ is a simple normal crossing (snc) divisor. A fundamental result of Deligne [Del71, Proposition 3.1.8] states that the cohomology of $U$ can be identified with the hypercohomology of the complex of logarithmic forms on the pair $(\widetilde{X},E)$. Specifically, let ${}_{\widetilde{X}}^{p}(\log E)$ be the sheaf of logarithmic $p$-forms, and let ${}_{\widetilde{X}}^{\bullet}(\log E)$ the complex of logarithmic forms. Then there is an isomorphism

$$H^{k}(U,\mathbb C)\cong\mathbb H^{k}\left(\widetilde{X},{}_{\widetilde{X}}^{\bullet}(\log E)\right),$$

which induces two filtrations on $H^{k}(U,\mathbb C)$. The first is the naive filtration associated to the complex ${}_{\widetilde{X}}^{\bullet}(\log E)$:

$$F^{p}H^{k}(U,\mathbb C)=\operatorname{im}\left(\mathbb H^{k}\left(\widetilde{X},\tau_{\geq p}{}_{\widetilde{X}}^{\bullet}(\log E)\right)\longrightarrow\mathbb H^{k}\left(\widetilde{X},{}_{\widetilde{X}}^{\bullet}(\log E)\right)\right),$$

where $\tau_{\geq p}{}_{\widetilde{X}}^{\bullet}(\log E)$ is the complex ${}_{\widetilde{X}}^{\bullet}(\log E)$ truncated in degree greater than or equal to $p$. The second filtration is induced at the level of sheaves: There is an increasing filtration on logarithmic $p$-forms given by

$$W_{l}{}_{\widetilde{X}}^{p}(\log E)={}_{\widetilde{X}}^{l}(\log E)\wedge{}_{\widetilde{X}}^{p-l}$$

descending to $\mathbb H^{k}(\widetilde{X},{}_{\widetilde{X}}^{\bullet}(\log E)$. These filtrations correspond to the canonical mixed Hodge structure on $H^{k}(U,\mathbb Q)$—$W_{\bullet}$ is the complexification of the rational weight filtration, and $F^{\bullet}$ is the Hodge filtration.

In particular, there is a non-canonical decomposition

(2.5)

$$H^{k}(U,\mathbb C)\cong\bigoplusop\displaylimits_{p+q=k}H^{q}\left(\widetilde{X},{}_{\widetilde{X}}^{p}(\log E)\right).$$

We note that the above construction applies to the regular locus $U$ of any proper Kähler variety $X$, where we may choose the smooth compactification to be a log-resolution $\widetilde{X}$ of $X$.

Deligne [Del71, Del74] also shows that the compactly supported cohomology of an algebraic variety carries a pure Hodge structure.

The compactly supported cohomology also carries a mixed Hodge structure. The easiest way to see this is to use Poincaré duality. Indeed, the Poincaré isomorphisms

$$H^{k}_{c}(U,\mathbb C)\cong\left(H^{2\dim U-k}(U,\mathbb C)\right)^{*}\otimes\mathbb C(-\dim U)$$

are isomorphisms of mixed Hodge structures. Noting that

$$H^{q}\left({}_{\widetilde{X}}^{p}(\log E)\right)\cong H^{\dim\widetilde{X}-q}\left(\widetilde{X},{}_{\widetilde{X}}^{\dim\widetilde{X}-p}(\log E)(-E)\right)^{*}$$

by Serre duality, this completely describes the mixed Hodge structure on $H^{k}_{c}(U,\mathbb Q)$. In particular, we have another non-canonical splitting

(2.6)

$$H_{c}^{k}(U,\mathbb C)\cong\bigoplusop\displaylimits_{p+q=k}H^{q}\left(\widetilde{X},{}_{\widetilde{X}}^{p}(\log E)(-E)\right).$$

Finally, let us consider an snc divisor $E=\sumop\displaylimits_{i=1}^{r}E_{i}$ with irreducible components $E_{i}$. For any subset $J\subset\left\{1,\ldots,r\right\}$, write $E_{J}=\bigcapop\displaylimits_{j\in J}E_{j}$ and let

(2.7)

$$E_{(p)}:=\coprodop\displaylimits_{|J|=p}E_{J}$$

be the $p$-fold intersections of the components. For each $p$, the various inclusion maps $E_{(p)}\hookrightarrow E_{(p-1)}$ induce a simplicial map $H^{k}(E_{(p-1)},\mathbb Q)\to H^{k}(E_{(p)},\mathbb Q)$. From this we obtain a complex

$$0\longrightarrow H^{k}(E_{(1)},\mathbb Q)\overset{\delta_{1}}{\longrightarrow}H^{k}(E_{(2)},\mathbb Q)\overset{\delta_{2}}{\longrightarrow}\cdots\xrightarrow{\delta_{p-1}}H^{k}(E_{(p)},\mathbb Q)\overset{\delta_{p}}{\longrightarrow}\cdots$$

which, for each $k$, computes graded pieces of the weight filtration on $H^{k}(E,\mathbb C)$:

$$\operatorname{gr}_{k}^{W}H^{k+r}(E,\mathbb C)=\ker\delta_{r+1}/\operatorname{im}\delta_{r}.$$

The $(p,q)$-Hodge pieces are obtained by applying the functor $H^{q}({}^{p})$ to the complex and taking cohomology. Precisely, there is an induced complex with morphisms

$$\delta^{p,q}_{r}\colon H^{q}\left(E_{(r)},{}_{E_{(r)}}^{p}\right)\longrightarrow H^{q}\left(E_{(r+1)},{}_{E_{(r+1)}}^{p}\right),$$

and

(2.8)

$$\operatorname{gr}_{k}^{W}H^{k+r}(E,\mathbb C)^{p,q}=\ker\delta_{r+1}^{p,q}/\operatorname{im}\leavevmode\nobreak\ \delta_{r}^{p,q}.$$