PERVERSE–HODGE COMPLEXES FOR LAGRANGIAN FIBRATIONS

. Perverse–Hodge complexes are objects in the derived category of coherent sheaves obtained from Hodge modules associated with Saito’s decomposition theorem. We study perverse–Hodge complexes for Lagrangian fibrations and propose a symmetry between them. This conjectural symmetry categorifies the “Perverse = Hodge” identity of the authors and specializes to Matsushita’s theorem on the higher direct images of the structure sheaf. We verify our conjecture in several cases by making connections with variations of Hodge structures, Hilbert schemes, and Looijenga–Lunts–Verbitsky Lie algebras.


Introduction 1.Perverse-Hodge symmetry
For a compact irreducible symplectic variety (1) M of dimension 2n with a Lagrangian fibration π : M → B, the decomposition theorem, cf.[BBD82], (1.1) provides important invariants for the topology of π.A perverse-Hodge symmetry was proven in [SY22], connecting the cohomology of the perverse sheaves P i with the Hodge numbers of M.
The identity (1.2) governs the cohomology of the Lagrangian base, the invariant cohomology of a nonsingular fiber of π, and the Gokapumar-Vafa theory of K3 surfaces; we refer to [SY22, FSY22, HLS + 21, HM22] for more discussions on Theorem 1.1 and its applications.
The purpose of this paper is to explore and propose a categorification of the perverse-Hodge symmetry.It suggests that Theorem 1.1 should conceptually be viewed as a cohomological shadow of a sheaf-theoretic symmetry for Lagrangian fibrations with possibly noncompact ambient spaces M. It is a mysterious phenomenon since all existing proofs of (1.2), cf.[SY22, HLS + 21, HM22], rely heavily on the global cohomological properties of compact irreducible symplectic manifolds, and they do not "explain" why such a categorification should exist.Our formulation uses perverse-Hodge complexes constructed from Hodge modules.By Saito's theory [Sai89,Sai90], the decomposition theorem (1.1) can be upgraded to an identity in the bounded derived category of Hodge modules.Let Q H M [2n] be the trivial Hodge module, i.e., the pure Hodge module associated with the shifted trivial local system Q M [2n].We have The Hodge module P H i consists of a regular holonomic (left-)D B -module P i equipped with a good filtration F • P i ; it corresponds to the perverse sheaf P i under the Riemann-Hilbert correspondence.The increasing filtration F • P i induces an increasing filtration on the de Rham complex The associated graded pieces are natural objects in the bounded derived category of coherent sheaves on B, Up to re-indexing and shifting, we define We call G i,k the perverse-Hodge complexes associated with the Lagrangian fibration π; here i is the perverse degree, and k is the Hodge degree.The object G i,k is nontrivial only if 0 ≤ i, k ≤ 2n.
Our main proposal is the following conjectural symmetry between perverse-Hodge complexes.
Conjecture 1.2.Let π : M → B be a Lagrangian fibration.We have As in Theorem 1.5, Conjecture 1.2 categorifies a refined version of Theorem 1.1 when M is a compact irreducible symplectic variety.By Proposition 3.3, it also recovers Matsushita's result [Mat05] on the higher direct images of O M .

Main results
We provide evidence for Conjecture 1.2 and verify it in several cases.

Smooth morphisms.
Our first theorem verifies Conjecture 1.2 when π : M → B is smooth.In fact, we obtain a stronger result in this case.
Theorem 1.3.Assume that π : M → B is smooth.The symplectic form σ on M together with a polarization induces an isomorphism at the level of complexes.
Theorem 1.3 is essentially a reformulation of a result of Donagi and Markman [DM96] on the polarized variation of Hodge structures associated with the family.Both complexes G i,k and G k,i have the same length, and a term-by-term isomorphism is constructed between them.The mystery of Conjecture 1.2 is an "extension" of this isomorphism to the singular fibers.As we see from Section 3.4, in general such an extension is complicated, and the derived category is essential for the formulation.
The decomposition theorem associated with (1.4) has many supports besides the full base B (n) .In particular, the isomorphism of Conjecture 1.2 in this case is not merely an extension of the isomorphism of Theorem 1.3 for variations of Hodge structures.Semisimple objects of the decomposition theorem (1.3) supported on the "boundary" of B (n) contribute nontrivially.

Global cohomology.
Lastly, we consider Lagrangian fibrations π : M → B associated with compact irreducible symplectic varieties. (2)Since B is projective, the (hyper-)cohomology groups of the perverse-Hodge complexes are finite-dimensional.
The following theorem shows that in this case Conjecture 1.2 holds cohomologically.
From another aspect, Theorem 1.5 suggests that, among all the symmetries encoded by the Looijenga-Lunts-Verbitsky algebra of M, the particular one inducing (1.5) can be lifted sheaf-theoretically.

Smooth morphisms and variations of Hodge structures
Throughout this section, we assume that π : M → B is smooth, so that the Hodge modules P H i are variations of Hodge structures.

Variations of Hodge structures
As a consequence of the Arnold-Liouville theorem, a nonsingular fiber of a Lagrangian fibration is a complex torus.In particular, the smooth map π : M → B is a family of abelian varieties.The key to understanding the topology of π is the variation of Hodge structures (2) As the base B is assumed to be nonsingular, by a result of Hwang [Hwa08] we know that B is a projective space.However, this fact will not be used in this paper.
it is polarized of weight 1 with associated holomorphic vector bundle B and the Hodge filtration are compatible via the Griffiths transversality relation This yields an O B -linear map between the graded pieces of (2.1) Here V i,1−i = gr i F V is a vector bundle describing the variation of H i,1−i (M b ) of the fibers M b with b ∈ B. For our purpose, we also consider the variation of Hodge structures where the i th piece is given by We denote by V i,j the graded piece gr F i V i+j .
Lemma 2.1.We have a canonical isomorphism of vector bundles Proof.The morphism is induced by the cup product and the compatibility with Hodge filtrations.It suffices to check that it is an isomorphism when restricting to each b ∈ B; this follows from the fact that M b is an abelian variety, so that we have

Symplectic form
We discuss the interplay between the symplectic form σ and the variation of Hodge structures V .By [Mat05, Lemma 2.6], the symplectic form σ and a polarization on M induce an isomorphism Combining (2.2) and (2.3), we obtain a morphism of vector bundles as the composition: B is symmetric with respect to the two factors of the target.
B via the polarization.Hence θ can be viewed as a section of The proposition follows from the cubic condition for Lagrangian fibration [DM96, Lemma 7.5] which says that the section corresponding to θ is induced by a section of Sym 3 Ω 1 B ; see also [Voi18,Theorem 4.4].□ More generally, for any k ≥ 1 we consider the morphism where the first map is induced by the Gauss-Manin connection ∇ : V → V ⊗ Ω 1 B and the second identity is given by Lemma 2.1.
Corollary 2.3.The composition is symmetric with respect to the second and third factors of the target.
Proof.We proceed by induction on k.The induction base is Lemma 2.1.Now assume that the statement holds for k − 1.We have We consider a local section s k of V k,0 which can be written as s k−1 ∧t with s k−1 and t local sections of V k−1,0 and V 1,0 , respectively.The image ∇(s k ) consists of two terms ∇(s k−1 ) ∧ s 1 and s k−1 ∧ ∇(s 1 ).We obtain from the induction hypothesis and the induction base that both of them are local sections of This completes the induction.□

Proof of Theorem 1.3
The main ingredients of the proof are (i) the isomorphism (2.3) induced by the symplectic form σ and a polarization, and (ii) the symmetry of Corollary 2.3, which follows from the Donagi-Markman cubic condition.
We first note that by Lemma 2.1 we have a canonical isomorphism by switching the second and third factors of the right-hand side of (2.4).Secondly, for any i, j, k, the Gauss-Manin connection of V induces an O B -linear morphism The following proposition shows the compatibility between the isomorphisms ι i,j,k and the morphisms ∇; it relies heavily on ingredient (ii).

Proposition 2.4. We have a commutative diagram
To simplify the notation, we write the morphism of the top horizontal arrow as B , where we suppress the isomorphisms induced by (2.3) and Lemma 2.1.In particular, for a local section Similarly, for the same element (2.7) we have To prove the commutativity of the diagram, it suffices to show that ∇(s ⊗ t ⊗ u) and ∇(s ⊗ u ⊗ t) coincide under the natural isomorphism switching the second and the third factors By Griffiths transversality, the morphism ∇ of (2.6) is linear with respect to the second factor on the left-hand side as it represents V 0,j .Therefore, we have and the wedge product with t and u are on the second and third factors, respectively.Hence the desired property is a consequence of Corollary 2.3, which states that (2.8) is symmetric with respect to the second and third factors.□ Lastly, we show that Theorem 1.3 follows from the commutative diagram of Proposition 2.4.Since π : M → B is smooth, the D B -module P i−n in the Hodge module P H i−n is the variation of Hodge structures V i , and the filtration F • P i−n is described as In particular, the de Rham complex of P i−n is and the associated perverse-Hodge complexes are We prove Theorem 1.3 by showing that the two complexes G i,k and G k,i match term by term via the isomorphisms (2.5).For convenience we may assume i ≤ k.As V i,j = 0 for j < 0, we find (2.9) On the other hand, we have V i,j = 0 for j > n by the fact that π is a family of abelian varieties of dimension n.
We see that the complexes (2.9) and (2.10) are of the same length and are both concentrated in degrees Then Proposition 2.4 yields an isomorphism where the commutative diagram guarantees that it is indeed an isomorphism of complexes.This completes the proof.□

Hodge modules
In order to extend the isomorphisms established in Section 2 to the singular fibers, we need to use Saito's theory of Hodge modules [Sai89,Sai90].We begin with some relevant properties of Hodge modules.Then we recall Matsushita's result [Mat05] on the higher direct images of O M ; we show in Proposition 3.3 that Matsushita's result is equivalent to the case k = 2n of Conjecture 1.2.

Hodge modules
Recall that a variation of Hodge structures of weight w on a nonsingular variety X is a triple Pure Hodge modules introduced by Saito [Sai89] are vast generalizations of variations of Hodge structures.As in (3.1), a pure Hodge module on X is a triple (3.2) where P is a regular holonomic D X -module, F • is a good filtration on P , and P is a perverse sheaf on X, such that P corresponds to P ⊗ Q C via the Riemann-Hilbert correspondence. (4)Such triples satisfy a number of technical conditions; in particular, one can define the notions of weight and polarization.
The following theorem by Saito [Sai90] provides a concrete description of polarizable pure Hodge modules on X as extensions of polarizable variations of Hodge structures.Theorem 3.1 (Saito).
(1) The category HM(X, w) of polarizable pure Hodge modules of weight w on X is abelian and semisimple.
(2) For any closed subvariety Z ⊂ X, a simple polarizable variation of Hodge structures of weight w − dim Z on a nonsingular open subset of Z can be uniquely extended to a simple object in HM(X, w).(3) All simple objects in HM(X, w) arise this way.
From now on, all Hodge modules are assumed to be pure and polarizable.For any simple Hodge module (3.2), we define its support to be the support of the simple perverse sheaf P .
Set dim X = d.The de Rham complex of a Hodge module P H = (P , F • ) is and is concentrated in degrees [−d, 0].The filtration F • P induces an increasing filtration whose k th graded piece is the complex of O X -modules (3) Traditionally the Hodge filtration is a decreasing filtration; the relation with the increasing filtration here is (4) For convenience, we sometimes only use the pair (P , F • ) to denote a pure Hodge module.
Note that gr F k P = 0 for k > 0 when P is given by a variation of Hodge structures, but this is not true for general Hodge modules.The functor gr k DR(−) extends naturally to the bounded derived category of Hodge modules taking values in D b Coh(X).

Decomposition theorem, Saito's formula, and duality
Let f : X → Y be a projective morphism between nonsingular varieties.For a Hodge module P H = (P , F • ) ∈ HM(X, w), Saito's decomposition theorem [Sai89] states that there is a decomposition in the bounded derived category of Hodge modules on Y , (3.3) with H i (f + P H ) ∈ HM(Y , w + i).Its compatibility with the functor gr F k DR(−) is given by the following formula (often known as Saito's formula; see [Sai89, Section 2.3.7]): The functor gr F k DR(−) is also compatible with Serre duality.Recall that dim X = d.For a Hodge module P H = (P , F • ) ∈ HM(X, w), we have where ω X is the dualizing sheaf of X; see [Sch16,Lemma 7.4].Now we consider a Lagrangian fibration π : M → B with dim M = 2n.For our purpose, we study the direct image π + Q H M [2n] of the trivial Hodge module A direct calculation yields Here conventionally Ω k M = 0 for k < 0. Formulas (3.3) and (3.4) then read Finally, since dim B = n and all the fibers of π : M → B have dimension n, we have gr F k DR(P i ) = 0 for k < max{−2n, −2n − i}.Applying the duality (3.5), we also have gr F k DR(P i ) = 0 for k > min{0, −i}.

Matsushita's theorem revisited
Let π : M → B be a Lagrangian fibration with dim M = 2n.Matsushita [Mat05] calculated the higher direct images of O M .Theorem 3.2 (Matsushita).For 0 ≤ i ≤ n, we have Proof.Matsushita's proof assumes that M is projective.However, as he explained in [Mat05, Remark 2.10], the projectivity for M is only used for Kollár's decomposition [Kol86] Since the decomposition now holds for any projective morphism as a consequence of Saito's theory of Hodge modules [Sai91], we may safely remove the projectivity assumption for M in Matsushita's theorem.□ Proposition 3.3.The case k = 2n of Conjecture 1.2 is equivalent to (3.8).In particular, Conjecture 1.2 holds when k = 2n.
Proof.When k = 2n, the desired isomorphism of Conjecture 1.2 is On the other hand, since gr F −2n DR(P i−n ) is always a sheaf (concentrated in degree 0), combining (3.6) and (3.7), we obtain that the left-hand side of (3.9) is Consequently, (3.9) is equivalent to (3.8).□ Remark 3.4.In view of (3.7), Conjecture 1.2 would provide a new recipe for calculating the higher direct images of Ω k M for all k, extending Matsushita's work.We could use the perverse-Hodge symmetry to trade the contributions of gr F −k DR(P i ) for all i for the contributions of gr F −i DR(P k−n ) for all i.The latter has the advantage of only involving a single Hodge module P H k−n .

An example
To illustrate the subtleties when extending Theorem 1.3 to the singular fibers, we consider the following basic example.
Let π : S → B be an elliptic fibration of a symplectic surface.We assume that π only has singular fibers with a double point, over a finite set D ⊂ B. Let j : U = B\D → B be the open embedding.
We look at the symmetry For the right-hand side, let (V , F • , V ) denote the variation of Hodge structures R 1 π U * Q S U on U .Then the Hodge module P H 0 on B is the minimal extension (j !* V , F • ), which can be described concretely using Deligne's canonical extension.
Recall that the canonical extension depends on a real interval [a, a+1) or (a, a+1] where the eigenvalues of the residue endomorphism should lie.In our situation, the monodromy around each point of D is unipotent (given by the matrix ( 1 1 0 1 ) in local coordinates), so the eigenvalues are necessarily integers.Let V be the canonical extension of V with respect to either [0, 1) or (−1, 0]; it is locally free of rank 2 on B. Schmid's theorem says that F • V := j * F • V ∩ V is a filtration by locally free subsheaves.By [Sai90, Section 3.10], we have where the D B -action is induced by Deligne's meromorphic connection on V ( * D), and It follows that for the right-hand side of (3.11), we have (3.12)gr F 0 DR(P 0 ) = In particular, as a complex it has two nontrivial terms.But ∇ is clearly surjective; to see its kernel, we do a calculation in local coordinates.Let t be the local coordinate of B near 0 ∈ D, and let α, β be a local trivialization of V .Since the monodromy matrix around 0 is ( 1 1 0 1 ), the residue matrix for V is ).In other words, we have We also have where f (t), g(t) are holomorphic functions with g(t) nonvanishing.From this we see that On the other hand, the map ∇ induces an isomorphism Combining (3.13) and (3.14), we deduce that the kernel of ∇ in (3.12) is Finally, by [Kol86, Theorem 2.6] and (3.8), we have which yields the desired isomorphism (3.11) only(!) in the derived category D b Coh(B).
Note that the proof in Section 4.1 works for all symplectic surfaces S with π : S → B and does not rely on information about the singular fibers.

Hilbert schemes of points
In this section we prove Theorem 1.4; it is completed by a series of compatibility results regarding the perverse-Hodge symmetry and natural geometric operations.

Surfaces
We first verify Theorem 1.4 for n = 1, where the Lagrangian fibration is an elliptic surface π : S → B. It suffices to prove Conjecture 1.2 for 0 ≤ i < k ≤ 2. The cases when k = 2 were covered by Proposition 3.3.Therefore, it remains to show the symmetry (3.11) for S, whose left-hand side is The right-hand side can be computed via the duality (3.5): , where the last isomorphism follows from (3.10).

Closed embeddings and finite morphisms
Let X be an irreducible quasi-projective variety of dimension d.Let (4.1) i∈Z be a finite sequence of Hodge modules X.We say that the Hodge modules in (4.1) are perverse-Hodge symmetric (PHS for short) on X if for any i, k ∈ Z we have (5) Clearly Conjecture 1.2 is equivalent to the statement that the Hodge modules P H i given by the decomposition theorem (1.3) are PHS on B. In general we say that a morphism f : is pure on X and the Hodge modules obtained from the decomposition theorem of f + Q H X [dim X] are PHS on Y .The following proposition shows the compatibility between the perverse-Hodge symmetry and pushforwards along closed embeddings and finite morphisms.

Proposition 4.1. Assume that the Hodge modules
(1) Let ι : Z → X be a closed embedding of codimension c.Then the Hodge modules (2) If f : Z → X is a finite surjective morphism with dim X = dim Z, then the Hodge modules Proof.We only prove (1) as ( 2) is completely parallel.For a closed embedding ι : Z → X, we have given by the assumption.□ (5) By [Sch16, Lemma 7.3], the functor gr k DR(−) is well defined for possibly singular varieties.

External products
Let X and Y be quasi-projective varieties, and let be Hodge modules on X and Y , respectively.We recall the following standard lemma concerning the external product Lemma 4.2.We have Proof.This follows from the fact that the (filtered) de Rham functor is compatible with taking external product (cf.[MSS11, Equation (1.4.1)]).□ Now we consider projective morphisms with X j nonsingular.For each f j , we have the Hodge modules Q H i,j obtained from the decomposition theorem We show the compatibility between the perverse-Hodge symmetry and products of varieties.
Proposition 4.3.If the morphisms (4.2) are PHS, then the product morphism we obtain each summand By Lemma 4.2 this further yields Using this decomposition and the fact that the Q H i,j are PHS, we see that there is a one-to-one correspondence between the summands in the decompositions of respectively.This completes the proof.□ Remark 4.4.If each f j : X j → Y j is a Lagrangian fibration, then the product f : X → Y is also a Lagrangian fibration.Hence Proposition 4.3 provides consistency checks for Conjecture 1.2; it shows that if each f j satisfies Conjecture 1.2, then their product satisfies it as well.
Remark 4.5.We note that the only use of the nonsingular assumption in Proposition 4.3 is that the trivial Hodge modules Q H X j [dim X j ] on X j are pure.

Symmetric products
Let Q H = (Q, F • ) be a Hodge module on X.Its symmetric product (Q H ) (n) was introduced in [MSS11], which defines a Hodge module on the symmetric product X (n) of the variety.Furthermore, such an operation is extended to the bounded derived category of Hodge modules on X. Proposition 4.6.If the Q H i are PHS on X, then the (Q H i ) (n) are PHS on X (n) .Proof.Let q : X n → X (n) be the S n -quotient map.For an object F • ∈ D b Coh(X), we similarly consider the symmetric product Now by Proposition 4.3, we know that the Hodge modules (Q H i ) ⊠n are PHS on X n .Proposition 4.1(2) further implies that the Hodge modules q + (Q H i ) ⊠n are PHS on X (n) .To prove the corresponding property for n) , it suffices to show that the natural isomorphism ⊠n is equivariant with respect to the S n -actions.It follows from [MSS11] that the (filtered) de Rham functor is compatible with the symmetric group action on X n ; more precisely, see [MSS11, proof of Proposition 1.5].□

Proof of Theorem 1.4
For our purpose, we describe the decomposition theorem associated with the morphism The first map f : S [n] → S (n) of the composition is semismall, and the associated decomposition theorem is calculated in [GS93], which we now review.For a partition of n, we use S (ν) to denote the variety We consider the stratification of the target variety S (n) by the combinatorial types of the points: there is a canonical finite surjective morphism

LLV (sub)algebras
Let M be a compact irreducible symplectic variety or, equivalently, a projective hyper-Kähler manifold.Assume dim M = 2n.We call an element α ∈ H 2 (M, C) of Lefschetz type if for any k ≥ 0, cupping with α k gives an isomorphism Such a class α induces an sl 2 -triple (L α , H, Λ α ) acting on H * (M, C).The LLV algebra g(M) is generated by all sl 2 -triples associated with Lefschetz type classes.As is shown in [Ver95,Ver96] and [LL97] independently, there is a natural isomorphism where b 2 (M) is the second Betti number of M.
Two subalgebras of g(M) played a crucial role in establishing Theorem 1.1.The first is Verbitsky's so(5) generated by the sl 2 -triples associated with the three Kähler classes ω I , ω J , ω K .By [Ver90], the weight decomposition of H * (M, C) under Verbitsky's so(5) coincides with the Hodge decomposition.Consider the Cartan subalgebra of this so(5) spanned by Then the Hodge decomposition (5.1) satisfies The second is the perverse so(5) introduced in [SY22], which concerns a Lagrangian fibration π : M → B. Let β ∈ H 2 (M, C) be the pull-back of an ample class on B, and let η ∈ H 2 (M, C) be a relatively ample class satisfying q M (η) = 0; here q M (−) is the Beauville-Bogomolov-Fujiki form on H 2 (M, C).The perverse so(5) is generated by the sl 2 -triples associated with here (−, −) M is the bilinear form associated with q M (−).
By [SY22, Theorem 3.1], the weight decomposition of H * (M, C) under the perverse so(5) takes the form where the P i are as in (1.1).In terms of the Cartan subalgebra spanned by we have

Perverse-Hodge algebra
Let π : M → B be a Lagrangian fibration with M a compact irreducible symplectic variety of dimension 2n.Since B is of Picard rank 1, the ample class ω I ∈ H 2 (M, C) admits a unique decomposition ω I = η + β with η, β as in the previous section.We also have ω J = σ + σ , where σ is the symplectic form; hence (ω J , η) M = (ω J , β) M = 0.In particular, the perverse so(5) can be generated by the sl 2 -triples associated with ω I = η + β, − √ −1(η − β), and ω J .We now consider the subalgebra g ⊂ g(M) generated by the sl 2 -triples associated with (5.3) We call it the perverse-Hodge algebra; it is naturally isomorphic to so(6) by the description of g(M) in [Ver95, Theorem 11.1].A Cartan subalgebra h ⊂ g is spanned by We have the weight decomposition (5.5) The perverse-Hodge algebra g contains both the perverse so(5) and Verbitsky's so(5) as subalgebras.Comparing (5.5) with (5.2) and (5.1), we find On the other hand, by Saito's formula (3.4) applied to the Hodge module P H i−n under the projection f : B → pt, we have The left-hand side is precisely the cohomology of the perverse-Hodge complex G i,k .We conclude that (5.6)

Proof of Theorem 1.5
Via (5.6), we have identified the cohomology groups H * (B, G i,k ) with the weight spaces of H * (M, C) under the perverse-Hodge algebra g.Theorem 1.5 is then equivalent to the symmetry of the weight spaces (5.7) The symmetry is a feature of so(6)-representations.More concretely, consider the subspace By [Ver95, Theorem 11.1] we have a natural isomorphism g ≃ so( V ).Up to renormalization, we may assume so that −Ω, η, σ , 1, β, σ form a standard isotropic basis of V .Under this basis, the three elements H, H P , H F in (5.4) are precisely (twice) the basis of h ⊂ g described in [FH91, Section 18.1].Let H * , H * P , H * F denote the dual basis of h * .By [FH91, Section 18.1, p. 271], the Weyl group of g is isomorphic to (Z/2Z) 2 ⋊ S 3 , where the symmetric group S 3 permutes the three coordinate axes of h * and the generators of (Z/2Z) 2 act diagonally by (1, −1, −1) and (−1, −1, 1).In particular, there is an element of the Weyl group exchanging H * P , H * F while fixing H * .Consequently, we obtain the symmetry of the weight spaces (5.7) through the Weyl group action.□ Remark 5.1.We have been kindly informed by Mirko Mauri that the isomorphism (5.7) can be obtained by combining Verbitsky's so(5)-action and the monodromy symmetry; see the proof of [HM23, Corollary 3.5].
Remark 5.2.Starting from the isomorphism H i+j−2n (B, G i,k ) ≃ H i+j−2n (B, G k,i ), we may sum over the index k.
On one hand, we get On the other hand, by (3.7) we have . We see that Theorem 1.5 refines Theorem 1.1.
Remark 5.3.One can collect the numbers h i,k,d := dim H i,k,d (M) and depict them in a 3-space.We call it the perverse-Hodge diamond of π : M → B. For example, the (d = 2n)-plane has the shape   To simplify the discussion we assume b 2 (M) ≥ 5. Then the perverse-Hodge algebra g can be upgraded to an so(7) by adding one more sl 2 -triple, namely the one associated with an element ρ ∈ H 2 (M, C) which is orthogonal to the four classes in (5.3) with respect to q M (−) and shares the same norm.The Weyl group of so(7) is the full symmetry group of the regular octahedron, and as such it acts on the perverse-Hodge diamond (whereas the Weyl group of so(6) acts as the subgroup of rotational symmetries).Meanwhile, it is expected that the perverse-Hodge diamond has precisely the shape of a regular octahedron, meaning that no nonzero numbers h i,k,d lie outside the convex hull of the six vertices h 0,0,0 , h 0,n,2n , h n,0,2n , h n,2n,2n , h 2n,n,2n , h 2n,2n,4n .