Perverse–Hodge complexes for Lagrangian fibrations

For a compact irreducible symplectic variety⁽¹⁾⁽¹⁾(1)We say that $M$ is irreducible symplectic if it is a compact Kähler manifold such that $H^{0}(M,{}_{M}^{2})$ is spanned by a nowhere degenerate $2$-form. $M$ of dimension $2n$ with a Lagrangian fibration $\pi\colon M\to B$, the decomposition theorem, cf. [BBD82],

(1.1)

$$R\pi_{*}{\mathbb{Q}}_{M}[2n]\simeq\bigoplusop\displaylimits_{i=-n}^{n}P_{i}[-i],\quad P_{i}={{p}{\mathcal{H}}}^{i}\left(R\pi_{*}{\mathbb{Q}}_{M}[2n]\right)\in\operatorname{Perv}(B)$$

provides important invariants for the topology of $\pi$. 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 $\pi$, 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.

Let $(M,\sigma)$ be a nonsingular quasi-projective symplectic variety of dimension $2n$. Here $\sigma$ is a closed nowhere degenerate holomorphic $2$-form on $M$. Let $\pi\colon M\to B$ be a proper Lagrangian fibration onto a nonsingular base $B$; i.e., the restriction of the symplectic form $\sigma$ to regular part of a fiber vanishes. Interesting examples of $\pi$ include Lagrangian fibrations of compact irreducible symplectic varieties, cf. [Bea91], and Hitchin’s integrable systems, cf. [Hit87a, Hit87b].

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 ${\mathbb{Q}}_{M}^{H}[2n]$ be the trivial Hodge module, i.e., the pure Hodge module associated with the shifted trivial local system ${\mathbb{Q}}_{M}[2n]$. We have

(1.3)

$$\pi_{+}{\mathbb{Q}}_{M}^{H}[2n]\simeq\bigoplusop\displaylimits_{i=-n}^{n}P_{i}^{H}[-i],\quad P_{i}^{H}={\mathcal{H}}^{i}\left(\pi_{+}{\mathbb{Q}}_{M}^{H}[2n]\right).$$

The Hodge module $P_{i}^{H}$ consists of a regular holonomic (left-)${\mathcal{D}}_{B}$-module ${\mathcal{P}}_{i}$ equipped with a good filtration $F_{\bullet}{\mathcal{P}}_{i}$; it corresponds to the perverse sheaf $P_{i}$ under the Riemann–Hilbert correspondence. The increasing filtration $F_{\bullet}{\mathcal{P}}_{i}$ induces an increasing filtration on the de Rham complex

$$\operatorname{DR}({\mathcal{P}}_{i})=\left[{\mathcal{P}}_{i}\longrightarrow{\mathcal{P}}_{i}\otimes{}^{1}_{B}\longrightarrow\cdots\longrightarrow{\mathcal{P}}_{i}\otimes{}^{n}_{B}\right]\![n].$$

The associated graded pieces are natural objects in the bounded derived category of coherent sheaves on $B$,

$$\operatorname{gr}^{F}_{k}\operatorname{DR}({\mathcal{P}}_{i})\in D^{b}\operatorname{Coh}(B).$$

Up to re-indexing and shifting, we define

$${\mathcal{G}}_{i,k}:=\operatorname{gr}^{F}_{-k}\operatorname{DR}({\mathcal{P}}_{i-n})[n-i].$$

We call ${\mathcal{G}}_{i,k}$ the perverse–Hodge complexes associated with the Lagrangian fibration $\pi$; here $i$ is the perverse degree, and $k$ is the Hodge degree. The object ${\mathcal{G}}_{i,k}$ is nontrivial only if $0\leq i,k\leq 2n$.

Our main proposal is the following conjectural symmetry between perverse–Hodge complexes.

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 ${\mathcal{O}}_{M}$.

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

As a consequence of the Arnold–Liouville theorem, a nonsingular fiber of a Lagrangian fibration is a complex torus. In particular, the smooth map $\pi\colon M\to B$ is a family of abelian varieties. The key to understanding the topology of $\pi$ is the variation of Hodge structures

$$V=R^{1}\pi_{*}{\mathbb{Q}}_{M};$$

it is polarized of weight $1$ with associated holomorphic vector bundle ${\mathcal{V}}=V\otimes_{\mathbb{Q}}{\mathcal{O}}_{B}$. The integrable connection $\nabla\colon{\mathcal{V}}\to{\mathcal{V}}\otimes{}_{B}^{1}$ and the Hodge filtration

(2.1)

$$0=F^{2}{\mathcal{V}}\subset F^{1}{\mathcal{V}}\subset F^{0}{\mathcal{V}}={\mathcal{V}}$$

are compatible via the Griffiths transversality relation

$$\nabla(F^{i}{\mathcal{V}})\subset F^{i-1}{\mathcal{V}}\otimes{}_{B}^{1}.$$

This yields an ${\mathcal{O}}_{B}$-linear map between the graded pieces of (2.1)

(2.2)

$$\overline{\nabla}\colon{\mathcal{V}}^{1,0}\longrightarrow{\mathcal{V}}^{\,0,1}\otimes{}_{B}^{1}.$$

Here ${\mathcal{V}}^{\,i,1-i}=\operatorname{gr}_{F}^{i}{\mathcal{V}}$ is a vector bundle describing the variation of $H^{i,1-i}({M_{b}})$ of the fibers $M_{b}$ with $b\in B$.

For our purpose, we also consider the variation of Hodge structures ${\mathcal{V}}^{k}=\wedge^{k}V$ of weight $k$. Its Hodge filtration is

$$0=F^{k+1}{\mathcal{V}}^{\mkern 1.0muk}\subset F^{k-1}{\mathcal{V}}^{\mkern 1.0muk}\subset\cdots\subset F^{0}{\mathcal{V}}^{\mkern 1.0muk}={\mathcal{V}}^{\mkern 1.0muk},$$

where the $i^{\mathrm{th}}$ piece is given by

$$F^{i}{\mathcal{V}}^{\mkern 1.0muk}=\sumop\displaylimits_{{i_{l}+i_{2}+\dots+i_{k}}=i}F_{i_{1}}{\mathcal{V}}\wedge F_{i_{2}}{\mathcal{V}}\wedge\cdots\wedge F_{i_{k}}{\mathcal{V}}.$$

We denote by ${\mathcal{V}}^{\,i,j}$ the graded piece $\operatorname{gr}_{i}^{F}{\mathcal{V}}^{\,i+j}$.

We discuss the interplay between the symplectic form $\sigma$ and the variation of Hodge structures ${\mathcal{V}}$.

By [Mat05, Lemma 2.6], the symplectic form $\sigma$ and a polarization on $M$ induce an isomorphism

(2.3)

$$\iota\colon{\mathcal{V}}^{\,0,1}\xrightarrow{\;\simeq\;}{}^{1}_{B},$$

which further yields

$$\wedge^{k}\iota\colon{\mathcal{V}}^{\,0,k}=\wedge^{k}{\mathcal{V}}^{\,0,1}\xrightarrow{\;\simeq\;}\wedge^{k}{}_{B}^{1}={}_{B}^{k}.$$

Combining (2.2) and (2.3), we obtain a morphism of vector bundles as the composition:

$$\theta=(\iota\otimes 1)\circ\overline{\nabla}\colon{\mathcal{V}}^{1,0}\longrightarrow{\mathcal{V}}^{\,0,1}\otimes{}_{B}^{1}\longrightarrow{}_{B}^{1}\otimes{}_{B}^{1}.$$

More generally, for any $k\geq 1$ we consider the morphism

$$\overline{\nabla}\colon{\mathcal{V}}^{\mkern 1.0muk,0}\longrightarrow{\mathcal{V}}^{\mkern 1.0muk-1,1}\otimes{}_{B}^{1}={\mathcal{V}}^{\mkern 1.0muk-1,0}\otimes{\mathcal{V}}^{\,0,1}\otimes{}_{B}^{1},$$

where the first map is induced by the Gauss–Manin connection $\nabla\colon{\mathcal{V}}\to{\mathcal{V}}\otimes{}_{B}^{1}$ and the second identity is given by Lemma 2.1.

The main ingredients of the proof are

1. (1)

   the isomorphism (2.3) induced by the symplectic form $\sigma$ and a polarization, and

2. (2)

   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

(2.4)

$${\mathcal{V}}^{\,i,j}\otimes{}_{B}^{k}={\mathcal{V}}^{\,i,0}\otimes\wedge^{j}{\mathcal{V}}^{\,0,1}\otimes\wedge^{k}{}_{B}^{1}.$$

Hence (2.3) induces an isomorphism of vector bundles

(2.5)

$$\iota_{i,j,k}\colon{\mathcal{V}}^{\,i,j}\otimes{}_{B}^{k}\xrightarrow{\;\simeq\;}{\mathcal{V}}^{\,i,k}\otimes{}_{B}^{j}$$

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 ${\mathcal{V}}$ induces an ${\mathcal{O}}_{B}$-linear morphism

$$\overline{\nabla}\colon{\mathcal{V}}^{\,i,j}\otimes{}_{B}^{k}\longrightarrow{\mathcal{V}}^{\,i-1,j+1}\otimes{}_{B}^{k+1}.$$

The following proposition shows the compatibility between the isomorphisms $\iota_{i,j,k}$ and the morphisms $\overline{\nabla}$; it relies heavily on ingredient 2.

Lastly, we show that Theorem 1.3 follows from the commutative diagram of Proposition 2.4. Since $\pi\colon M\to B$ is smooth, the $\mathcal{D}_{B}$-module $\mathcal{P}_{i-n}$ in the Hodge module $P_{i-n}^{H}$ is the variation of Hodge structures ${\mathcal{V}}^{\,i}$, and the filtration $F_{\bullet}{\mathcal{P}}_{i-n}$ is described as

$$F_{k}{\mathcal{P}}_{i-n}=F^{-k}{\mathcal{V}}^{\,i}.$$

In particular, the de Rham complex of $\mathcal{P}_{i-n}$ is

$$\operatorname{DR}({\mathcal{P}}_{i-n})=\left[{\mathcal{V}}^{\,i}\xrightarrow{{\;\nabla\;}}{\mathcal{V}}^{\,i}\otimes{}^{1}_{B}\xrightarrow{{\;\nabla\;}}\cdots\xrightarrow{{\;\nabla\;}}{\mathcal{V}}^{\,i}\otimes{}^{n}_{B}\right]\![n],$$

and the associated perverse–Hodge complexes are

$${\mathcal{G}}_{i,k}=\left[{\mathcal{V}}^{\mkern 1.0muk,i-k}\xrightarrow{\;\overline{\nabla}\;}{\mathcal{V}}^{\mkern 1.0muk-1,i-k+1}\otimes{}^{1}_{B}\xrightarrow{\;\overline{\nabla}\;}\cdots\xrightarrow{\;\overline{\nabla}\;}{\mathcal{V}}^{\mkern 1.0muk-n,i-k+n}\otimes{}^{n}_{B}\right]\![2n-i].$$

We prove Theorem 1.3 by showing that the two complexes ${\mathcal{G}}_{i,k}$ and ${\mathcal{G}}_{k,i}$ match term by term via the isomorphisms (2.5). For convenience we may assume $i\leq k$. As ${\mathcal{V}}^{\,i,j}=0$ for $j<0$, we find

(2.9)

$${\mathcal{G}}_{i,k}=\left[{\mathcal{V}}^{\,i,0}\otimes{}_{B}^{k-i}\xrightarrow{\;\overline{\nabla}\;}{\mathcal{V}}^{\,i-1,1}\otimes{}_{B}^{k-i+1}\xrightarrow{\;\overline{\nabla}\;}\cdots\xrightarrow{\;\overline{\nabla}\;}{\mathcal{V}}^{\mkern 1.0muk-n,i-k+n}\otimes{}^{n}_{B}\right]\![2n-k].$$

On the other hand, we have ${\mathcal{V}}^{\,i,j}=0$ for $j>n$ by the fact that $\pi$ is a family of abelian varieties of dimension $n$. Consequently, the complex ${\mathcal{G}}_{k,i}$ is of the form

(2.10)

$${\mathcal{G}}_{k,i}=\left[{\mathcal{V}}^{\,i,k-i}\xrightarrow{\;\overline{\nabla}\;}{\mathcal{V}}^{\,i-1,k-i+1}\otimes{}_{B}^{1}\xrightarrow{\;\overline{\nabla}\;}\cdots\xrightarrow{\;\overline{\nabla}\;}{\mathcal{V}}^{\mkern 1.0muk-n,n}\otimes{}^{i-k+n}_{B}\right]\![2n-k].$$

We see that the complexes (2.9) and (2.10) are of the same length and are both concentrated in degrees $[k-2n,i-n]$. Then Proposition 2.4 yields an isomorphism

$$(\iota_{i,0,k-i},\iota_{i-1,1,k-i+1},\dots,\iota_{k-n,i-k+n,n})\colon{\mathcal{G}}_{i,k}\xrightarrow{\;\simeq\;}{\mathcal{G}}_{k,i},$$

where the commutative diagram guarantees that it is indeed an isomorphism of complexes. This completes the proof. ∎

Recall that a variation of Hodge structures of weight $w$ on a nonsingular variety $X$ is a triple

(3.1)

$$V^{H}=({\mathcal{V}},F_{\bullet},V),\quad\nabla(F_{k}{\mathcal{V}})\subset F_{k+1}{\mathcal{V}}\otimes{}_{X}^{1},$$

where $V$ is a (${\mathbb{Q}}$-)local system, $F_{\bullet}$ is an increasing filtration, and ${\mathcal{V}}=V\otimes_{\mathbb{Q}}{\mathcal{O}}_{X}$, such that the restriction $({\mathcal{V}}_{x},F_{\bullet},V_{x})$ to each $x\in X$ is a pure Hodge structure of weight $w$.⁽³⁾⁽³⁾(3)Traditionally the Hodge filtration is a decreasing filtration; the relation with the increasing filtration here is $F_{-k}=F^{k}$. We say that $V^{H}$ is polarizable if it admits a morphism

$$Q\colon V\times V\longrightarrow{\mathbb{Q}}(w)=(2\pi i)^{-w}{\mathbb{Q}}$$

inducing a polarization on each stalk $V_{x}$.

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)

$$P^{H}=({\mathcal{P}},F_{\bullet},P),\quad\nabla(F_{k}{\mathcal{P}})\subset F_{k+1}{\mathcal{P}}\otimes{}_{X}^{1},$$

where ${\mathcal{P}}$ is a regular holonomic ${\mathcal{D}}_{X}$-module, $F_{\bullet}$ is a good filtration on ${\mathcal{P}}$, and $P$ is a perverse sheaf on $X$, such that ${\mathcal{P}}$ corresponds to $P\otimes_{\mathbb{Q}}{\mathbb{C}}$ via the Riemann–Hilbert correspondence.⁽⁴⁾⁽⁴⁾(4)For convenience, we sometimes only use the pair $({\mathcal{P}},F_{\bullet})$ to denote a pure Hodge module. 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.

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 $P^{H}=({\mathcal{P}},F_{\bullet})$ is

$$\operatorname{DR}({\mathcal{P}})=\left[{\mathcal{P}}\xrightarrow{\;\nabla\;}{\mathcal{P}}\otimes{}^{1}_{X}\xrightarrow{\;\nabla\;}\cdots\xrightarrow{\;\nabla\;}{\mathcal{P}}\otimes{}^{d}_{X}\right]\![d]$$

and is concentrated in degrees $[-d,0]$. The filtration $F_{\bullet}{\mathcal{P}}$ induces an increasing filtration

$$F_{k}\operatorname{DR}({\mathcal{P}})=\left[F_{k}{\mathcal{P}}\xrightarrow{\;\nabla\;}F_{k+1}{\mathcal{P}}\otimes{}^{1}_{X}\xrightarrow{\;\nabla\;}\cdots\xrightarrow{\;\nabla\;}F_{k+d}{\mathcal{P}}\otimes{}^{d}_{X}\right]\![d]$$

whose $k^{\mathrm{th}}$ graded piece is the complex of ${\mathcal{O}}_{X}$-modules

$$\operatorname{gr}^{F}_{k}\operatorname{DR}({\mathcal{P}})=\left[\operatorname{gr}^{F}_{k}{\mathcal{P}}\xrightarrow{\;\overline{\nabla}\;}\operatorname{gr}^{F}_{k+1}{\mathcal{P}}\otimes{}^{1}_{X}\xrightarrow{\;\overline{\nabla}\;}\cdots\xrightarrow{\;\overline{\nabla}\;}\operatorname{gr}^{F}_{k+d}{\mathcal{P}}\otimes{}^{d}_{X}\right]\![d].$$

Note that $\operatorname{gr}^{F}_{k}{\mathcal{P}}=0$ for $k>0$ when ${\mathcal{P}}$ is given by a variation of Hodge structures, but this is not true for general Hodge modules. The functor $\operatorname{gr}_{k}\operatorname{DR}(-)$ extends naturally to the bounded derived category of Hodge modules taking values in $D^{b}\operatorname{Coh}(X)$.

Let $f\colon X\to Y$ be a projective morphism between nonsingular varieties. For a Hodge module $P^{H}=(\mathcal{P},F_{\bullet})\in\operatorname{HM}(X,w)$, Saito’s decomposition theorem [Sai89] states that there is a decomposition in the bounded derived category of Hodge modules on $Y$,

(3.3)

$$f_{+}P^{H}\simeq\bigoplusop\displaylimits_{i}\mathcal{H}^{i}\left(f_{+}P^{H}\right)[-i],$$

with $\mathcal{H}^{i}(f_{+}P^{H})\in\operatorname{HM}(Y,w+i)$. Its compatibility with the functor $\operatorname{gr}^{F}_{k}\operatorname{DR}(-)$ is given by the following formula (often known as Saito’s formula; see [Sai89, Section 2.3.7]):

(3.4)

$$Rf_{*}\operatorname{gr}^{F}_{k}\operatorname{DR}(\mathcal{P})\simeq\operatorname{gr}^{F}_{k}\operatorname{DR}(f_{+}{\mathcal{P}})\simeq\bigoplusop\displaylimits_{i}\operatorname{gr}^{F}_{k}\operatorname{DR}(\mathcal{H}^{i}(f_{+}{\mathcal{P}}))[-i].$$

The functor $\operatorname{gr}^{F}_{k}\operatorname{DR}(-)$ is also compatible with Serre duality. Recall that $\dim X=d$. For a Hodge module $P^{H}=({\mathcal{P}},F_{\bullet})\in\operatorname{HM}(X,w)$, we have

(3.5)

$$R\mathcal{H}\mathrm{om}_{{\mathcal{O}}_{X}}\left(\operatorname{gr}^{F}_{k}\operatorname{DR}({\mathcal{P}}),\omega_{X}[d]\right)\simeq\operatorname{gr}^{F}_{-k-w}\operatorname{DR}({\mathcal{P}}),$$

where $\omega_{X}$ is the dualizing sheaf of $X$; see [Sch16, Lemma 7.4].

Now we consider a Lagrangian fibration $\pi\colon M\to B$ with $\dim M=2n$. For our purpose, we study the direct image $\pi_{+}{\mathbb{Q}}_{M}^{H}[2n]$ of the trivial Hodge module

$${\mathbb{Q}}_{M}^{H}[2n]=({\mathcal{O}}_{M},F_{\bullet}),\quad F_{-1}{\mathcal{O}}_{M}=0\subset F_{0}{\mathcal{O}}_{M}={\mathcal{O}}_{M}.$$

A direct calculation yields

(3.6)

$$\operatorname{gr}^{F}_{-k}\operatorname{DR}({\mathcal{O}}_{M})\simeq{}_{M}^{k}[2n-k].$$

Here conventionally ${}_{M}^{k}=0$ for $k<0$. Formulas (3.3) and (3.4) then read

$$\pi_{+}{\mathbb{Q}}_{M}^{H}[2n]\simeq\bigoplusop\displaylimits_{i=-n}^{n}P^{H}_{i}[-i],\quad P^{H}_{i}=(\mathcal{P}_{i},F_{\bullet})\in\operatorname{HM}(B,2n+i)$$

and

(3.7)

$$R\pi_{*}{}_{M}^{k}[2n-k]\simeq\operatorname{gr}^{F}_{-k}\operatorname{DR}(f_{+}{\mathcal{O}}_{M})\simeq\bigoplusop\displaylimits_{i=-n}^{n}\operatorname{gr}^{F}_{-k}\operatorname{DR}({\mathcal{P}}_{i})[-i].$$

Finally, since $\dim B=n$ and all the fibers of $\pi\colon M\to B$ have dimension $n$, we have $\operatorname{gr}_{k}^{F}\operatorname{DR}({\mathcal{P}}_{i})=0$ for $k<\max\{-2n,-2n-i\}$. Applying the duality (3.5), we also have $\operatorname{gr}_{k}^{F}\operatorname{DR}({\mathcal{P}}_{i})=0$ for $k>\min\{0,-i\}$.

Let $\pi\colon M\to B$ be a Lagrangian fibration with $\dim M=2n$. Matsushita [Mat05] calculated the higher direct images of ${\mathcal{O}}_{M}$.

To illustrate the subtleties when extending Theorem 1.3 to the singular fibers, we consider the following basic example.

Let $\pi\colon S\to B$ be an elliptic fibration of a symplectic surface. We assume that $\pi$ only has singular fibers with a double point, over a finite set $D\subset B$. Let $j\colon U=B\backslash D\hookrightarrow B$ be the open embedding.

We look at the symmetry

(3.11)

$$\operatorname{gr}^{F}_{-1}\operatorname{DR}({\mathcal{P}}_{-1})[1]\simeq\operatorname{gr}^{F}_{0}\operatorname{DR}({\mathcal{P}}_{0})$$

proposed by Conjecture 1.2. Since the Hodge module $P_{-1}^{H}$ is the trivial Hodge module ${\mathbb{Q}}_{B}^{H}[1]$, the left-hand side of (3.11) is ${}_{B}^{1}[1]$. For the right-hand side, let $({\mathcal{V}},F_{\bullet},V)$ denote the variation of Hodge structures $R^{1}\pi_{U*}{\mathbb{Q}}_{S_{U}}$ on $U$. Then the Hodge module $P_{0}^{H}$ on $B$ is the minimal extension $(j_{!*}{\mathcal{V}},F_{\bullet})$, 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 $(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix})$ in local coordinates), so the eigenvalues are necessarily integers. Let $\overline{{\mathcal{V}}}$ be the canonical extension of ${\mathcal{V}}$ with respect to either $[0,1)$ or $(-1,0]$; it is locally free of rank $2$ on $B$. Schmid’s theorem says that $F_{\bullet}\overline{{\mathcal{V}}}:=j_{*}F_{\bullet}{\mathcal{V}}\cap\overline{{\mathcal{V}}}$ is a filtration by locally free subsheaves. By [Sai90, Section 3.10], we have

$$j_{!*}{\mathcal{V}}={\mathcal{D}}_{B}\cdot\overline{{\mathcal{V}}}\subset\overline{{\mathcal{V}}}(*D),$$

where the ${\mathcal{D}}_{B}$-action is induced by Deligne’s meromorphic connection on $\overline{{\mathcal{V}}}(*D)$, and

$$F_{k}j_{!*}{\mathcal{V}}=\sumop\displaylimits_{i\geq 0}F_{i}{\mathcal{D}}_{B}\cdot F_{k-i}\overline{{\mathcal{V}}}.$$

It follows that for the right-hand side of (3.11), we have

(3.12)

$$\operatorname{gr}_{0}^{F}\operatorname{DR}({\mathcal{P}}_{0})=\left[\frac{\overline{{\mathcal{V}}}+F_{1}{\mathcal{D}}_{B}\cdot F_{-1}\overline{{\mathcal{V}}}}{F_{-1}\overline{{\mathcal{V}}}}\xrightarrow{\;\overline{\nabla}\;}\frac{F_{1}{\mathcal{D}}_{B}\cdot\overline{{\mathcal{V}}}+F_{2}{\mathcal{D}}_{B}\cdot F_{-1}\overline{{\mathcal{V}}}}{\overline{{\mathcal{V}}}+F_{1}{\mathcal{D}}_{B}\cdot F_{-1}\overline{{\mathcal{V}}}}\otimes{}_{B}^{1}\right]\![1].$$

In particular, as a complex it has two nontrivial terms. But $\overline{\nabla}$ is clearly surjective; to see its kernel, we do a calculation in local coordinates. Let $t$ be the local coordinate of $B$ near $0\in D$, and let $\alpha,\beta$ be a local trivialization of $\overline{{\mathcal{V}}}$. Since the monodromy matrix around $0$ is $(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix})$, the residue matrix for $\overline{{\mathcal{V}}}$ is $(\begin{smallmatrix}0&1/2\pi\sqrt{-1}\\ 0&0\end{smallmatrix})$. In other words, we have

$$\nabla\alpha=0,\quad\nabla\beta=\frac{1}{2\pi\sqrt{-1}}\alpha\otimes\frac{dt}{t}.$$

We also have

$$F_{-1}\overline{{\mathcal{V}}}=\langle f(t)\alpha+g(t)\beta\rangle,$$

where $f(t),g(t)$ are holomorphic functions with $g(t)$ nonvanishing. From this we see that

$$F_{1}{\mathcal{D}}_{B}\cdot\overline{{\mathcal{V}}}=\overline{{\mathcal{V}}}+F_{1}{\mathcal{D}}_{B}\cdot F_{-1}\overline{{\mathcal{V}}},$$

hence

(3.13)

$$\overline{\nabla}\left(\frac{\overline{{\mathcal{V}}}}{F_{-1}\overline{{\mathcal{V}}}}\right)=0.$$

On the other hand, the map $\overline{\nabla}$ induces an isomorphism

(3.14)

$$\overline{\nabla}\colon\frac{\overline{{\mathcal{V}}}+F_{1}{\mathcal{D}}_{B}\cdot F_{-1}\overline{{\mathcal{V}}}}{\overline{{\mathcal{V}}}}\xrightarrow{\;\simeq\;}\frac{F_{1}{\mathcal{D}}_{B}\cdot\overline{{\mathcal{V}}}+F_{2}{\mathcal{D}}_{B}\cdot F_{-1}\overline{{\mathcal{V}}}}{F_{1}{\mathcal{D}}_{B}\cdot\overline{{\mathcal{V}}}}\otimes{}_{B}^{1}$$

sending $\frac{1}{t}\alpha$ to $-\frac{1}{t^{2}}\alpha\otimes dt$. Combining (3.13) and (3.14), we deduce that the kernel of $\overline{\nabla}$ in (3.12) is

$$\ker(\overline{\nabla})=\frac{\overline{{\mathcal{V}}}}{F_{-1}\overline{{\mathcal{V}}}}=\operatorname{gr}_{0}^{F}\overline{{\mathcal{V}}}.$$

Finally, by [Kol86, Theorem 2.6] and (3.8), we have

$$\operatorname{gr}_{0}^{F}\overline{{\mathcal{V}}}\simeq R^{1}\pi_{*}{\mathcal{O}}_{S}\simeq{}_{B}^{1},$$

which yields the desired isomorphism (3.11) only(!) in the derived category $D^{b}\operatorname{Coh}(B)$.

Note that the proof in Section 4.1 works for all symplectic surfaces $S$ with $\pi\colon S\to B$ and does not rely on information about the singular fibers.

We first verify Theorem 1.4 for $n=1$, where the Lagrangian fibration is an elliptic surface $\pi\colon S\to B$. It suffices to prove Conjecture 1.2 for

$$0\leq i<k\leq 2.$$

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

$$\operatorname{gr}_{-1}^{F}\operatorname{DR}({\mathcal{P}}_{-1})[1]\simeq{}_{B}^{1}[1].$$

The right-hand side can be computed via the duality (3.5):

$$\operatorname{gr}^{F}_{0}\operatorname{DR}({\mathcal{P}}_{0})\simeq R\mathcal{H}\mathrm{om}_{{\mathcal{O}}_{B}}\left(\operatorname{gr}^{F}_{-2}\operatorname{DR}({\mathcal{P}}_{0}),\omega_{B}[1]\right)\simeq{}_{B}^{1}[1],$$

where the last isomorphism follows from (3.10).

Let $X$ be an irreducible quasi-projective variety of dimension $d$. Let

(4.1)

$$\left\{Q^{H}_{i}=({\mathcal{Q}}_{i},F_{\bullet})\right\}_{i\in{\mathbb{Z}}}$$

be a finite sequence of Hodge modules on $X$. We say that the Hodge modules in (4.1) are perverse–Hodge symmetric (PHS for short) on $X$ if for any $i,k\in{\mathbb{Z}}$ we have

$$\operatorname{gr}^{F}_{-k}\operatorname{DR}({\mathcal{Q}}_{i-d})[d-i]\simeq\operatorname{gr}^{F}_{-i}\operatorname{DR}({\mathcal{Q}}_{k-d})[d-k].$$

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 $f\colon X\to Y$ is PHS if the trivial Hodge module ${\mathbb{Q}}^{H}_{X}[\dim X]$ is pure on $X$ and the Hodge modules obtained from the decomposition theorem of $f_{+}{\mathbb{Q}}^{H}_{X}[\dim X]$ are PHS on $Y$.

The following proposition shows the compatibility between the perverse–Hodge symmetry and push-forwards along closed embeddings and finite morphisms.