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Perverse-Hodge complexes for Lagrangian fibrations and symplectic resolutions

Zhengze Xin

TL;DR

The paper investigates perverse-Hodge complexes for Lagrangian fibrations on holomorphic symplectic varieties, including singular and non-compact cases. It establishes a symplectic Hard Lefschetz-type theorem for the intersection cohomology Hodge module IC_M under a perverse-coherence condition and proves symmetry of the associated perverse-Hodge complexes G_{i,k} in the presence of a symplectic resolution, generalizing prior smooth-case results. A key technical contribution is showing that IC_X is strongly coherent m-perverse on étale-locally symplectic spaces, enabling an étale-local extension of symplectic Lefschetz actions to graded de Rham complexes and yielding a numerical Perverse=Hodge equality without relying on the BBF form. The results are applied to singular Higgs moduli spaces, leveraging local models as hypertoric varieties to obtain the relative HL theorem for Hitchin fibrations and, when resolutions exist, the perverse-Hodge symmetry, thereby linking deep Hodge-theoretic structures to moduli problems. Together, these developments offer a categorified perspective on P=W-type phenomena in singular settings and suggest new avenues for understanding moduli spaces with symplectic singularities.

Abstract

We study perverse-Hodge complexes for Lagrangian fibrations on holomorphic symplectic varieties. We prove the symplectic Hard Lefschetz type theorem and the symmetry of perverse-Hodge complexes when the symplectic variety admits symplectic resolutions, therefore generalize the previous result by Schnell in the smooth case verifying a conjecture by Shen-Yin. Along the way, we study the perverse coherent properties of the intersection complex Hodge modules on symplectic varieties. As an application, we obtain an alternative proof of the numerical "perverse=Hodge" result by Felisetti-Shen-Yin, without using the Beauville-Bogomolov-Fujiki form. We also apply our results to study singular Higgs moduli spaces over reduced curves using results by Mauri-Migliorini on the local structure.

Perverse-Hodge complexes for Lagrangian fibrations and symplectic resolutions

TL;DR

The paper investigates perverse-Hodge complexes for Lagrangian fibrations on holomorphic symplectic varieties, including singular and non-compact cases. It establishes a symplectic Hard Lefschetz-type theorem for the intersection cohomology Hodge module IC_M under a perverse-coherence condition and proves symmetry of the associated perverse-Hodge complexes G_{i,k} in the presence of a symplectic resolution, generalizing prior smooth-case results. A key technical contribution is showing that IC_X is strongly coherent m-perverse on étale-locally symplectic spaces, enabling an étale-local extension of symplectic Lefschetz actions to graded de Rham complexes and yielding a numerical Perverse=Hodge equality without relying on the BBF form. The results are applied to singular Higgs moduli spaces, leveraging local models as hypertoric varieties to obtain the relative HL theorem for Hitchin fibrations and, when resolutions exist, the perverse-Hodge symmetry, thereby linking deep Hodge-theoretic structures to moduli problems. Together, these developments offer a categorified perspective on P=W-type phenomena in singular settings and suggest new avenues for understanding moduli spaces with symplectic singularities.

Abstract

We study perverse-Hodge complexes for Lagrangian fibrations on holomorphic symplectic varieties. We prove the symplectic Hard Lefschetz type theorem and the symmetry of perverse-Hodge complexes when the symplectic variety admits symplectic resolutions, therefore generalize the previous result by Schnell in the smooth case verifying a conjecture by Shen-Yin. Along the way, we study the perverse coherent properties of the intersection complex Hodge modules on symplectic varieties. As an application, we obtain an alternative proof of the numerical "perverse=Hodge" result by Felisetti-Shen-Yin, without using the Beauville-Bogomolov-Fujiki form. We also apply our results to study singular Higgs moduli spaces over reduced curves using results by Mauri-Migliorini on the local structure.

Paper Structure

This paper contains 16 sections, 37 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overview of Hodge modules
  4. Graded de Rham complexes
  5. Perverse-Hodge complexes and symmetry
  6. Geometry of symplectic varieties
  7. Perverse coherent t-structures
  8. Perverse coherent t-structures
  9. Intersection complexes of coherent sheaves
  10. Proof of the main theorem
  11. Support conditions
  12. Relative symplectic Hard Lefschetz theorem
  13. Symmetry of perverse-Hodge complexes
  14. Numerical "Perverse=Hodge"
  15. Singular Higgs moduli space
  16. ...and 1 more sections

Key Result

Theorem 1.1

If the intersection cohomology Hodge module $\operatorname{IC}_M$ is strongly coherent m-perverse (see Definition weaklyperverse), i.e. it satisfies the following support condition on its graded de Rham: for $j\ge 1$, then the action by the reflexive symplectic form $\sigma\in H^0(M,\Omega^{[2]}_M)$ satisfies symplectic Hard Lefschetz theorem for $\operatorname{IC}_M$, i.e is an isomorphism.

Theorems & Definitions (67)

  • Theorem 1.1: Symplectic Hard Lefschetz, Theorem \ref{['semiperversesymphardlef']}
  • Theorem 1.2: Perverse-Hodge symmetry, Theorem \ref{['sympP=H']}
  • Theorem 1.3: Numerical "Perverse=Hodge", Corollary \ref{['numP=H']}
  • Theorem 1.4: Theorem \ref{['5.4']}
  • Theorem 1.5: Theorem \ref{['perv=Hodgehiggs']}
  • Theorem 2.1: saito1988modules, Theorem 5.3.1
  • Corollary 2.2: Saito's Decomposition theorem
  • Definition 2.3
  • Lemma 2.4: kebekus2021extending, Proposition 4.10
  • Lemma 2.5: saito1988modules, § 2.4.3
  • ...and 57 more