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Local multiplicativity of perverse filtrations

Zili Zhang

TL;DR

Let $f:S\to C$ be a proper surjective morphism from a smooth Kähler surface to a smooth curve. The authors define the local perverse filtration on fibers of the induced map $f^{[n]}:S^{[n]}\to C^{(n)}$ and show that it is multiplicative on each fiber if and only if $f$ is an elliptic fibration, linking fiberwise local data to the global geometry. They develop an analytic Douady-space framework to obtain a Göttsche-type decomposition of $R f^{[n]}_* \mathbb{Q}_{S^{[n]}}[2n]$ and describe the perverse filtration on $H^*(S^{[n]})$ via decompositions associated with partitions of $n$, compatible with products and symmetric powers. The paper also situates these results within the P=W program for Hitchin-type moduli spaces, clarifying when perverse and weight filtrations interact multiplicatively and providing a route to understanding the topology of Hitchin fibers through local models. Altogether, the work extends multiplicativity of perverse filtrations from Hilbert schemes to the analytic setting and reveals the elliptic-fibration condition as the precise obstruction to local multiplicativity on collision loci.

Abstract

Let $f:S\to C$ be a proper surjective morphism from a smooth Kähler surface to a smooth curve. We show that the local perverse filtration associated with the induced map $S^{[n]}\to C^{(n)}$ is multiplicative on each fiber if and only if $f$ is an elliptic fibration.

Local multiplicativity of perverse filtrations

TL;DR

Let be a proper surjective morphism from a smooth Kähler surface to a smooth curve. The authors define the local perverse filtration on fibers of the induced map and show that it is multiplicative on each fiber if and only if is an elliptic fibration, linking fiberwise local data to the global geometry. They develop an analytic Douady-space framework to obtain a Göttsche-type decomposition of and describe the perverse filtration on via decompositions associated with partitions of , compatible with products and symmetric powers. The paper also situates these results within the P=W program for Hitchin-type moduli spaces, clarifying when perverse and weight filtrations interact multiplicatively and providing a route to understanding the topology of Hitchin fibers through local models. Altogether, the work extends multiplicativity of perverse filtrations from Hilbert schemes to the analytic setting and reveals the elliptic-fibration condition as the precise obstruction to local multiplicativity on collision loci.

Abstract

Let be a proper surjective morphism from a smooth Kähler surface to a smooth curve. We show that the local perverse filtration associated with the induced map is multiplicative on each fiber if and only if is an elliptic fibration.
Paper Structure (12 sections, 14 theorems, 77 equations)

This paper contains 12 sections, 14 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Multiplicativity of perverse filtrations and $P=W$
  3. Local multiplicativity of Hitchin-type morphisms
  4. Outline
  5. Acknowledgements
  6. Perverse filtrations for Kähler manifolds
  7. Cohomology and compactly supported cohomology
  8. Perverse decompositions
  9. Local perverse filtrations on fibers
  10. Functoriality of local perverse filtrations on fibers
  11. Douady Spaces
  12. Local multiplicativity of Hilbert schemes of fibered surfaces

Key Result

Theorem 1.2

Let $n\ge 2$, $f:S\to C$, and $f^{[n]}:S^{[n]}\to C^{(n)}$ as above.

Theorems & Definitions (25)

  • Conjecture 1.1
  • Theorem 1.2: Theorem \ref{['4.3']}
  • Theorem 1.3: Theorem \ref{['4.4']}
  • Proposition 2.1
  • Corollary 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 15 more