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Equivariant multiplicities and mirror symmetry for Hilbert schemes

Alexandre Minets, Filip Živanović

Abstract

Following Hausel-Hitchin, we investigate core Lagrangians and upward flows in Hilbert schemes of points on elliptic surfaces. We compute the scheme-theoretic multiplicities of core Lagrangians, as well as the equivariant multiplicities of the very stable ones. Furthermore, we extend the notion of equivariant multiplicity to wobbly components and compute it for Hilbert schemes of two points. Inspired by Eisenstein series functor in Dolbeault Langlands correspondence, we propose that upward flows of very stable ideals are mirror dual to modified Procesi bundles, and justify this claim through numerical checks. Finally, we make some conjectures about extending this picture to wobbly upward flows.

Equivariant multiplicities and mirror symmetry for Hilbert schemes

Abstract

Following Hausel-Hitchin, we investigate core Lagrangians and upward flows in Hilbert schemes of points on elliptic surfaces. We compute the scheme-theoretic multiplicities of core Lagrangians, as well as the equivariant multiplicities of the very stable ones. Furthermore, we extend the notion of equivariant multiplicity to wobbly components and compute it for Hilbert schemes of two points. Inspired by Eisenstein series functor in Dolbeault Langlands correspondence, we propose that upward flows of very stable ideals are mirror dual to modified Procesi bundles, and justify this claim through numerical checks. Finally, we make some conjectures about extending this picture to wobbly upward flows.
Paper Structure (44 sections, 58 theorems, 143 equations, 4 tables)

This paper contains 44 sections, 58 theorems, 143 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Equivariant multiplicities
  4. Hilbert schemes
  5. Mirror symmetry
  6. Organization
  7. Core Lagrangians
  8. Pre-integrable systems
  9. Flows and Lagrangians
  10. Products and finite covers
  11. Flow order
  12. Very stable points
  13. Equivariant multiplicities
  14. $K$-theoretic prerequisites
  15. Equivariant multiplicity of coherent sheaves
  16. ...and 29 more sections

Key Result

Theorem A

Let $F\in \pi_0(\mathcal{M}^{\mathbb{T}})$ be a fixed component, and $p\in F$. We have $\gamma_p \leq \mu_F(1)$, and the equality holds if and only if $p$ is very stable. $F$ contains a very stable point if and only if $m_F(t)=\mu_F(t)$.

Theorems & Definitions (147)

  • Theorem A: \ref{['prop:HH-mult', 'prop:v-stable-HHvsMZ']}
  • Theorem B: \ref{['cor:HilbTE-mult']}, \ref{['prop:vst-eq-mult-parab', 'prop:mult-pain']}
  • Theorem C
  • Theorem D: \ref{['thm:eq-index']}
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 137 more