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Locally trivial monodromy of moduli spaces of sheaves on Abelian surfaces

Ludovica Buelli

Abstract

The aim of this work is to give a description of the locally trivial monodromy group of irreducible symplectic varieties arising from moduli spaces of semistable sheaves on Abelian surfaces with non-primitive Mukai vector. The outcome is that the locally trivial monodromy group of a singular moduli space of this type is isomorphic to the monodromy group of a smooth moduli space, extending Markman's and Mongardi's description to the non-primitive case. As a consequence, we also prove the SYZ conjecture for any singular moduli space of this type.

Locally trivial monodromy of moduli spaces of sheaves on Abelian surfaces

Abstract

The aim of this work is to give a description of the locally trivial monodromy group of irreducible symplectic varieties arising from moduli spaces of semistable sheaves on Abelian surfaces with non-primitive Mukai vector. The outcome is that the locally trivial monodromy group of a singular moduli space of this type is isomorphic to the monodromy group of a smooth moduli space, extending Markman's and Mongardi's description to the non-primitive case. As a consequence, we also prove the SYZ conjecture for any singular moduli space of this type.
Paper Structure (41 sections, 58 theorems, 210 equations, 1 table)

This paper contains 41 sections, 58 theorems, 210 equations, 1 table.

Table of Contents

  1. Preliminaries
  2. Singular symplectic varieties
  3. Moduli spaces of sheaves on Abelian surfaces
  4. Locally trivial deformations and monodromy operators
  5. Locally trivial deformations of symplectic varieties
  6. Locally trivial monodromy operators
  7. The monodromy group of smooth moduli spaces of sheaves on Abelian surfaces
  8. An injective morphism from Mon2lt(Kv(S,H)) to Mon2(Kw(S,H))
  9. The embedding Kw(S,H) → Kv(S,H)
  10. Action on the second integral cohomology
  11. Action on monodromy operators
  12. The cases $k=1$ and $k=2$
  13. A groupoid representation
  14. Deformations of $(m,k)-$triples and their groupoid
  15. Fourier-Mukai equivalences and their groupoid
  16. ...and 26 more sections

Key Result

Theorem A.1

Let $X$ be an irreducible symplectic variety that is locally trivial deformation equivalent to a moduli space $K_{v}(S,H)$, where $S$ is an Abelian surface, $v=mw$ is a Mukai vector with $m>1$ and $w$ primitive and such that $w^2>4$, and $H$ is a $v-$generic polarization. Then

Theorems & Definitions (154)

  • Theorem A.1: Theorem \ref{['mainthm']}, Corollary \ref{['maincor']}
  • Theorem B.1: Corollary \ref{['corinjmon']}, Corollary \ref{['maincor']}
  • Theorem A.2
  • Definition 1.1.1
  • Remark 1.1.2
  • Proposition 1.1.3
  • Definition 1.2.1
  • Remark 1.2.2
  • Remark 1.2.3
  • Definition 1.2.4
  • ...and 144 more