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On the moduli space of simple sheaves on singular K3 surfaces

Barbara Fantechi, Rosa M. Miró-Roig

TL;DR

This work generalizes Mukai's result by extending the symplectic structure on the moduli of simple sheaves from smooth K3 surfaces to reduced singular K3 surfaces via perfect sheaves. It introduces the relative and absolute moduli frameworks for simple perfect sheaves, proves a canonical, global $2$-form from Serre duality, and shows that smoothability of the pair $(X,L)$ yields a closed form, making the moduli space symplectic; two constructions, extension bundles and generalized syzygy bundles, transport Lagrangian/isotropic subspaces between moduli of different ranks, with the extension-construction universally valid on singular $X$ and the syzygy-construction requiring smoothability for closedness. The paper then develops smoothability criteria for polarized K3 surfaces, including an explicit criterion for isolated lci singularities and a detailed treatment of quartics in $\mathbb{P}^3$, showing how these conditions guarantee the existence of deformations to smooth K3 pairs and hence preservation of the symplectic framework in families. Overall, the results provide a robust bridge between singular and smooth K3 geometry in the study of moduli of sheaves, with concrete criteria and constructions for obtaining symplectic and Lagrangian structures in singular settings.

Abstract

Mukai proved that the moduli space of simple sheaves on a smooth projective K3 surface is symplectic, and in \cite{FM2} we gave two constructions allowing one to construct new locally closed Lagrangian/isotropic subspaces of the moduli from old ones. In this paper, we extend both Mukai's result and our construction to reduced projective K3 surfaces; for the former we need to restrict our attention to perfect sheaves. There are two key points where we cannot get a straightforward generalization. In each, we need to prove that a certain differential form on the moduli space of simple, perfect sheaves vanishes, and we introduce a smoothability condition to complete the proof.

On the moduli space of simple sheaves on singular K3 surfaces

TL;DR

This work generalizes Mukai's result by extending the symplectic structure on the moduli of simple sheaves from smooth K3 surfaces to reduced singular K3 surfaces via perfect sheaves. It introduces the relative and absolute moduli frameworks for simple perfect sheaves, proves a canonical, global -form from Serre duality, and shows that smoothability of the pair yields a closed form, making the moduli space symplectic; two constructions, extension bundles and generalized syzygy bundles, transport Lagrangian/isotropic subspaces between moduli of different ranks, with the extension-construction universally valid on singular and the syzygy-construction requiring smoothability for closedness. The paper then develops smoothability criteria for polarized K3 surfaces, including an explicit criterion for isolated lci singularities and a detailed treatment of quartics in , showing how these conditions guarantee the existence of deformations to smooth K3 pairs and hence preservation of the symplectic framework in families. Overall, the results provide a robust bridge between singular and smooth K3 geometry in the study of moduli of sheaves, with concrete criteria and constructions for obtaining symplectic and Lagrangian structures in singular settings.

Abstract

Mukai proved that the moduli space of simple sheaves on a smooth projective K3 surface is symplectic, and in \cite{FM2} we gave two constructions allowing one to construct new locally closed Lagrangian/isotropic subspaces of the moduli from old ones. In this paper, we extend both Mukai's result and our construction to reduced projective K3 surfaces; for the former we need to restrict our attention to perfect sheaves. There are two key points where we cannot get a straightforward generalization. In each, we need to prove that a certain differential form on the moduli space of simple, perfect sheaves vanishes, and we introduce a smoothability condition to complete the proof.
Paper Structure (16 sections, 23 theorems, 47 equations)

This paper contains 16 sections, 23 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Notation and background material
  3. Smoothable K3 surfaces and pairs
  4. Moduli of simple perfect sheaves on surfaces
  5. Moduli of simple perfect sheaves in families
  6. Ext groups and trace maps
  7. Symplectic structure on moduli of perfect simple sheaves on K3
  8. Moduli of perfect simple sheaf on singular K3 surfaces
  9. Symplectic structure on moduli of simple perfect sheaves on smoothable K3 surfaces
  10. Lagrangian submanifolds of $\mathop{\mathrm{PSpl}}\nolimits(r;L,c_2)$
  11. Extension bundles
  12. Generalized syzygy bundles
  13. Smoothability criteria for polarized K3 surfaces
  14. Smoothability for polarized schemes
  15. Smoothability of polarized K3 surfaces with isolated singularities
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a K3 surface, $L\in \mathop{\mathrm{Pic}}\nolimits(X)$ and $r,c_2$ integers with $r\ge 1$.

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 51 more