Lagrangian subspaces of the moduli space of simple sheaves on K3 surfaces

TL;DR

This paper explores the geometry of the moduli space of simple sheaves on K3 surfaces, showing how certain subspaces are Lagrangian or isotropic, and constructs families of bundles via syzygy and extension methods.

Contribution

It introduces a novel construction linking Lagrangian and isotropic subspaces of the moduli space to new components via syzygy and extension bundles.

Findings

Constructs families of syzygy and extension bundles on K3 surfaces.

Establishes that these constructions induce embeddings between moduli components.

Shows that Lagrangian and isotropic subspaces correspond to similar subspaces in different moduli components.

Abstract

Let $X$ be a K3 surface and let $\text{Spl}(r;c_1,c_2)$ be the moduli space of simple sheaves on $X$ of fixed rank $r$ and Chern classes $c_1$ and $c_2$. Under suitable assumptions, to a pair $(F,W)$ (respectively, $(F,V)$) where $F\in \text{Spl}(r;c_1,c_2)$ and $W\subset H^0(F)$ (resp.~$V^*\subset H^1(F^*)$) is a vector subspace, we associate a simple syzygy bundle (resp.~extension bundle) on $X$. We show that both syzygy bundles and extension bundles can be constructed in families and that the induced morphism to a different component of the moduli of simple sheaves is a locally closed embedding. We show that this construction associates to every Lagrangian (resp.~isotropic) algebraic subspace of $\text{Spl}(r;c_1,c_2)$ an induced Lagrangian (resp.~isotropic) algebraic subspace of a different component of the moduli of simple sheaves.