Author: Eiko

Time: 2025-01-28 11:03:31 - 2025-01-28 11:03:31 (UTC)

Duality Involution On Symplectic Moduli Spaces

(S.H): Polarized K3 or abelian surfaces defined over $\mathbb{C}$, the main object in this topic is the moduli spaces of sheaves $M(r,c_1,c_2)$, (rank, chern classes) space of Gieseker $H$-semistable sheaves on $S$. This is a projective variety with a holomorphic symplectic form.

The next main object is duality involution. When $c_1=0$ and $M^{\mu -st}\ne \varnothing$ ($\mu$-stable part), we can consider the duality involution by taking the dual of a sheaf

$$D:\mathcal{F}\mapsto {\mathcal{F}}^{\vee }\in \mathrm{Aut}(M(r,0,c_2))$$

• $D$ is regular on $M^{\mu -st}$.

• $D$ is symplectic.

• $D$ is nontrivial if $r\ge 3$.

Motivation: construct of Irreducible Holomorphic Symplectic (IHS) Variety

$X$ smooth projective variety with holomorphic symplectic form $\sigma$. For classification, we may assume that $X$ is IHS (i.e. $X$ is simply connected and $H^0(X,{\mathrm{\Omega }}_X^2)=\mathbb{C}\sigma$). By B-B decomposition, known IHS manifolds up to deformation:

• ${\mathrm{K}3}^{[n]}$

• Generalized Kummer $K_n(A)$

• OG10

• OG6

Alternative Approach: Partially Resolve Finite Quotients of Symplectic Varieties

For example Fujiki 1983, Kawatani 2009, $S^n/G$ and $K(A)\to K{3}^{[n]}/G$.

Results

Theorem 1. Assume $\rho (S)=1$ and $r\ge 3$. Then $D$ extends to a regular involution on the whole $M(r,0,c_2)$. This is equivalent to $c_2=2r$ if $S$ is K3, and $c_2=2$ if $S$ is abelian.

Remark. Even if these conditions are satisfied, the extended involution is not given by the duality on $M(r,0,c_2)$.

Theorem 2. Consider

• Sis K3

• $M(3,0,6)$: singular of dimension $20$.