Shifted Poisson structures and moduli spaces of complexes

TL;DR

This paper explores the derived algebraic geometry of moduli stacks of complexes over Calabi-Yau varieties, revealing shifted Poisson structures and their connections to known geometric and algebraic structures, especially in the elliptic case.

Contribution

It introduces a shifted Poisson structure on the moduli stack of complexes over Calabi-Yau varieties and relates it to classical Poisson structures in elliptic geometry and algebra.

Findings

Moduli stack of complexes has a (1-d)-shifted Poisson structure for Calabi-Yau variety of dimension d.

In dimension 1, the moduli stack admits a foliation by 0-shifted symplectic substacks.

Recovers known Poisson structures on elliptic curve related moduli spaces and connects them to elliptic Sklyanin algebras.

Abstract

In this paper we study the moduli stack of complexes of vector bundles (with chain isomorphisms) over a smooth projective variety $X$ via derived algebraic geometry. We prove that if $X$ is a Calabi-Yau variety of dimension $d$ then this moduli stack has a $(1-d)$-shifted Poisson structure. In the case $d=1$, we construct a natural foliation of the moduli stack by $0$-shifted symplectic substacks. We show that our construction recovers various known Poisson structures associated to complex elliptic curves, including the Poisson structure on Hilbert scheme of points on elliptic quantum projective planes studied by Nevins and Stafford, and the Poisson structures on the moduli spaces of stable triples over an elliptic curves considered by one of us. We also relate the latter Poisson structures to the semi-classical limits of the elliptic Sklyanin algebras studied by Feigin and Odesskii.