Poisson geometry of the moduli of local systems on smooth varieties

TL;DR

This paper explores the Poisson geometric structures on the moduli spaces of G-local systems on smooth algebraic varieties, extending known results from curves to higher dimensions and analyzing the impact of boundary conditions.

Contribution

It introduces the Poisson structure on the moduli of local systems for higher-dimensional varieties and constructs symplectic leaves via local monodromies, highlighting the role of strictness.

Findings

Moduli spaces carry natural Poisson structures as derived stacks

Symplectic leaves are constructed by fixing local monodromies at infinity

Strictness phenomenon arises with non-trivial crossings at the divisor at infinity

Abstract

We study the moduli of G-local systems on smooth but not necessarily proper complex algebraic varieties. We show that, when suitably considered as derived algebraic stacks, they carry natural Poisson structures, generalizing the well known case of curves. We also construct symplectic leaves of this Poisson structure by fixing local monodromies at infinity, and show that a new feature, called strictness, appears as soon as the divisor at infinity has non-trivial crossings.