Smoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients

TL;DR

This paper develops a method to compute the cohomology of moduli spaces formed as quotients by non-reductive group actions, revealing their topological structure and providing explicit formulas for their Poincaré series.

Contribution

It introduces a new approach to analyze non-reductive GIT quotients, including smoothness results and explicit cohomological formulas for moduli spaces.

Findings

Fixed point sets for non-reductive actions are smooth under certain conditions

Quotients have at worst finite quotient singularities

Explicit Poincaré series formulas are derived

Abstract

We establish a method for calculating the Poincar\'e series of moduli spaces constructed as quotients of smooth varieties by suitable non-reductive group actions; examples of such moduli spaces include moduli spaces of unstable vector or Higgs bundles on a smooth projective curve, with a Harder-Narasimhan type of length two. To do so, we first prove a result concerning the smoothness of fixed point sets for suitable non-reductive group actions on smooth varieties. This enables us to prove that quotients of smooth varieties by such non-reductive group actions, which can be constructed using Non-Reductive GIT via a sequence of blow-ups, have at worst finite quotient singularities. We conclude the paper by providing explicit formulae for the Poincar\'e series of these non-reductive GIT quotients.