Moment maps and cohomology of non-reductive quotients

TL;DR

This paper develops a moment map framework for non-reductive GIT quotients, enabling the computation of their cohomology and intersection pairings, and applies it to conjectures in algebraic geometry.

Contribution

It introduces a moment map approach for non-reductive quotients and derives cohomological formulas, extending classical GIT techniques to new settings.

Findings

Formulation of a moment map for non-reductive group actions.

Explicit formulas for Betti numbers and cohomology rings of non-reductive quotients.

Residue formula for intersection pairings on non-reductive GIT quotients.

Abstract

Let $H$ be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety $X$. Given an ample linearisation of the action and an associated Fubini-Study K\"ahler form which is invariant for a maximal compact subgroup $Q$ of $H$, we define a notion of moment map for the action of $H$, and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient $X/\!/H$ introduced by B\'erczi, Doran, Hawes and Kirwan in terms of this moment map. Using this description we derive formulas for the Betti numbers of $X/\!/H$ and express the rational cohomology ring of $X/\!/H$ in terms of the rational cohomology ring of the GIT quotient $X/\!/T^H$, where $T^H$ is a maximal torus in $H$. We relate intersection pairings on $X/\!/H$ to intersection pairings on $X/\!/T^H$, obtaining a residue formula for these pairings on $X/\!/H$ analogous to the residue formula of Jeffrey-Kirwan. As an application, we announce a proof of the Green-Griffiths-Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree.