Moment maps and cohomology of non-reductive quotients

Abstract

Let $H$ H be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety $X$ X . Given an ample linearisation of the action and an associated Fubini–Study Kähler form which is invariant for a maximal compact subgroup $Q$ Q of $H$ H , we define a notion of moment map for the action of $H$ H , and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient $X/\!/H$ X / / H introduced in (Bérczi et al. in J. Topol. 11(3):826–855, 2018) in terms of this moment map. Using this description we derive formulas for the Betti numbers of $X/\!/H$ X / / H and express the rational cohomology ring of $X/\!/H$ X / / H in terms of the rational cohomology ring of the GIT quotient $X/\!/T^{H}$ X / / T H , where $T^{H}$ T H is a maximal torus in $H$ H . We relate intersection pairings on $X/\!/H$ X / / H to intersection pairings on $X/\!/T^{H}$ X / / T H , obtaining a residue formula for these pairings on $X/\!/H$ X / / H analogous to the residue formula of (Jeffrey and Kirwan in Topology 34(2):291–327, 1995). As an application, we announce a proof of the Green–Griffiths–Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree.