Moment maps and cohomology of non-reductive quotients - Bérczi, G; Kirwan, F (Springer, 2023)
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 Kähler 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 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 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 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.