Moment maps and cohomology of non-reductive quotients

• 成果类型: Article; Early Access
• 署名作者: Berczi, Gergely; Kirwan, Frances
• 署名单位: Aarhus University; University of Oxford
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01218-0
• 发表日期: 2023
• 关键词: geometric invariant-theory; multiple-point formulas; thom polynomials; localization; HYPERBOLICITY; SPACES

摘要

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 Kahler 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 (B & eacute;rczi 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.