Moment maps and cohomology of non-reductive quotients

2019/09/25 by Gergely Bérczi, Frances Kirwan, Bérczi, Gergely +1 · 1 citation
Mathematics · #14F43 #14L24 #53D20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.11495

openalex publication_date 2019/09/25 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

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 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 by Bérczi, 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.

Cited by

• Deformation theory of parabolic representation pairs

Related

• Refinements of the Morse stratification of the normsquare of the moment map
• Projective completions of graded unipotent quotients
• Projectivity of moment map quotients
• Trivializations of moment maps
• The Decomposition Theorem and the Intersection Cohomology of Quotients in Algebraic Geometry