Moment maps and cohomology of non-reductive quotients

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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 metric which is invariant for a maximal compact subgroup $Q$ of $H$, we define a notion of moment map for the action of $H$, and show that when the (non-reductive) GIT quotient $X/\!/H$ introduced by B\'erczi, Doran, Hawes and Kirwan exists, it can be identified with the quotient by $Q$ of a suitable level set for this moment map. When semistability coincides with stability for the action of $H$, we derive formulas for the Betti numbers of $X/\!/H$ and we 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 hyperbolicity conjectures for projective hypersurfaces with polynomial degree.
Original languageEnglish
PublisherArXiv
Publication statusPublished - 25 Sep 2019

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31 pages

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