# ArXived articles on non-reductive geometry

Gergely Bérczi presents two new articles with his collaborator on arXiv

2019.09.27 | Jane Jamshidi

Associate Professor Gergely Bérczi has jointly with Professor Frances Kirwan (University of Oxford) submitted two new results on arXiv.

In the first paper written in collaboration with entitled "Moment maps and cohomology of non-reductive quotients" they study rational cohomology and cohomological intersection numbers on non-reductive quotients. 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 metric which is invariant for a maximal compact subgroup Q of H, they define a notion of moment map for the action of H, and show that when the (non-reductive) GIT quotient X//H introduced by Bérczi, 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, they derive formulas for the Betti numbers of X//H and they express the rational cohomology ring of X//H in terms of the rational cohomology ring of the GIT quotient X//TH, where TH is a maximal torus in H. They relate intersection pairings on X//H to intersection pairings on X//TH, obtaining a residue formula for these pairings on X//H analogous to the residue formula of Jeffrey-Kirwan. As an application, they announce a proof of the Green-Griffiths-Lang and Kobayashi hyperbolicity conjectures for projective hypersurfaces with polynomial degree.

Link to the paper on arXiv: https://arxiv.org/pdf/1909.11495.pdf

In the second paper written also in collaboration with Professor Frances Kirwan entitled "Non-reductive geometric invariant theory and hyperbolicity" they prove the Green-Griffiths-Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree using intersection theory for non-reductive geometric invariant theoretic quotients and recent work of Riedl and Yang.

Link to the paper on arXiv: https://arxiv.org/pdf/1909.11417.pdf

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