Research output: Working paper › Research

**Moment maps and cohomology of non-reductive quotients.** / Bérczi, Gergely; Kirwan, Frances.

Research output: Working paper › Research

Bérczi, G & Kirwan, F 2019 'Moment maps and cohomology of non-reductive quotients' ArXiv. <https://arxiv.org/abs/1909.11495>

Bérczi, G., & Kirwan, F. (2019). *Moment maps and cohomology of non-reductive quotients*. ArXiv. https://arxiv.org/abs/1909.11495

Bérczi G, Kirwan F. 2019. Moment maps and cohomology of non-reductive quotients. ArXiv.

Bérczi, Gergely and Frances Kirwan *Moment maps and cohomology of non-reductive quotients*. ArXiv. 2019.,

Bérczi G, Kirwan F. Moment maps and cohomology of non-reductive quotients. ArXiv. 2019 Sep 25.

Bérczi, Gergely ; Kirwan, Frances. / **Moment maps and cohomology of non-reductive quotients**. ArXiv, 2019.

@techreport{34d7598d89ac4da5adcf9f484bca32d6,

title = "Moment maps and cohomology of non-reductive quotients",

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 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. ",

keywords = "math.AG, 14L24, 14F43",

author = "Gergely B{\'e}rczi and Frances Kirwan",

note = "31 pages",

year = "2019",

month = sep,

day = "25",

language = "English",

publisher = "ArXiv",

type = "WorkingPaper",

institution = "ArXiv",

}

TY - UNPB

T1 - Moment maps and cohomology of non-reductive quotients

AU - Bérczi, Gergely

AU - Kirwan, Frances

N1 - 31 pages

PY - 2019/9/25

Y1 - 2019/9/25

N2 - 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.

AB - 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.

KW - math.AG

KW - 14L24, 14F43

M3 - Working paper

BT - Moment maps and cohomology of non-reductive quotients

PB - ArXiv

ER -