GIT for $\hat U$-actions on algebraic $\mathbb C$-schemes

Qiao, Yikun

doi:10.48550/arxiv.2204.13884

Cited by 1 publication

(2 citation statements)

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“…The same result is proved in [16] under a different condition from ( * ), which is also called 'stability coincides with semistability for the Û action' (see [16] Remark 2.2 for a comparison). In [64] Y. Qiao studies the relationship between these and alternative geometric notions for stability to coincide with semi-stability.…”

Section: The û -Theoremmentioning

confidence: 99%

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Moment maps and cohomology of non-reductive quotients

Bérczi,

Kirwan

2023

Invent. math.

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Let $H$ H be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety $X$ X . Given an ample linearisation of the action and an associated Fubini–Study Kähler form which is invariant for a maximal compact subgroup $Q$ Q of $H$ H , we define a notion of moment map for the action of $H$ H , and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient $X/\!/H$ X / / H introduced in (Bérczi 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 $X/\!/H$ X / / H and express the rational cohomology ring of $X/\!/H$ X / / H in terms of the rational cohomology ring of the GIT quotient $X/\!/T^{H}$ X / / T H , where $T^{H}$ T H is a maximal torus in $H$ H . We relate intersection pairings on $X/\!/H$ X / / H to intersection pairings on $X/\!/T^{H}$ X / / T H , obtaining a residue formula for these pairings on $X/\!/H$ 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.

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Section: The û -Theoremmentioning

confidence: 99%

“…The blow-up construction in the reductive case can be modified in suitable nonreductive settings as described in [17,64]. Suppose that H = U R Û has internally graded unipotent radical U and that we have a well-adapted linear action of H on X with respect to L as at Definition 4.19.…”

Section: Blow-ups To Ensure Semistability Coincides With Stabilitymentioning

confidence: 99%

Moment maps and cohomology of non-reductive quotients

Bérczi,

Kirwan

2023

Invent. math.

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GIT for $\hat U$-actions on algebraic $\mathbb C$-schemes

Cited by 1 publication

References 9 publications

Moment maps and cohomology of non-reductive quotients

Moment maps and cohomology of non-reductive quotients

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