Degrees $d \geqslant \big( \sqrt{n}\, \log\, n\big)^n$ and $d \geqslant \big( n\, \log\, n\big)^n$ in the Conjectures of Green-Griffiths and of Kobayashi

Merker, Joel; Ta, The-Anh

doi:10.48550/arxiv.1901.04042

Cited by 2 publications

(3 citation statements)

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“…Siu [71] and Brotbek [19] proved Kobayashi hyperbolicity for projective hypersurfaces of sufficiently high (but not effective) degree. Based on the work of Brotbek effective degree bounds were worked out by Deng [22], Demailly [21], Merker and The-Anh Ta [59]. The best known bound based on these existing techniques is (n log n) n .…”

Section: Applicationsmentioning

confidence: 99%

“…Following the strategy developed by Demailly [20] and Siu [69][70][71][72] for generic hypersurfaces X ⊆ P n+1 of high degree, and using techniques of Demailly [20], the first effective lower bound for the degree of a generic hypersurface in the GGL conjecture was given by Diverio, Merker and Rousseau [25], where the conjecture for generic projective hypersurfaces X ⊆ P n+1 of degree deg(X) > 2 n 5 was confirmed. The current best bound for the Green-Griffiths-Lang Conjecture is deg(X) > ( √ n log n) n due to Merker and The-Anh Ta [59].…”

Section: Conjecture 23 (Green-griffiths-lang Conjecture 1979) Any Pro...mentioning

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: Applicationsmentioning

confidence: 99%

Section: Conjecture 23 (Green-griffiths-lang Conjecture 1979) Any Pro...mentioning

confidence: 99%

Moment maps and cohomology of non-reductive quotients

Bérczi,

Kirwan

2023

Invent. math.

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“…This lower bound improves the one stated in [Dem12], but is unfortunately far from being optimal. Better bounds -still probably non optimal -have been obtained in [Dar16] and [MTa19].…”

Section: B Compact Case (No Boundary Divisor)mentioning

confidence: 99%

On the existence of logarithmic and orbifold jet differentials

Campana¹,

Darondeau²,

Demailly³

et al. 2021

Preprint

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We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V ), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle T X . In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the corresponding directed structure V D in the general orbifold case -with a special attention to the compact case D = 0 and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green-Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.

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Degrees $d \geqslant \big( \sqrt{n}\, \log\, n\big)^n$ and $d \geqslant \big( n\, \log\, n\big)^n$ in the Conjectures of Green-Griffiths and of Kobayashi

Cited by 2 publications

References 10 publications

Moment maps and cohomology of non-reductive quotients

Moment maps and cohomology of non-reductive quotients

On the existence of logarithmic and orbifold jet differentials

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