Algebraic Differential Equations for Entire Holomorphic Curves in Projective Hypersurfaces of General Type: Optimal Lower Degree Bound

Merker, Joël

doi:10.1007/978-3-319-11523-8_4

Cited by 10 publications

(6 citation statements)

References 42 publications

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“…Those can be seen as higher order analogues of symmetric differential forms and provide obstructions to the existence of entire curves [31,58,14,15]. A fruitful way to produce jet differential equations on a given variety is to use the Riemann-Roch theorem (see for instance [31,51]) or Demailly's holomorphic Morse inequalities [13] (see for instance [22,23], and also [42,16]). However, our proof relies on another construction described below.…”

Section: Introductionmentioning

confidence: 99%

On the hyperbolicity of general hypersurfaces

Brotbek

2017

Publ.math.IHES

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In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in P n are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen such that the geometry of their higher order jet spaces can be related to the geometry of a universal family of complete intersections. To do so, we introduce a Wronskian construction which associates a (twisted) jet differential to every finite family of global sections of a line bundle.

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Section: Introductionmentioning

confidence: 99%

On the hyperbolicity of general hypersurfaces

Brotbek

2017

Publ.math.IHES

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“…, f (k) ) = 0. Merker [34] proved the same for projective hypersurfaces in P n+1 of degree at least n + 3 using algebraic Morse inequalities. Darondeau [12] adapted techniques of the present paper to study algebraic degeneracy of entire curves in complements of smooth projective hypersurfaces.…”

Section: Introductionmentioning

confidence: 83%

“…Then (34) and (35) gives Lemma 5.16. Combined with ( 28) and ( 23) gives the desired Proposition 5.12:…”

Section: Estimation Of the Coefficients P N−l (N A δ)mentioning

confidence: 89%

“…More recently, Demailly formulated a generalised version of the GGL conjecture for directed manifolds

(X, V)

, where

V \subseteq T_{X}

is a subbundle, and proved — using holomorphic Morse inequalities and probabilistic methods — that for any projective directed manifold

(X, V)

with

K_{V}

big there is a differential equation

P

of order

k ≫ 1

such that any entire curve

f

must satisfy

P (f, f^{'}, \dots, f^{(k)}) = 0

. Merker proved the same for projective hypersurfaces in

{double-struckP}^{n + 1}

of general type using Riemann–Roch–Hirzebruch. Darondeau adapted techniques of the present paper to study algebraic degeneracy of entire curves in complements of smooth projective hypersurfaces.…”

Section: Introductionmentioning

confidence: 98%

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Towards the Green–Griffiths–Lang conjecture via equivariant localisation

Bérczi

2018

Proc. London Math. Soc.

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Green, Griffiths and Lang conjectured that for every complex projective algebraic variety X of general type there exists a proper algebraic subvariety of X containing all nonconstant entire holomorphic curves f:C→X. Using equivariant localisation we develop an iterated residue formula for cohomological pairings on the Demailly–Semple jet bundle. We apply this formula and a strategy of Demailly to give affirmative answer to the Green–Griffiths–Lang conjecture for generic projective hypersurfaces X⊂Pn+1 of degree degfalse(Xfalse)⩾n9n.

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“…In higher dimensions, Diverio [Div08,Div09] proved the existence of sections of H 0 (X, E GG k,m V * ⊗ Ç(−1)) whenever X is a hypersurface of P n+1 C of high degree d d n , assuming k n and m m n . More recently, Merker [Mer10] was able to treat the case of arbitrary hypersurfaces of general type, i.e. d n + 3, assuming this time k to be very large.…”

Section: Introductionmentioning

confidence: 99%

Applications of Pluripotential Theory to Algebraic Geometry

Demailly

2013

Lecture Notes in Mathematics

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Plurisubharmonic functions and positive currents are an essential tool of modern complex analysis. Since their inception by Oka and Lelong in the mid 1940's, major applications to algebraic and analytic geometry have been developed in many directions. One of them is the Bochner-Kodaira technique, providing very strong existence theorems for sections of holomorphic vector bundles with positive curvature via L 2 estimates; one can mention here the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, Andreotti-Vesentini, Grauert, Hörmander, Bombieri, Skoda and OhsawaTakegoshi in the course of more than 4 decades. Another development is the theory of holomorphic Morse inequalities (1985), which relate certain curvature integrals with the asymptotic cohomology of large tensor powers of line or vector bundles, and bring a useful complement to the Riemann-Roch formula.We describe here the main techniques involved in the proof of holomorphic Morse inequalities (chapter I) and their link with Monge-Ampère operators and intersection theory (chapter II). The last two chapters III, IV provide applications to the study of asymptotic cohomology functionals and the Green-Griffiths-Lang conjecture. The latter conjecture asserts that every entire curve drawn on a projective variety of general type should satisfy a global algebraic equation; via a probabilistic curvature calculation, holomorphic Morse inequalities imply that entire curves must at least satisfy a global algebraic differential equation.

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Algebraic Differential Equations for Entire Holomorphic Curves in Projective Hypersurfaces of General Type: Optimal Lower Degree Bound

Cited by 10 publications

References 42 publications

On the hyperbolicity of general hypersurfaces

On the hyperbolicity of general hypersurfaces

Towards the Green–Griffiths–Lang conjecture via equivariant localisation

Applications of Pluripotential Theory to Algebraic Geometry

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