Hyperbolic algebraic varieties and holomorphic differential equations
type: Item
instance of: scholarly article
title: Hyperbolic algebraic varieties and holomorphic differential equations
zbMATH Open document ID: 1264.32022
author: Jean-Pierre Demailly
published in: Acta Mathematica Vietnamica
publication date: 2013-01-31T00:00:00Z
full work available at URL: Toc ACTA 4 37.htm
review text: The long and exhaustive article reviews and explains recent methods and results in the theory of hyperbolic algebraic varieties, including several new results of the author which are focussed on some of the most challenging conjectures about hyperbolicity of algebraic manifolds of general type, the conjectures of Green-Griffiths and Lang. The conjectures of Green-Griffiths state that a projective algebraic variety \(X\) is of general type if and only if \(X\) is measure hyperbolic, and that \(X\) then contains a proper subvariety \(Y\) such that \(f(\mathbb C)\subset Y\) for every non-constant holomorphic map \(f:\mathbb C\rightarrow X\). An affirmative solution for these conjectures would imply that every very generic hypersurface in \(\mathbb P^{n+1}\), \(n\geq 3\), of degree \(d\geq 2n+1\) is hyperbolic, see [\textit{L. Ein}, Invent. Math. 94, No. 1, 163--169 (1988; Zbl 0701.14002); Math. Ann. 289, No. 3, 465--471 (1991; Zbl 0746.14019)] and [\textit{C. Voisin}, J. Differ. Geom. 44, No. 1, 200--213 (1996; Zbl 0883.14022)]. The author studies hyperbolicity problems in a more general setting. He works in the class of directed manifolds and transfers classical hyperbolicity concepts for complex varieties to this category. Special emphasis lies on the application of the Ahlfors-Schwarz lemma, jets of curves, Semple jet bundles, k-jet bundles, holomorphic Morse inequalities, jet differentials and k-jet metrics with negative curvature. A (compact resp. projective) directed manifold is understood as a pair \((X,V)\) of a (compact resp. projective) connected complex manifold \(X\) and an irreducible closed analytic subspace \(V\) of the holomorphic tangent bundle \(T_X\) such that \(V\cap T_{X,x}\) is a linear subspace of \(T_{X,x}\) for every \(x\in X\). Based on the Brody criterion it turns out that a compact directed manifold \((X,V)\) is hyperbolic iff every holomorphic map \(f:\mathbb C\rightarrow X\) with \(f'(\mathbb C)\subset V\) is constant. A projective directe…
reviewed by: Eberhard Oeljeklaus
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 14J40
zbMATH DE Number: 6133189
zbMATH Keywords: Kobayashi hyperbolic variety
zbMATH Keywords: directed manifold
zbMATH Keywords: jet bundle
zbMATH Keywords: Chern connection
zbMATH Keywords: variety of general type
zbMATH Keywords: Green-Griffiths conjecture
zbMATH Keywords: Lang conjecture
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6133189