Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

Abstract

Abstract. These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995. The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi. A convenient framework for this is the category of "directed manifolds", that is, the category of pairs (X, V ) where X is a complex manifold and V a holomorphic subbundle of T X . If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C → X tangent to V (Brody's criterion). We describe a construction of projectivized kjet bundles P k V , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature. More precisely, π k : P k V → X is a tower of projective bundles over X and carries a canonical line bundle O P k V (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on O P k V (−1) for k large enough. The direct images (π k ) ⋆ O P k V (m) can be viewed as bundles of algebraic differential operators of order k and degree m, acting on germs of curves and invariant under reparametrization. Following an approach initiated by Green and Griffiths, we establish a basic Ahlfors-Schwarz lemma in the situation when O P k V (−1) has a (possibly singular) metric of negative curvature, and we infer that every nonconstant entire curve f : C → V tangent to V must be contained in the base locus of the metric. This basic result is then used to obtain a proof of the Bloch theorem, according to which the Zariski closure of an entire curve in a complex torus is a translate of a subtorus. Our hope, supported by explicit Riemann-Roch calculations and other geometric considerations, is that the Semple bundle construction should be an efficient tool to calculate the base locus. Necessary or sufficient algebraic criteria for hyperbolicity are then obtained in terms of inequalities relating genera of algebraic curves drawn on the variety, and singularities of such curves. We finally describe some techniques introduced by Siu in value distribution theory, based on a use of meromorphic connections. These techniques have been developped later by Nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3-space; thanks to a suitable concept of "partial projective projection" and the associated Wronskian operators, substantial improvements on Nadel's degree estimate will be achieved here.

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

“…In his seminal paper [15], Demailly suggested a strategy to the Green-Griffiths conjecture through the investigation of the invariant jet differentials. These are sections of a bundle, whose fibres are canonically isomorphic to the invariant ring…”

“…• In the seminal paper [15] Demailly-using ideas of Green, Griffiths and Blochworks out a strategy for projective hypersurfaces.…”

“…The strategy of the proof [15,17,18,24,49]. Proving the algebraic degeneracy of holomorphic curves on X means finding a non zero polynomial P on X such that all entire curves f : C → X satisfy P( f (C)) = 0.…”

“…These polynomial functions are called jet differentials, and their use can be traced back to the work of Bloch [12], Cartan [13], Ahlfors [1], Green and Griffiths [24], Siu [49]. Their ideas were extended in the seminal paper of Demailly [15], by Diverio, Merker and Rousseau [17] and by the author [6].…”

Existence of global invariant jet differentials on projective hypersurfaces of high degree

“…This bundle reflects better the geometry of entire curves since it just takes care of the image of such curves and not of the way they are parametrized: following [Dem95], we will denote it E k,m T * X . The general philosophy is that global holomorphic sections of E k,m T * X vanishing on a fixed ample divisor give rise to global algebraic differential equations that every entire holomorphic curve must satisfy.…”