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Recent results on the Kobayashi and Green-Griffiths-Lang conjectures

  • Special Feature: The Takagi Lectures
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Japanese Journal of Mathematics Aims and scope

Abstract

The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory—especially through the concepts of curvature and positivity which are central themes in Kodaira’s contributions to mathematics. The aim of these lectures is to present recent results concerning the geometric side of the problem. The Green-Griffiths-Lang conjecture stipulates that for every projective variety X of general type over ℂ, there exists a proper algebraic subvariety Y of X containing all entire curves f : ℂ → X. Using the formalism of directed varieties and jet bundles, we show that this assertion holds true in case X satisfies a strong general type condition that is related to a certain jet-semi-stability property of the tangent bundle TX. It is possible to exploit similar techniques to investigate a famous conjecture of Shoshichi Kobayashi (1970), according to which a generic algebraic hypersurface of dimension n and of sufficiently large degree d ⩾ dn in the complex projective space ℙn+1 is hyperbolic: in the early 2000’s, Yum-Tong Siu proposed a strategy that led in 2015 to a proof based on a clever use of slanted vector fields on jet spaces, combined with Nevanlinna theory arguments. In 2016, the conjecture has been settled in a different way by Damian Brotbek, making a more direct use of Wronskian differential operators and associated multiplier ideals; shortly afterwards, Ya Deng showed how the proof could be modified to yield an explicit value of dn. We give here a short proof based on a substantial simplification of their ideas, producing a bound very similar to Deng’s original estimate, namely \({d_n} = \left\lfloor {{1 \over 3}{{(en)}^{2n + 2}}} \right\rfloor \).

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  1. Université de Grenoble-Alpes, Institut Fourier (Mathématiques) UMR 5582 du C.N.R.S., 100 rue des Maths, 38610, Gières, France

    Jean-Pierre Demailly

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Communicated by: Toshiyuki Kobayashi

Contribution to the 16th Takagi Lectures in celebration of the 100th anniversary of K. Kodaira’s birth

This article is based on the 16th Takagi Lectures that the author delivered at The University of Tokyo on November 28 and 29, 2015.

Work supported by the advanced ERC grant ALKAGE No. 670846 started in September 2015.

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Demailly, JP. Recent results on the Kobayashi and Green-Griffiths-Lang conjectures. Jpn. J. Math. 15, 1–120 (2020). https://doi.org/10.1007/s11537-019-1566-3

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  • DOI: https://doi.org/10.1007/s11537-019-1566-3

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