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ALKAGE: Algebraic and KEnhler geometry.

EU contribution: EUR 1 809 345

Objective: The purpose of this project is to study basic questions in algebraic and KEnhler geometry. It is well known that the structure of projective or KEnhler manifolds is governed by positivity or negativity properties of the curvature tensor. However, many fundamental problems are still wide open. Since the mid 1980~s, I have developed a large number of key concepts and results that have led to important progress in transcendental algebraic geometry. Let me mention the discovery of holomorphic Morse inequalities, systematic applications of LA estimates with singular hermitian metrics, and a much improved understanding of Monge-Ampere equations and of singularities of plurisuharmonic functions. My first goal will be to investigate the Green-Griffiths-Lang conjecture asserting that an entire curve drawn in a variety of general type is algebraically degenerate. The subject is intimately related to important questions concerning Diophantine equations, especially higher dimensional generalizations of Faltings~ theorem - the so-called Vojta program. One can rely here on a breakthrough I made in 2010, showing that all such entire curves must satisfy algebraic differential equations. A second closely related area of research of this project is the analysis of the structure of projective or compact KEnhler manifolds. It can be seen as a generalization of the classification theory of surfaces by Kodaira, and of the more recent results for dimension 3 (Kawamata, KollEir, Mori, Shokurov, ...) to other dimensions. My plan is to combine powerful recent results obtained on the duality of positive cohomology cones with an analysis of the instability of the tangent bundle, i.e. of the Harder-Narasimhan filtration. On these ground-breaking questions, I intend to go much further and to enhance my national and international collaborations. These subjects already attract many young researchers and postdocs throughout the world, and the grant could be used to create even stronger interactions.

Project completion date : 2020-09-01 12:00:00

Major organization : UNIVERSITE JOSEPH FOURIER GRENOBLE 1

Address : BP 53, 38041 Grenoble

cedex 9

Country :France

Url : https://www.ujf-grenoble.fr/

Financier : European Union (EU),

Financier address : European Union (EU)

Rue de la Loi 200/Wetstraat 200,

B-1049 Bruxelles/Brussels,

Belgium

Tel: 32-2-2999696, 2993085

Fax: 32-2-2961749

Url: ec.europa.eu/

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Publication:Mena Report
Date:Jul 16, 2015
Words:375
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