ERC ALKAGE: description of the project

The purpose of the ALKAGE project is to study basic questions in algebraic and Kähler geometry. It is well known that the structure of projective or Kähler 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, J.-P. Demailly has developed a large number of key concepts and results that have led to important progress in transcendental algebraic geometry. Let us mention the discovery of holomorphic Morse inequalities, systematic applications of L² estimates with singular hermitian metrics, and a much improved understanding of Monge-Ampère equations and of singularities of plurisuharmonic functions. The first goal of the project 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 made in 2010, showing that all such entire curves must satisfy algebraic differential equations. A second closely related area of research of the project is the analysis of the structure of projective or compact Kähler manifolds. The ultimate goal would be a generalization of the classification theory of surfaces by Kodaira, and of the more recent results for dimension 3 (Kawamata, Kollár, Mori, Shokurov, ...), to higher dimensions. The general 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, it should be possible to go much further and to enhance the national and international collaborations. These subjects already attract many young researchers and postdocs throughout the world, and the ERC grant will be used to create even stronger interactions.





A post-doctoral grant at Institut Fourier - Université Grenoble Alpes, supported by the ALKAGE program, will be available in 2018-2019, with a possibility of renewal during the next year 2019-2020. Applicants are expected to work in the general area of the program, namely complex algebraic and analytic geometry. Institut Fourier has a strong research team and runs a regular seminar in this field. More specifically, ALKAGE focuses on various aspects of Kähler geometry, on structure theorems for projective and compact Kähler manifolds, on the study of entire curves drawn in projective or quasi-projective algebraic varieties. The applicants should send a curriculum vitae, a letter of motivation explaining their research projects, and all usual documents (passport or ID document, diploms, recommendation letters) to The deadline for the application has been set on March 31, 2018. The Post-Doc position will normally start on September 1 or October 1, 2018.







ALKAGE workshop, May 14-18, 2018 at Institut Fourier

A one week workshop will be organized from May 14 to May 18 at Institut Fourier. The general themes covered include: vanishing theorems, structure of algebraic varieties, Kähler geometry, special metrics, Monge-Ampère equations and pluripotential theory, entire functions, holomorphic foliations.







Dr Young-Jun Choi, a former student of Professor Kang-Tae Kim and a former researcher at KIAS in Seoul, has joined Institut Fourier as a PostDoc supported by the ERC ALKAGE on February 1, 2016. His recent work is concerned with Kähler geometry, especially the study of variations of pseudoconvex domains via Kähler-Einstein metrics, and the semipositivity of fiberwise Ricci flat Kähler metrics on Calabi-Yau fibrations.




Long Li

Long Li got his PhD at SUNY at Stony Brook with Professor Xiuxion Chen as advisor in 2014. His PhD thesis was concerned with proofs of the Bando-Mabuchi’s uniqueness theorem of Kähler Einstein metrics on Fano manifolds from a new perspective, based on the convexity of Ding-functional. Since then, he has been a post-doctor at McMaster University. His main interest are in several complex variables, Kähler geometry, pluripotential thoery. He has written several expert papers on each of these subjects. He arrived at Institut Fourier in September 2016.




Tao Zheng

Tao Zheng got his PhD in 2013 at the Academy of Mathematics and systems science (Academy of Sciences, Beijing), under the supervision of Professor Hongcang Yang. His main domain of research is (complex) differential geometry, with contributions in the study of eigenvalue estimates, Chern-Ricci and harmonic-Ricci flow, isoperimetric inequalities. He arrived at Institut Fourier in September 2016.




Philipp Naumann

Philipp Naumann defended his PhD in 2016 at Philipps-Universität Marburg, in the area of complex differential geometry, under the supervision of Professor Georg Schumacher. His work is concerned with Kähler geometry, especially the calculation of the curvature of higher direct images, and the study of the twisted Hodge filtration (curvature of the determinant). He joined Institut Fourier in September 2017.



Grégoire Menet

Dr Grégoire Menet is a former student of Professor Dimitri Markoushevich at Lille and defended his PhD in July 2014, entitled "Integral cohomology and lagrangian fibrations on certain singular holomorphic symplectic manifolds". He will bring advanced expertise in Hodge theory and various other aspects of complex geometry such as mirror symmetry and holomorphic symplectic manifolds. Grégoire Menet has been hired in April 2018 and will join Institut Fourier on September 1st, 2018.



Tim Kirschner

Tim Kirschner  defended in 2012 a PhD thesis entitled "Irreducible symplectic complex spaces" at the University of Bayreuth, under the supervision of Thomas Peternell. His work concerns compact Kähler manifolds with zero first Chern class and compact complex spaces that share an analogous property. The techniques combine certain differential geometric tools such as holonomy, with general methods of complex geometry, deformation theory, etc. He is also interested in understanding Lagrangian fibrations in the general context of holomorphic symplectic manifolds. Tirm Kirschner was hired in April 2018 and will join Institut Fourier in January 2019.


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