Algebraic and Kähler geometry

The goal of the ALKAGE project was to address a number of fundamental questions in algebraic and analytic geometry, in the following three directions (work packages):

1) Entire curves, hyperbolic algebraic manifolds and the Green-Griffiths-Lang conjecture

2) Further investigations on the curvature of complex varieties and Kähler geometry

3) L² vanishing and extension theorems, and the abundance conjecture

All three directions are connected with major aspects of contemporary research in complex geometry. There are active developments pursued in a large number of mathematical departments worldwide, and improvements of the related techniques are eagerly expected. In the long term, these questions might also have far reaching consequences in other domains of mathematics such as arithmetic geometry.

At the term of the project, the major scientific achievements include new L² extension theorems of Ohsawa-Takegoshi type that hold for non necessarily reduced subvarieties, and for line bundles satisfying optimal semipositivity conditions. This work of the PI conducted in 2015-2018 led to collaboration with S.Matsumura and JY.Cao and to important further contributions by collaborators and visitors (YoungJun Choi - Mario Chan, ChenYu Chi, Sheng Rao, …). In 2017, the remarkable PhD thesis of Ya Deng produced an effective version of Damian Brotbek’s recent solution of the Kobayashi conjecture (1970), stating that general algebraic hypersurfaces of high degree in complex projective space are Kobayashi hyperbolic. In 2018, the PI obtained a different and simpler proof of the same fundamental result, using geometric techniques for jet differential bundles that he continuously developed since 1995. In the context of non Kähler complex geometry, jointly with F. Campana and Th. Peternell, a complete proof was given that every complex structure on S6 has zero algebraic dimension (a result claimed already in 1998, but with an incomplete proof). In the direction of the abundance conjecture, the PI introduced the concept of “Bergman bundles”, which provide Hilbert bundle analogues of ample vector bundles in the non algebraic setting, on general compact complex manifolds. Finally, in 2020, a new approach was developed for the study of the Griffiths conjecture on the positivity of ample vector bundles. An outcome was the introduction of a new concept of Monge-Ampère volume for vector bundles, for which Siarhei Finski obtained deep asymptotic formulas.

The ALKAGE project mostly ran as expected, even though the COVID pandemic considerably slowed down some operations after February 2020, such as invitations and visits to other centers. In total, 7 PostDocs have been hired on ALKAGE grants during 2 year periods (one of them left after 1 year and a half, and another one after 1 year, as they were offered junior research positions abroad); 6 PhD students were supported, two of them by grants offered during the whole three years of their PhD research activity, and 5 PhD theses were defended before end of 2021.

During the period 2015-2021, very substantial work was conducted in the major research directions planned in the three work packages of the project described above. The PI, his PhD students, PostDocs and local collaborators all contributed to the activity by producing high level research papers, participating to conferences and delivering seminar talks or summer school lectures. The complete reference list includes about 80 publications. Numerous short expert visits (from one-two day visits to several months visits) to Institut Fourier have been supported. A substantial part of the budget has been used to support missions of the PI and his collaborators to various French and foreign institutions.

In total, four ALKAGE workshops of approximately 30 participants, and two bigger international conferences have been organized during the 6 years of the project. These events, each of a duration of one week, gave the occasion of many profitable exchanges, leading e.g. to a new collaboration of the PI with F. Campana, L. Darondeau and E. Rousseau. As mentioned in the previous section, the results that were obtained include important advances on several longstanding mathematical problems such as the Kobayashi conjecture from 1970.

The remarkable work conducted by Ya Deng during his PhD and by Siarhei Finski during his PostDoc allowed them to be both recruited by CNRS as Chargés de Recherche, in February and October 2021, respectively in Nancy and at École Polytechnique. All other younger collaborators of the ALKAGE project found junior academic positions after leaving Grenoble university: as assistant professors, YoungJun Choi at Pusan university (Korea), Tao Zheng at the Beijing Institute of Technology (China), Long Li at ShanghaiTech university (China), Philipp Naumann at Bayreuth Universität (Germany); Grégoire Menet, as ATER at Université de Lille; Juliana Restrepo, as PostDoc at Université Aix-Marseille.

One of the very long term goals of the ALKAGE related research would be a general classification theory of projective varieties and compact Kähler spaces. There is still a very long route to reach a complete understanding of the structure of such varieties. Among the many steps required, one can cite basic vanishing, non vanishing, and extension theorems, in connection with an analysis of fundamental fibrations or with the "curvature" properties of the tangent sheaf. The results obtained by the PI and his students, postdocs and collaborators are substantial steps in these directions.

Periodic Reporting for period 4 - ALKAGE (Algebraic and Kähler geometry)

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