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Gromov-Hausdorff convergence in Kähler geometry – GRACK

Gromov-Hausdorff limits in Kähler geometry: from Riemannian geometry to algebraic and non-Archimedean geometry.

Over the last few years, Kähler geometry has known spectacular progress, bringing together algebraic geometry, non-linear partial differential equations and Riemannian geometry on possibly singular spaces. The present project, gathering recognized experts covering this large spectrum of mathematics, aims to contribute to the ensuing burst of research activity by attacking several major problems in the field.

Existence, limits and degenerations of Kähler-Einstein metrics: the Yau-Tian-Donaldson and Kontsevich-Soibelman conjectures and the Song-Tian program.

The Calabi problem seeks to describe compact complex manifolds admitting a Kähler-Einstein metric. While the answer has long been understood in the case of negative or zero curvature, the case of positive curvature (Fano manifolds) turned out to considerably more complicated. It gave rise to the Yau-Tian-Donaldson conjecture, which relates the existence of a Kähler-Einstein metric on a Fano manifold to the purely algebro-geometric condition of K-stability. <br />The spectacular resolution of this conjecture by Chen-Donaldson-Sun, and the deep and difficult techniques involved, have opened a new era in the subject, with numerous first class results appearing in the last few years. The initial impetus for this project was the desire to create a French group of recognized experts covering the vast mathematical spectrum involved: Kähler geometry (Kähler-Einstein metrics and constant scalar curvature Kähler metrics), Riemannian geometry and global analysis (Gromov-Hausdorff convergence and Ricci flow), pluripotential theory (plurisubharmonic functions and degenerate complex Monge-Ampère equations), algebraic geometry (Minimal Model Program and Geometric Invariant Theory), and, more recently, non-Archimedean geometry (pluripotential theory on Berkovich spaces). <br />Singular (log terminal) Fano varieties appear naturally as metric and algebro-geometric limits, and an essential goal of the project is to extend to this setting the Yau-Tian-Donaldson conjecture. A second central objective is to contribute to the Kontsevich-Soibelman conjecture, which deals with the limit behavior as metric spaces of a maximal degeneration of polarized Calabi-Yau manifolds. Finally, we hope to make progress on the Song-Tian program, which makes a remarkable connection between the Kähler-Ricci flow and the Minimal Model Program.

The core of the project’s activities consists in organizing workshops and schools where the latest key developments of the field are presented. This is done through mini-courses by group members as well as invited international experts, which enable us to reach a global expertise, quickly attack more reasonable problems, and try to make progress in the direction of the major conjectures we aim at.
One initial motivation for the project was to analyze in detail the resolution of the Yau-Tian-Donaldson conjecture by Chen-Donaldson-Sun, a mathematical tour de force involving a wide range of techniques, and building in particular in a crucial way on the Cheeger-Colding theory describing Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded below. This approach was further exploited by Datar-Szekelyhidi to obtain an equivariant version of the conjecture with respect to a reductive group action, which proved to be essential in concretely testing K-stability and thereby obtaining new examples of Kähler-Einstein Fano manifolds.
While the detailed study of these works remains of course an essential aspect of the project, new perspectives have appeared in the Fall of 2015, when the project coordinator, Sébastien Boucksom, and his collaborators Robert Berman and Mattias Jonsson, have obtained a new proof of the Yau-Tian-Donaldson conjecture, relying on a totally different variational approach in which Gromov-Hausdorff limits are replaced by non-Archimedean Berkovich spaces. This new flexible approach appears to lead to generalizations more easily, in particular to the case of singular varieties, and is naturally at the heart of the current research activity of some of the group members.

The main result obtained so far is a new proof of the Yau-Tian-Donaldson conjecture by the project coordinator, Sébastien Boucksom, and his collaborators Robert Berman and Mattias Jonsson. Relying on a variational approach and a non-Archimedean interpretation of K-stabililty, this new approach is significantly simpler than the one by Chen-Donaldson-Sun, and naturally leads to various generalizations.
Boucksom and Jonsson have also progressed towards the Kontsevich-Soibelman conjecture. For a maximal degeneration of polarized Calabi-Yau manifolds, this conjecture describes the limit in the sense of metric spaces as a tropicalization of the family, realized as the essential skeleton of the associated Berkovich space. Thanks to the construction of a “hybrid” compactification putting together the complex family and its Berkovich space, Boucksom and Jonsson have managed to confirm the convergence at the level of the associated volume forms.
In a different direction, Junyan Cao, on his own and in joint work with Mihai Paun, has obtained new remarkable results on the structure of manifolds with nef anticanonical bundle. This class of manifolds, an algebro-geometric version of those with semipositive Ricci curvature, is the object of an important conjecture of Campana-Demailly-Peternell, reducing their study to the the case of rationally connected and Calabi-Yau manifolds. Cao has very recently established the local triviality of the Albanese map of any manifold with nef anticanonical bundle, thereby reducing the conjecture to the case of simply connected manifolds.
Finally, Philippe Eyssidieux and Vincent Guedj, together with their collaborator Ahmed Zeriahi, have made progress in the understanding of the long time behavior of the Kähler-Ricci flow, by generalizing to the case of minimal models with log terminal singularities an important previous result of Song-Tian.

The new proof of the Yau-Tian-Donaldson conjecture by Berman-Boucksom-Jonsson opens many new perspectives. In particular, the interpretation of K-stability in terms of non-Archimedean geometry enables to import in this context pluripotential theory on Berkovich spaces and the non-Archimedean version of the Calabi conjecture previously established by Boucksom-Favre-Jonsson. The extension of the new approach to the case of “twisted” Kähler-Einstein metrics and to the equivariant case with respect to a reductive group action seem to be within reach, and are the object of intense research activity by some members of the group. Gromov-Hausdorff limits of Kähler-Einstein manifolds nevertheless remain a central theme in the project, in particular through the Kontsevich-Soibelman conjecture, for which it can be hoped that the new point of view developped by Boucksom-Jonsson will lead to progress in the near future.

S.Boucksom, M.Jonsson. Tropical and non-Archimedean limits of degenerating families of volume forms. J. Ecole Polytechnique 4 (2017), 87-139

S.Boucksom, T.Hisamoto, M.Jonsson. Uniform K-stability and asymptotics of energy functionals in Kähler geometry. Preprint arXiv:1603.01026, to appear in J. Eur. Math. Soc.

C.Boyer, H.Huang, E.Legendre, C.W.Tønnesen-Friedman. The Einstein-Hilbert functional and the Sasaki-Futaki invariant. IMRN 2016.

J.Cao. Albanese maps of projective manifolds with nef anticanonical bundles. Preprint arXiv:1612.05921.

J.Cao, J.P.Demailly, S.I.Matsumura. A general extension theorem for cohomology classes on non reduced analytic spaces. Preprint arXiv:1703.00292.

J.Cao, M.Paun. Kodaira dimension of algebraic fiber spaces over abelian varieties. Invent. Math. 207 (2017), no. 1, 345-387.

G.Carron. Geometric inequalities for manifolds with Ricci curvature in the Kato class. Preprint arXiv:1612.03027.

S.Diverio, S.Trapani. Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle. Preprint arXiv:1606.01381.

P.Eyssidieux, V.Guedj, A.Zeriahi. Weak solutions to degenerate complex Monge-Ampère flows II. Adv. Math. 293 (2016), 37-80.

P.Eyssidieux, V.Guedj, A.Zeriahi. Convergence of weak Kähler-Ricci flows on minimal models of positive Kodaira dimension. Preprint arxiv 1604.07001.

V.Guedj, S.Sahin, A.Zeriahi. Choquet-Monge-Ampère Classes. To appear in Potential Analysis.

The present projet lies at the cross-roads of complex algebraic geometry and Riemannian geometry, the unifying theme being the study of limits of Kähler manifolds.

Kähler geometry has recently known dramatic progress through the resolution, in an important special case (the "anticanonically polarized" case), of the Yau-Tian-Donaldson conjecture, generally regarded as one of the most important conjectures in the field.

The diversity of techniques involved, extending across algebraic geometry, complex analysis, fully non linear partial differential equations, pluripotential theory, and the Riemannian geometry of possibly singular spaces, makes it a very challenging task to understand all aspects of the proof. It is at the same time clear that the new methods introduced are bound to play a major role in the forthcoming developments of Kähler geometry, as illustrated internationally by numerous meetings and worskhops dedicated to these results.

It is therefore of crucial importance and most timely to gather a French group of experts in the different fields involved, in order to study in detail and develop the consequences of this proof. This is one of the main goals of the present project.

The above result belongs to the general problem of comparing algebro-geometric compactifications of moduli spaces to their Riemannian analogues, where polarized varieties are endowed with "canonical" Kähler metrics. Many recent and exciting developments also pertain to this general question. Song and Tian's program for the Kähler-Ricci flow, which involves the latest technology of the Minimal Model Program in birational geometry, asks for a deepened understanding of the Gromov-Hausdorff convergence both when the flow encounters a finite time singularitiy and in the long time limit on a minimal model. In a different vein, the Ströminger-Yau-Zaslow conjecture, inspired from Mirror Symmetry, led Kontsevich-Soibelman and Gross-Siebert to make a precise conjectural description of Gromov-Hausdorff limits of maximally degenerating families of Calabi-Yau manifolds.

We propose to investigate these problems, whose common feature is to require a deep understanding of techniques belonging to Riemannian geometry on the one hand and complex geometry on the other hand. Our team consists of mathematicians belonging to these two communities, who are eager to learn from each other in order to develop fruitful collaborations.

Project coordination

Sebastien Boucksom (Centre de mathématiques Laurent Schwartz)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.


IMT-UPS Institut de Mathématiques de Toulouse - Université Toulouse III Paul Sabatier
CMLS Centre de mathématiques Laurent Schwartz

Help of the ANR 234,125 euros
Beginning and duration of the scientific project: September 2015 - 48 Months

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