Real Monge-Ampère and Kähler geometry of homogeneous spaces – MARGE

The project MARGE is a project in fundamental Mathematics on the existence of canonical metrics, a central problem in complex geometry. The quest for canonical metrics on complex manifolds is a natural generalization of the metric aspects of Riemann's uniformization theorem. Major breakthroughs in the field were always preceded or accompanied with examples providing key insight or illustrating behaviors leading to further research.

In the case of Kähler-Einstein metrics, as in the case of Calabi's extremal metrics, varieties equipped with a complex Lie group action have played a key role. In seminal works of Calabi, Koiso-Sakane, etc, cohomogeneity one varieties such as Hirzebruch surfaces, whose geometry is read off from one-variable functions, provided the first non-trivial examples. In decisive work of Donaldson, Wang-Zhu, etc, the study of toric varieties allowed to prove effective characterizations of existence or non-existence of canonical metrics.

The project’s first goal is to explore a new generation of examples, almost-homogeneous manifolds, to understand their Kähler geometry, to build new explicit canonical metrics and to exhibit new pathological behaviors. From the initial steps in that direction, it appears clearly that the relation between complex and real Monge-Ampère equations is a crucial aspect of this, hence our second goal is to study such relations.

Previous work of the coordinator, Thibaut Delcroix, on the Kähler and complex algebraic geometry of multiplicity free varieties, containing both cohomogeneity one manifolds and toric varieties, form the foundations of the project. The tasks to be accomplished during the project are diverse: to push further the understanding of these multiplicity free varieties to prove an effective Yau-Tian-Donaldson conjecture on such varieties, to construct new Calabi-Yau metrics on non-compact homogeneous spaces, derive a geometric interpretation of the lack of Kähler-Einstein or constant scalar curvature Kähler metrics via geometric flows, obtain a construction of non-abelian Calabi ansatz and relate notions of stability of a variety equipped with a group action to its quotient, transfer recent techniques for the study of degenerate complex Monge-Ampère equations to their real analogues with potential applications to the SYZ conjecture, and finally participate in the development of non-Kähler geometry thanks to an approach by examples.

Une part importante du financement sera dédiée au recrutement de postdocs et de stagiaires de Master, qui viendront renforcer l’équipe initialement composée de cinq jeunes chercheurs aux talents complémentaires en géométrie différentielle et algébrique complexe, géométrie symplectique, analyse géométrique et théorie de Lie. Des rencontres régulières et une conférence internationale seront également organisées.

A major part of the funding will be used to recruit postdocs and Master students to reinforce the team, initially comprised of five young researchers with complementary skills in differential and algebraic complex geometry, symplectic geometry, geometric analysis and Lie theory. Regular meetings and an international conference will also be organized.