Stability in Kähler Geometry

Much of complex geometry aims to explore the relationship between an algebro-geometric stability condition (K-stability) and the existence of canonical Kähler metrics (constant scalar curvature Kähler metrics); this is the Yau-Tian-Donaldson conjecture. I will describe how this problem is the “infinite-dimensional” analogue of a classical “finite-dimensional” problem, linking the existence of zeroes of moment maps (as used in symplectic geometry) to stability in the sense of geometric invariant theory. I will then give an interpretation of the scalar curvature as a moment map (originally due to Donaldson and Fujiki), and will explain how this is the main philosophical motivation behind the link with K-stability. The focus will be on ideas rather than proofs, and there will not really be any new results, but the philosophy behind this talk is largely based on recent joint work with Michael Hallam.

Following recent trends in K-stability and Kähler geometry an interesting question is the existence of optimal destabilizers. A second question of interest is to clarify the relationship between the new emerging variants of stability, especially uniform stability versus its original non-uniform counterparts. A third recently popularized question is the idea of exhibiting “chamber decompositions” for stability in Kähler geometry, inspired by Bridgeland stability. In this talk we will investigate and clarify many of these questions in the simplified context of the J-equation on surfaces, in particular demonstrating the existence of a finite number of destabilizing curves for the deformed Hermitian Yang-Mills (dHYM) and J-equations, providing effective existence criteria and analogs to Bridgeland chamber decompositions in these settings. We also characterize optimal destabilizers for the dHYM and J-equations, and clarify the relationship between J-stability and uniform J-stability. This is joint work with Sohaib Khalid, SISSA.

I will report on a recent joint work with Benoît Claudon and Patrick Graf. We show that given a compact klt pair (X,D) with standard coefficients such that K_X+D is ample or numerically trivial, then equality holds in the orbifold Miyaoka-Yau inequality for (X,D) if and only if (X,D) is uniformized by either the unit ball (ample case) or the affine space (flat case). Stability of the orbifold tangent bundle plays a key role in the proof.

Let (X,L) be a polarized manifold.  The cscK metrics in the first Chern class of L are critical points for T. Mabuchi’s K-energy map M. An important recent result of X.X. Chen and J.Cheng shows that cscK metrics exist iff M is proper. In 2021 the speaker showed that M is proper iff (X,L) is asymptotically stable. The stability condition is closely related to Mumford’s stability as well as Tian and Donaldson’s “K-(semi)stability”. The speaker will give a non-technical account of the many areas of mathematics involved in the proof, with an emphasis on the historical development of the subject. Also, he hopes to discuss the surprising connection to arithmetic geometry in the spirit of Arakelov, Faltings, Bismut-Gillet-Soule.

In this talk, we consider the problem of finding a canonical representative of a complexified Kähler class on a compact complex manifold, motivated by fundamental constructions in mirror symmetry. I will describe a new geometric PDE, obtained by coupling the deformed Hermitian Yang-Mills and the constant scalar curvature equations, whose solutions will give the required representative of the class. We will then consider a natural variational framework in which to study the equation, and prove the existence of solutions in some special cases. Based on joint work with Jacopo Stoppa.

Several years ago, Chi Li introduced the normalized volumes of valuations in his work on K-stability. The stable degeneration conjecture, due to Li and Xu, predicts that every klt singularity has a canonical “stable degeneration” induced by the minimizers of the normalized volume functions. I’ll talk about the recent solution of this conjecture, focusing on its connection to certain finite generation property of valuations. Based on joint work with Chenyang Xu.