Any Complex Geometry

In this talk, I will discuss some recent work on positive scalar curvature Kähler (PSCK) metrics. I will first introduce a variational approach to PSCK metrics based on prescribing the scalar curvature measure. Based on this approach, I will explain the link between K-stability and PSCK metrics, and how quantitative data underlying K-stability also detect the existence of PSCK metrics in a given Kähler class.

Using Monge-Ampère equations to obtain auxiliary functions, Guo-Phong-Tong proved $L^\infty$ esimates for Monge-Ampère and Hessian equations on compact Kähler manifolds. These estimates are uniform in a bounded subset of the Kähler cone, provided certain cohomological constants - involving the volume - are bounded. This makes the estimates especially useful for dealing with degenerations to big and nef classes. However, if the limiting class is only nef and of numerical dimension $1<k<n$, the cohomological constant is not under control. In this case, we propose to adapt the technique using auxiliary Hessian equations. I will explain the modifications needed and some proof-of-concept results.

The effective Yau–Tian–Donaldson conjecture concerns the development of techniques for concretely testing the existence of canonical metrics on a Kähler manifold.
After briefly discussing some current strategies, we will present an approach of a “toric symplectic” nature for projective bundles over a curve. We will then show that, for this class of manifolds, the existence of an extremal metric is equivalent to a notion of stability for a certain moment polytope. We will illustrate this condition through classes of examples for which it can be explicitly verified, thereby obtaining new “Calabi dream manifolds”, that is, manifolds admitting an extremal metric in every Kähler class.

We establish sharp inequalities for two-dimensional systolic invariants of metrics with positive scalar curvature: the 2-systole and the spherical 2-systole of compact Kähler manifolds, and the stable 2-systole of Riemannian metrics on a general class of Spin-c manifolds and their products. These bounds attain equality precisely for complex projective space CPn equipped with the Fubini-Study metric. Moreover, we prove an algebraic characterization of manifolds admitting Kähler metrics with non-negative total scalar curvature, which implies a conjecture of Gromov for Kähler metrics. Finally, we obtain uniform bounds for the Gromov width, volume, and higher stable systoles of Kähler manifolds in terms of their scalar curvature.

Non-Archimedean (NA) geometry can be viewed as a degeneration of complex geometry. In recent years, it has found applications in the existence problem for canonical metrics via K-stability, particularly for Fano varieties. For Calabi--Yau varieties, while the existence problem on the projective ones is classical, much less is understood in the affine setting. In this talk, we will discuss some recent progress on this problem and focus on an NA perspective. In particular, we present a solution to a corresponding NA existence problem in certain cases using Berkovich spaces. We also explain how this NA solution relates to the original complex analytic problem.