Any Complex Geometry

Motivated by the Chen–Sun description of analytic tangent cones of Hermitian–Yang–Mills connections, we develop a valuation-theoretic framework for algebraic tangent cones of torsion-free sheaves. Replacing the blow-up valuation, which corresponds to the local behaviour of smooth Kähler metrics, by finitely generated valuations, we construct degenerations via Rees algebras and introduce a slope stability theory for the associated graded modules. We define an instability functional and prove the existence of an optimal degeneration for quasi-regular valuations. Moreover, we show that its Harder–Narasimhan graded object is uniquely determined up to grading twists.

Kähler quantization provides a bridge between infinite-dimensional geometric objects in Kähler geometry and finite-dimensional data arising from spaces of holomorphic sections. In this talk, I will first review this correspondence in the ample case, where it is well understood and plays a central role in the study of canonical metrics.

I will then explain how this picture can be extended beyond the ample setting, where smooth positively curved metrics are no longer available. In particular, I will describe how the Monge–Ampère energy can still be recovered from finite-dimensional approximations in the semipositive and big setting. Finally, if time permits, I will outline the idea of the proof.

The uniqueness of infinity plays a crucial role in understanding solutions to geometric PDEs and the geometric/topological properties of manifolds with Ricci curvature lower bounds. A major progress is made by Colding--Minicozzi who studied one type of infinity, called the asymptotic cones, of a Ricci flat manifold with Euclidean volume growth and proved the uniqueness of asymptotic cones if one cross section of the asymptotic cones is smooth. Their result generalizes an earlier uniqueness result by Cheeger—Tian which requires integrability of the cross section among other things. 

In this talk I will talk about some recent results on the study of the similar uniqueness problem of another type of infinity called the asymptotic limit spaces for Ricci flat manifold with linear volume growth, following the paths of Cheeger--Tian and Colding--Minicozzi for cones. Here one considers translation limits instead of rescaling limits and the limits are no longer cones but cylinders. I will highlight the similarities and differences between the two settings. This is joint work with Zetian Yan.

For a given family of Kähler–Einstein manifolds, understanding the structure of their degenerations from both differential-geometric and algebro-geometric perspectives is a fundamental problem.

In the case of K3 surfaces—two-dimensional Calabi–Yau manifolds—it is a classical result that non-collapsing Gromov–Hausdorff limits correspond precisely to the convergence of the points in the moduli space.

In this talk, based on arXiv:2512.16320, I will describe how the bubbling phenomena arising in non-collapsing degenerations can be characterized in terms of explicit algebro-geometric data of the family.

Pseudo Calabi Flow is a gradient flow of K-energy respect to an infinite dimension Riemannian structure on metrics moduli space. In this talk, we briefly introduce a C^0 stability result which is a generalization of work done by Xiuxiong Chen and Kai Zheng. The main idea of this improvement is a new estimate on a system of PDEs. This is a joint work with Jingrui Cheng.

In this talk, we first review recent progress on the positive mass theorem (PMT) for asymptotically flat (AF) manifolds, which is fundamental in scalar curvature geometry and general relativity. Then, we would introduce the corresponding statements and progress for asymptotically hyperbolic (AH) manifolds. By the concept of initial data sets, one can prove AH PMT with arbitrary ends. Part of the discussion is joint work with Prof. Greg Galloway. 

In recent years, much work has been devoted to extending the Ricci flow theory to singular initial data and to understanding the geometry of the resulting flow near singular points. In this talk, we will consider Ricci flows whose initial data is a Kähler space with isolated conical singularities. We will see how expanding solitons naturally arise as models for the flow near the singular set, and discuss the relationship between our solutions and the Kähler–Ricci flow through singularities constructed by Song and Tian. This is joint work with Longteng Chen and Max Hallgren.

As a first step towards a refined description of the asymptotic of the Weil-Petersson metric on the moduli space of polarized Calabi-Yau manifolds we investigate the concrete case of abelian varieties by linking such asymptotic with the multi-scale collapsing limits of the parametrized flat tori, as explicitly classified by Odaka. This is a joint work with Yanbo Fang.

On a compact Kähler manifold X, a Kähler-Ricci flow (KRF) is immortal when the canonical bundle of X is numerically effective. In this case, assuming the abundance conjecture and intermediate Kodaira dimension, the collapsing behavior of the normalized KRF, as already known, poses difficulty on uniform curvature estimates. We extend known C^0 estimates on the scalar curvature (by Song-Tian) and Ricci curvature (recently by Hein-Lee-Tosatti) to orders up to 2, and explain the failure for higher orders in general.