Roughly speaking, an ALF metric of real dimension 2n should be a complete metric such that its asymptotic cone is of dimension 2n−1, the volume growth of this metric is of the order of 2n − 1 and its sectional curvature tends towards 0 near the infinity. I will give examples of ALF Calabi-Yau metrics of real dimension greater than 4. Our first example is that the Taub-NUT deformation of a hyperkählerian cone with respect to a locally free circle action is hyperkählerian ALF. The second example is that a special class of complete Calabi-Yau metric on C^n constructed by Apostolov and Cifarelli is ALF. Based on these examples, I will explain how to produce more ALF Calabi-Yau metrics on some resolutions of known examples modeled on them. In particular, there exist ALF Calabi-Yau metrics on the canonical bundles of classical homogeneous Fano contact manifolds.
We classify all possible ends of Hermitian non-Kahler gravitational instantons. Our main results build on an observation for the SU(infinity) Toda equation and a careful analysis of the collapsing geometry at infinity.
In 2019 Conlon--Deruelle--Sun [CDS24] proved that gradient Kähler--Ricci shrinkers (M, X, ω) with quadratic curvature decay are asymptotic to a Kähler cone, and M is a resolution of the algebraic variety underlying the cone. Using the techniques from [Ber15, BWN14] we construct geodesics in the space of Kähler potentials and use them show that two different shrinkers (M, X, ω₁), (M, X, ω₂) with the same soliton vector field must be equal up to biholomorphism. This generalizes the main result of [TZ00] to the case of shrinkers with quadratic curvature decay.
In this talk, I'll discuss a possible non-Archimedean approach to solving the Yau–Tian–Donaldson conjecture. I will give a brief idea of how to make a step towards implementing this approach, generalizing to the transcendantal setting a result of Chi Li.
Any complex submanifold of a Kahler manifold is stableminimal. Micallef previously showed that the converse is true in the genus zero case via the Poincare-Birkhoff-Grothendieck theorem, but Fraser-Schoen recently extended these results to the genus 1 case using Atiyah's classification. I'll try to explain the Fraser-Schoen result at the least.
In this talk, I will present my works on general inverse $\sigma_k$ equations in Kähler geometry. Some classical examples are the complex Monge–Ampère equation, the J-equation, the complex Hessian equation, and the deformed Hermitian–Yang–Mills equation.
In this talk, I will present my works on general inverse $\sigma_k$ equations in Kähler geometry. Some classical examples are the complex Monge–Ampère equation, the J-equation, the complex Hessian equation, and the deformed Hermitian–Yang–Mills equation.
I will present a recent result by Cho and Choi. They prove that on compact normal Kähler spaces, the solutions to complex Monge-Ampère equations with Lp densities are continuous. Moreover, the result can be applied to obtain continuous approximations of psh functions in the singular setting.
In this talk, we'll discuss a complex version of Alexandrov-Bakelman-Pucci's maximum principle.
Homogeneous complex Monge-Ampere (HCMA) equations become a central topic in understanding uniqueness and existence of canonical metrics in Kähler classes. Under the setting of ALE Kähler manifolds, one of the main difficulties is to understand the asymptotic behaviors of solutions to HCMA equations. In this talk, I will give an introduction to canonical metric problems under the setting of ALE Kähler manifolds. I will present a new result on the asymptotic behavior of HCMA solutions and outline the proof of the result.
I will present the work of Naber-Valtorta-Edelen on rectifiable Reigenberg theorem. The key ingredient is the study of neck regions, which also appeared for degeneration of Einstein metrics or Yang-Mills connections.
We carry out a gluing construction for collapsing warped-QAC Calabi-Yau manifolds in $\mathbb{C}^{n+2}, n\geq 2$. This gluing theorem verifies a conjecture by Yang Li on the behavior of the warped QAC Calabi-Yau metrics on affine quadrics when two singular fibers of a holomorphic fibration go apart. We will also discuss a bubble tree structure for those metrics.
We prove the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold that satisfy a determinant domination condition. This viscosity solution is shown to be unique when the right hand is strictly monotone increasing in terms of the solution. When the right hand side does not depend on the solution, we reduces it to the strict monotonicity of the solvability constant.
In this talk, we describe a gluing constructions of families of Ricci-flat Kähler metrics on crepant resolutions and on polarized smoothings of Calabi-Yau manifolds with isolated conical singularities as obtained by Hein-Sun and we obtain asymptotic expansions in terms of the parameters of degeneration. This work is part of my PhD thesis.
We prove parabolic versions of several known gap theorems in classical Yang-Mills theory. This is joint work with Alex Waldron.
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.
We begin with an elliptic K3 surface equipped with a parabolic automorphism $T$ preserving the fibration over $\mathbb{P}^1$. A natural object associated with this dynamical system is the limiting current generated by iterating $T$, often called a canonical current or Green current.
The aim of the talk is to relate the potential of this canonical current to an Archimedean height pairing for fiberwise cohomologically trivial differential forms. Using recent analytic results of Dai–Yoshikawa on the asymptotics of small eigenvalues, we establish the precise asymptotic behavior of this pairing near singular fibers and derive regularity results for the potential of the canonical current. Finally, we discuss a new approach to small-eigenvalue asymptotics based on auxiliary Monge–Ampère equations, which extends the Dai–Yoshikawa picture to higher-dimensional Kähler degenerations.