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.