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Song Sun: Lectures

March 16, 2023
TITLE: Singularity formations in Kahler geometry

ABSTRACT: When a sequence of Kahler-Einstein metrics develop a non-collapsing singularity in the Gromov-Hausdorff sense, it has been known that under natural assumptions the singularity is algebro-geometric. To study the singular behavior of the metric the 1st order approximation is given by tangent cones, which are also known to be algebro-geometric. To get more dynamic information on the singularity formation process one is lead to study rescaled limits, which we call “bubbles”. They are non-compact complete Calabi-Yau spaces associated to the degeneration. I will review the above backgrounds and discuss some new results on these bubbles, which also generate questions related to algebraic geometry and Riemannian geometry.

September 9, 2022
TITLE: Gravitational instantons and del Pezzo surfaces

ABSTRACT: I will talk about the classification of gravitational instantons of type ALH* in terms of weak del Pezzo surfaces. This is based on joint work with Hein, Viaclovsky, and Zhang, arXiv 2111.09287.

Slides of Lecture

September 12, 2021
TITLE: Collapsing geometry of 4 dimensional hyperkahler manifolds, Part 2

ABSTRACT: This is a continuation of the talk by Ruobing Zhang at the Simons Foundation annual meeting in the previous week. Our work is about understanding the structure of lower dimensional Gromov-Hausdorff limits of 4 dimensional hyperkahler manifolds, with applications to the moduli compactification of K3 metrics and classification of gravitational instantons. We will explain some technical aspects of our work, in particular the role played by the hyperkahler equation and the Cheeger-Fukaya-Gromov collapsing theory (in the 1980/90s) in Riemannian geometry .

January 7, 2020
TITLE: Singular Hermitian-Yang-Mills connections and reflexive sheaves

ABSTRACT: In higher dimensional gauge theory it is important to understand essential singularities of finite energy Yang-Mills instantons. I will discuss joint work with Xuemiao Chen (University of Maryland), which studies the model case of Hermitian-Yang-Mills connections, and the results say that the singularities have unique tangent cones which can be characterized in terms of certain local algebro-geometric invariants of reflexive sheaves.

April 9, 2019
TITLE: Small complex structure limit of Calabi-Yau hypersurfaces

ABSTRACT: This will be a synopsis of the talks I gave on the previous two meetings. The main purpose is to explain a relatively complete result for understanding the metric collapsing of a generic splitting of a family of smooth Calabi-Yau hypersurfaces into the union of two Fano hypersurfaces.

January 8, 2019 and January 10, 2019
TITLE: Degeneration of Calabi-Yau metrics under complex structure degenerations

ABSTRACT: We will describe some progress towards understanding complex structure limit of Calabi-Yau metrics via glueing constructions. The main ingredient involves constructing Calabi-Yau metrics with torus symmetry (with fixed loci), by studying the dimension reduced equation (i.e. Gibbons-Hawking ansatz and its non-linear generalization due to Matessi) and Dirac type solutions to the linearized equation. Based on arXiv:1807.09367 (with Hein, Viaclovsky, Zhang) and more recent results and work in progress (with Zhang).

September 13, 2018
TITLE: Collapsing of hyperkahler metrics on K3 surfaces

ABSTRACT: Hyperkahler metrics on K3 surfaces are prototypical examples of compact Ricci-flat metrics with special holonomy. I will explain some known results in this field and describe a new gluing construction, joint with Hans-Joachim Hein, Jeff Viaclovsky and Ruobing Zhang, of a family of hyperkahler metrics on K3 surfaces with multiscale collapsing phenomenon.

September 13, 2017
TITLE: Singularities of Hermitian-Yang-Mills connections

ABSTRACT: In this talk we discuss how to understand the tangent cones of Hermitian-Yang-Mills connections in terms of the local algebraic data on the underlying reflexive sheaf. (Joint work with Xuemiao Chen)

January 10, 2017 and January 12, 2017
TITLE: Isolated conical singularities of Calabi-Yau varieties

ABSTRACT: I will talk about joint work with Hans-Joachim Hein on the conical behavior of Ricci-flat metrics on Calabi-Yau varieties with certain types of isolated singularities. Interesting examples include hypersurfaces with nodal singularities. In the more technical part I will try to sketch the main steps involved in the proof.
January 10:

January 12:

September 9, 2016
TITLE: Degeneration of Calabi-Yau metrics

ABSTRACT: The complex structure moduli space of Calabi-Yau manifolds can be compactified using the Gromov-Haudorff topology, and a central question is to understand the structure of these Gromov-Hausdorff limits. We will focus on the “non-collapsing” case, and explain the connection with algebraic geometry (joint work with S. Donaldson), and the recent example of a compact Calabi-Yau manifold with isolated conical singularities (joint work with H. Hein). A well-known application of the latter is the existence of special Lagrangian spheres on the smoothing of nodal Calabi-Yau varieties.