Schedule & Abstract

Schedule

August 27 (Tuesday)

10:00 - 11:00

Guolei Zhong (Center for Complex Geometry, IBS)

Title: Periodic points for meromorphic self-maps of Fujiki varieties

11:20 - 12:20

Tuen Wai Ng (The University of Hong Kong)

Title: Brody curves on the degree six Fermat surface

12:20 - 14:00

Lunch

14:00 - 15:00

Ye-Won Luke Cho (Gyeongsang National University)

Title: Introduction to singular Kähler-Einstein metrics

15:20 - 16:20

Zhining Liu (Center for Complex Geometry, IBS)

Title: Some recent results on the global topology of singular spaces

16:40 - 17:10

Hyunsoo Ahn (KAIST, Student speaker)

Title: Convergence speed for Fekete points on uniformly polynomially cuspidal sets

August 28 (Wednesday)

10:00 - 11:00

Hoseob Seo (Center for Complex Geometry, IBS)

Title: On approximations of toric plurisubharmonic functions and currents

11:20 - 12:20

Ngoc Cuong Nguyen (KAIST)

Title: Complex Hessian measures with respect to a background Hermitian form

12:20 - 14:00

Lunch

Abstracts

Hyunsoo Ahn (KAIST, Student speaker)

Title: Convergence speed for Fekete points on uniformly polynomially cuspidal sets

Abstract: We obtain the convergence speed for Fekete measures of compact uniformly polynomially cuspidal sets to the equilibrium measures associated to those sets. Here, uniformly polynomially cuspidal sets are introduced by Pawłucki and Pleśniak. This is done by showing that these sets are (\mathscr{C}^{\alpha}, \mathscr{C}^{\alpha'})-regular in the sense of Dinh, Ma and Nguyen. This is a joint work with Ngoc Cuong Nguyen.

Ye-Won Luke Cho (Gyeongsang National University)

Title: Introduction to singular Kähler-Einstein metrics

Abstract: In 2009, Eyssidieux-Guedj-Zeriahi constructed singular Kähler-Einstein metrics on a compact klt pair with negative or trivial first Chern class, generalizing the works of Aubin and Yau. Their work was largely motivated by the Minimal Model Program developed by Birkar-Cascini-Hacon-Mckernan, and it led to the existence of the SKE metric of negative scalar curvature on the canonical model of a smooth projective variety of general type. Later, the notion of the SKE metric also found some applications in the uniformization theory of complex varieties. In this talk, I shall summarize the construction of SKE metrics and my recent work with Y.-J. Choi on the regularity of SKE potentials.

Zhining Liu (Center for Complex Geometry, IBS)

Title: Some recent results on the global topology of singular spaces

Abstract: Given a germ (X,x), its local singularities dictate its local fundamental group \pi_1^{\mathrm{reg}}(X,x). In the global counter part, it is expected that for a compact singular space with mild singularities the positivity of its anti-canonical class controls its global (orbifold) fundamental group. L. Braun's theorems on the finitenss of local fundamental groups of germs of klt singularities and the finiteness of orbifold fundamental groups of klt weak Fano varieites are emblematic examples of the aforementioned principals. In this talk, I will present two recent results on the orbifold fundamental groups of singular spaces: a) The orbifold fundamental group of a projective varieties with quotient singularities and with nef anti-canonical class is virtually nilpotent; b) The orbifold fundamental group of a log Calabi-Yau surface is virtually metabelian. These results are obtained via methods of quite different natures. Part of the talk is based on a joint work with C. Gachet and J. Moraga and the other part a paper by myself.

Tuen Wai Ng (The University of Hong Kong)

Title: Brody curves on the degree six Fermat surface

Abstract: A holomorphic map from the complex line to the n-dimensional complex projective space is called a Brody curve if its spherical derivative is bounded. In 2010, Eremenko applied potential theory to study Brody curves omitting n hyperplanes in general position and showed that these curves have growth order at most one, normal type. In this talk, we will characterize Brody curves on the degree six Fermat surface in the three dimensional complex projective space based on Eremenko's potential theoretical method. This is a joint work with Sai Kee Yeung.

Ngoc Cuong Nguyen (KAIST)

Title: Complex Hessian measures with respect to a background Hermitian form

Abstract: We develop potential theory for m-subharmonic functions with respect to a Hermitian metric on a Hermitian manifold. First, we show that the complex Hessian operator is well-defined for bounded functions in this class. This allows to define the m-capacity and then showing the quasi-continuity of m-subharmonic functions. Thanks to this we derive other results parallel to those in pluripotential theory such as the equivalence between polar sets and negligible sets. The theory is then used to study the complex Hessian equation on compact Hermitian manifold with boundary, with the right hand side of the equation admitting a bounded subsolution. This is an extension of a recent result of Collins and Picard dealing with classical solutions. This is a joint work with S. Kołodziej.

Hoseob Seo (Center for Complex Geometry, IBS)

Title: On approximations of toric plurisubharmonic functions and currents

Abstract: When a plurisubharmonic function on a complete Reinhardt domain is invariant under the torus action, convex analysis plays a crucial role in analyzing its properties. For instance, the multiplier ideal sheaf, the log canonical threshold of it at the origin and the existence of equisingular approximations with analytic singularities can be described in terms of its Legendre transformation. In this talk, we will review the correspondence between toric positive closed currents and real positive closed currents and then show the existence of equisingular approximations of a toric positive closed (1, 1)-current converging in the Hartogs sense.

Guolei Zhong (Center for Complex Geometry, IBS)

Title: Periodic points for meromorphic self-maps of Fujiki varieties

Abstract: As one of the fundamental problems in complex dynamics, the equi-distribution of periodic points is expected to impose rather strong ergodic properties on the dynamical system. In the past decade, Dinh, Nguyễn and Truong proved that, given a meromorphic self-map of a compact Kähler manifold with a dominant topological degree (i.e., the last dynamical degree is strictly larger than the other ones), the set of isolated periodic points is asymptotically equi-distributed with respect to the equilibrium measure. In this talk, after a review on the background, we extend the previous results to the (possibly singular) Fujiki variety. As a by-product, we give an affirmative answer to a conjecture of Shou-Wu Zhang. This is based on a joint work with Tien-Cuong Dinh.

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