Seminar on Complex Analysis and Complex Geometry - Fall 2014

Wednesday, September 24, 2014, 11am-12noon, PGH 646

Title: A generalization of the Schmidt subspace theorem

Speaker: Gordon Heier, UH

Abstract: We will discuss a generalization of the Schmidt subspace theorem (and Cartan's Second Main Theorem) in the setting of not numerically equivalent divisors. This is joint work with Aaron Levin.

Wednesday, October 22, 2014, 11am-12noon, PGH 646

Title: Normal Family and Gauss Curvature estimate of minimal surfaces in R^m

Speaker: Xiaojun Liu, University of Shanghai for Science and Technology, Shanghai, China

Abstract: In this talk, we first extend Zalcman's principle of normality to the families of holomorphic mappings from Riemann surfaces to a compact Hermitian manifold. We then use this principle to derive an estimate for Gauss curvatures of the minimal surfaces in R^m whose Gauss maps satisfy some property $\mathcal P$. As a consequence, we give the Gauss curvature estimate for minimal surfaces in R^m whose Gauss maps intersect hyperplanes in P^{m-1} with high multiplicities.

Wednesday, October 29, 2014, 11am-12noon, PGH 646

Title: Interpolation for weighted Bergman space and effective very ampleness

Speaker: Jujie Wu, Tongji University, Shanghai, China

Abstract: In this talk, we shall use interpolating techniques and Poincare series to give an effective criterion of very ampleness of the pluri-canonical bundles for compact quotients of bounded symmectric domain.

Wednesday, November 5, 2014, 3pm-4pm, PGH 646 (Colloquium)

Title: On the curvature tensors of Hermitian manifolds

Speaker: Fangyang Zheng, Ohio State University

Abstract: In recent years, the geometry of Hermitian manifolds has regenerated interests, with the intent of pushing analysis on Kaehler manifolds to general Hermitian ones, and also with the study of non-Kaehler Calabi-Yau manifolds from string theory.

For a given Hermitian metric on a complex manifold, there are two canonical connections associated with the metric, namely, the Hermitian (aka Chern) connection r and the Riemannian (aka Levi-Civita) connection. The former is the unique connection that is compatible with both the metric and the complex structure, while the latter is the unique torsion-free connection that is compatible with the metric. These two connection coincide precisely when the metric is Kaehler.

In this talk, we will explore properties and behaviors of the curvature tensors of the Hermitian and the Riemannian connection of a Hermitian manifold, and examine some conditions on the curvature that will lead to the Kaehlerness of the metric.

MONDAY, November 10, 2014, 11am-12noon, PGH 646

Title: Holomorphic mappings into compact complex manifolds

Speaker: Do Duc Thai, Hanoi National University of Education, Vietnam

Abstract: The purpose of this talk is to present a second main theorem with the explicit truncation level for holomorphic mappings of C (or of a compact Riemann surface) into a compact complex manifold sharing divisors in subgeneral position. Finally, we will give some applications of the above main theorems. Namely, we show a unicity theorem for holomorphic curves of a compact Riemann surface into a compact complex manifold sharing divisors in N-subgeneral position. Moreover, we also generalize the Five-Point Theorem of Lappan to a normal family from an arbitrary hyperbolic complex manifold to a compact complex manifold. Here is a work joining with Vu Duc Viet.

Wednesday, November 12, 2014, 11am-12noon, PGH 646

Title: A Lang Exceptional Set for Integral Points

Speaker: Paul Vojta, UC Berkeley

Abstract: In 1986 and 1991, Serge Lang defined holomorphic, diophantine, and geometric exceptional sets of a complete variety over $\mathbb C$, over a number field, or over a field of characteristic zero, respectively, and conjectured that they should coincide when defined.

This talk will consider the possibility of extending this definition to holomorphic curves or integral points in quasi-projective varieties. A central question that arises is, given an abelian (or semiabelian) variety $A$ and a Zariski-closed subset $Z$ of codimension $\ge 2$, can one find a nonconstant holomorphic curve in $A\setminus Z$ with Zariski-dense image, or a Zariski-dense set of integral points on $A\setminus Z$? It is quite easy to prove this result for holomorphic curves, but for integral points the question remains open (unless you ``cheat'' and change the definitions).