Seminar on Complex Analysis and Complex Geometry - Spring 2013

Wednesday, February 13, 2013, 12noon, PGH 646

Title: Higher convexity of the (co)amoeba complement and the Generalized order map

Speaker: Mounir Nisse, Texas A&M University

Abstract: Amoebas (resp. coamoebas) are the image under the logarithmic (resp. argument) map of algebraic (or even analytic) subvarieties of the complex algebraic torus. they inherit some algebraic, geometric, and topological properties of the variety itself. A global statement generalizing convexity of amoeba complement components was found by Andr\'e Henriques, and he proved a weaker version of this property for the complement of amoebas. First, using Bochner-Martinelli form and the residue theory, we show a stronger version of this result, which complete the generalization of the $k$-convexity of the amoeba complement in higher codimension. Moreover, we prove the same property for the complement of the closure of coamoebas in the real torus. Also, we define the generelized order maping in higher codimension, which is already well known for hypersurfaces by Forsberg, Passare and Tsikh. (This is a joint work with Frank Sottile.)

Wednesday, February 20, 2013, 12noon, PGH 646

Title: Regularity of canonical operators and the Nebenh\"ulle of Hartogs domains

Speaker: Yunus Zeytuncu, Texas A&M University

Abstract: In this talk, we discuss how to extract geometric information about a domain $\Omega$ in $\mathbb{C}^n$ by using analytic properties of some canonical operators on $\Omega$. Let $\mathbb{D}$ denote the unit disk in $\mathbb{C}$ and let $\phi(z)$ be a bounded subharmonic function on $\mathbb{D}$. We consider the pseudoconvex complete Hartogs domains in $\mathbb{C}^2$ of the form $$\Omega=\{(z,w)\in \mathbb{C}^2 : z\in \mathbb{D} \text{ and } |w|< e^{-\phi(z)}\}.$$ Let $N_1$ denote the $\bar{\partial}$-Neumann operator on $L^2_{(0,1)}(\Omega)$ and $\mathbf{B}_{\Omega}$ denote the Bergman projection on $L^2(\Omega)$. In this talk, we relate the Sobolev regularity properties of $N_1$ and $\mathbf{B}_{\Omega}$ to the Nebenh\"ulle of $\Omega$ and the Stein neighborhood bases of $\Omega$.

Wednesday, March 6, 2013, 12noon, PGH 646

Title: Surfaces with big cotangent bundle

Speaker: Erwan Rousseau, Université de Provence, Marseille, France

Abstract: Surfaces of general type with positive second Segre number are known to have big cotangent bundle. We give a new criterion ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle. (Joint work with X. Roulleau.)

Wednesday, March 6, 2013, 3pm, PGH 646, Colloquium

Title: Complex hyperbolicity, differential equations and automorphic forms

Speaker: Erwan Rousseau, Université de Provence, Marseille, France

Abstract: I will explain how the study of entire curves in complex projective manifolds is related to differential equations and automorphic forms.

Wednesday, March 27, 2013, 12noon, PGH 646

Title: Hypergeometric functions and Mahler measures of polynomials

Speaker: Matt Papanikolas, Texas A+M University

Abstract: Hypergeometric functions play important roles in capturing periods of algebraic varieties via solutions of their Picard-Fuchs equations. Also, examples of formulas that relate Fourier coefficients of classical modular forms to values of finite field hypergeometric functions have been known for some time. When these two types of examples are linked, the Mahler measure of the defining polynomials are often related to special values of these modular L-functions. We will present new formulas relating finite field hypergeometric functions, eigenvalues of modular forms, Mahler measures, and special values of L-functions.

Thursday, April 4, 2013, 4pm-5pm, PGH 646 (NOTE THE SPECIAL DATE AND TIME)

Title: Vertical Brauer groups and del Pezzo surfaces of degree 4

Speaker: Anthony Várilly-Alvarado, Rice University

Abstract: Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space. They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface. I will explain how to construct, for every non-trivial, non-constant element A of Br X, a rational genus-one fibration X --> P^1 such that A is ``vertical'' for this map. This implies, for example, that if there is a cohomological obstruction to the existence of a point on X arising from A, then there is a genus-one fibration X --> P^1 where none of the fibers are locally soluble, giving a concrete, geometric way of ``seeing'' a Brauer-Manin obstruction. The construction also gives a fast, practical algorithm for computing the Brauer group of X. Conjecturally, this gives a mechanical way of testing for the existence of rational points on these surfaces. This is joint work with Bianca Viray.

Wednesday, April 24, 2013, 11am, PGH 646

Title: The proof of Kodaira's embedding theorem I (graduate student lecture)

Speaker: Ananya Chaturvedi, UH

Abstract: See title.

Wednesday, April 24, 2013, 12noon, PGH 646

Title: The proof of Kodaira's embedding theorem II (graduate student lecture)

Speaker: Angelynn Alvarez, UH

Abstract: See title.