Seminar on Complex Analysis and Complex Geometry - Fall 2012

Wednesday, September 19, 2012, 11am, PGH 646

Title: On a conjecture of Demailly-Peternell-Schneider

Speaker: Qi Zhang, University of Missouri-Columbia

Abstract: Projective varieties with numerically effective anticanonical bundles appear naturally in the Minimal Model Program. In the surface case, these objects include del-Pezzo surfaces, K3 surfaces, Abelian surfaces and so on. In this talk, we give an affirmative answer to an open question posed by Demailly-Peternell-Schneider in 2001. Let $f:X\rw Y$ be a surjective morphism from a log canonical pair $(X,D)$ onto a $\bQ$-Gorenstein variety $Y$. If $-(K_X+D)$ is nef, we show that $-K_Y$ is pseudo-effective. This is a joint work with M. Chen.

Wednesday, September 19, 2012, 3pm, Colloquium, PGH 646

Title: On projective varieties with nef anti-canonical divisor

Speaker: Qi Zhang, University of Missouri-Columbia

Abstract: A projective variety X over an algebraically closed field k is the zero-locus of some finite family of homogeneous polynomials with coefficients in k. It turns out that the geometrical (topological) properties of a projective variety are closely related to its algebraic structure of the meromorphic functions--the Kodaira dimension. In this talk, we shall discuss some recent development on the projective varieties with non-positive Kodaira dimension, in particular, the varieties with nef anti-canonical divisor.

Wednesday, October 3, 2012, 11am, PGH 646

Title: Distribution of rational points on algebraic varieties

Speaker: Sho Tanimoto, Rice University

Abstract: In this talk, I will introduce Manin's conjecture and theory of Height zeta functions. Then I will discuss some geometric issues arising in this context and introduce the notion of balanced line bundles.

Wednesday, October 10, 2012, 11am, PGH 646

Title: Ample, big, nef: the concepts and criteria

Speaker: Min Ru, UH

Abstract: This will be an introductory lecture on notions of positivity in complex geometry, aimed at graduate students.

Wednesday, October 17, 2012, 11am, PGH 646

Title: Application of Representation Theory to Punctual Hilbert Schemes of K3 Surfaces

Speaker: Letao Zhang, Rice University

Abstract: We compute the graded character formula of the Mumford-Tate group representation on the cohomology ring of the Hilbert schemes of n points on K3 surfaces. Also, we find the way to compute the generating series for the number of canonical Hodge classes of degree 2n.

Wednesday, October 24, 2012, 11am, PGH 646

Title: Ample, big, nef: the concepts and criteria, II

Speaker: Min Ru, UH

Abstract: This will be an introductory lecture on notions of positivity in complex geometry, aimed at graduate students.

Wednesday, October 31, 2012, 11am, PGH 646

Title: Ample, big, nef: the concepts and criteria, III

Speaker: Min Ru, UH

Abstract: This will be an introductory lecture on notions of positivity in complex geometry, aimed at graduate students.

Wednesday, November 7, 2012, 11am, PGH 646

Title: Ample, big, nef: the concepts and criteria, IV

Speaker: Min Ru, UH

Abstract: This will be an introductory lecture on notions of positivity in complex geometry, aimed at graduate students.

Wednesday, November 14, 2012, 11am, PGH 646

Title: Bubbles in Complex Algebraic Geometry

Speaker: Steven Lu, University of Quebec at Montreal

Abstract: In analysis, the failure in ``compactness" in some space of maps to a compact analytic object $X$ is explained by the existence of bubbles in $X$. In holomorphic geometry (resp. algebraic geometry) bubbles are given by nonconstant holomorphic functions from $\bf C$, i.e. Brody curves, (resp. from ${\bf CP}^1$, i.e. rational curves) to $X$. We will discuss a non compact version of bubbles for quasi projective varieties $X$ and their effect on the Ricci curvature of $X$. This is joint work with De-Qi Zhang.

Wednesday, November 14, 2012, 3pm, Colloquium, PGH 646

Title: Generalizing the Little Picard Theorem

Speaker: Steven Lu, University of Quebec at Montreal

Abstract: The Little Picard Theorem says that a complex analytic function defined everywhere on $\bf C$, can miss at most one complex value. Its standard proofs are all based on the fact that $\C$ minus two point is hyperbolic (in the sense of negative curvature as is the case of the unit disk). The higher dimensional generalization of hyperbolicity, at least in the birational context, is that of general type (almost everywhere negative curvature). We will define the opposite notion to that of general type, that of being special, and discuss our result that any object defined by complex polynomials (a variety) $X$ has a decomposition as a fiber space over a base object of general type whose fibers are special. A conjectural generalization of the Little Picard theorem would then be that there exist an entire function with values in $X$ not contained in any subvariety in $X$ if and only if $X$ is special. We will conclude by our verification of the conjecture for $X$ that is of maximal albanese dimension, which is the case for $\C$ minus two points. This is joint work with Jorg Winkelmann.