SCV Seminar

Monday, Sep. 8, 08	Sonmez Sahutoglu (University of Michigan) Compactness of Hankel Operators and Analytic Discs in the Boundary of Pseudoconvex Domains: Using several complex variables techniques, we want to investigate the interplay between the geometry of the boundary and compactness of Hankel Operators. Let $\beta$ be a function smooth up to the boundary on a smooth bounded pseudoconvex domain $\Omega\subset\C^n$. We show that if $\Omega$ is convex or the Levi form of $\partial\Omega$ is of rank at least $n-2$ then compactness of the Hankel operator $H_\beta$ implies that $\beta$ is holomorphic "along" analytic discs in the boundary. Furthermore, when $\Omega$ is convex in $\C^2$, we show that the condition on $\beta$ is necessary and sufficient for compactness of $H_\beta$. This is a joint work with Zeljko Cuckovic.
Thursday, Sep. 18, 08 3pm~4pm	Peter Ebenfelt (University of California, San Diego) Cauchy-Riemann (CR) transversality of holomorphic mappings: A holomorphic mapping is CR transversal to a CR submanifold $M'$ of complex space if it is transversal to its complex tangent space. CR transversality can be forced on a mapping by imposing the condition that it sends another CR submanifold $M$ into $M'$ and the CR structure of $M$ (or $M'$) satisfies certain conditions. For instance, a consequence of the classical Hopf lemma is that any nonconstant mapping of a strictly pseudoconvex hypersurface into another is necessarily CR transversal. In this talk, we will discuss a number of recent, more general results in this direction.
Monday, Sep. 22, 08	Zhou Zhang (University of Michigan) Complex Monge-Amp\`ere Equations over closed K\"ahler Manifolds: In recent years, pluripotential theory has given more life to the study of complex Monge-Amp\`ere equation and its parabolic version, K\"ahler-Ricci flow. More precisely, the argument, originated by S. Kolodziej, coming from the study of Monge-Amp\`ere operator, has provided new (and better) way to achieve some key estimates. It's role in degenerate case is of special interests as indicated by many people's results. The fundamental works on this subject by Bedford and Taylor are heavily relied on.
Monday, Sep. 29, 08	Elizabeth Wulcan (University of Michigan) On Bochner-Martinelli type residue currents: Bochner-Martinelli type residue currents were introduced by Passare, Tsikh and Yger as a generalization of the classical Coleff-Herrera residue currents. In this talk we will discuss a geometric decomposition of these currents in terms of valuations. This is a joint work (in progress) with Mattias Jonsson.
Monday, Oct. 13, 08	David Barrett (University of Michigan) Holomorphic projection and duality for domains in complex projective space: We show that the efficiency of a natural pairing between certain projectively invariant Hardy spaces on dual strongly C-linearly convex real hypersurfaces in complex projective space is measured by the norm of the corresponding Leray transform.
Monday, Oct. 27, 08	Mattias Jonsson (University of Michigan) Integrability in holomorphic dynamics: Integrability for a dynamical system means that some quantity is preserved by the dynamics. I will survey some aspects of this notion in (discrete-time) holomorphic dynamics, with emphasis on holomorphic selfmaps of the complex projective plane.
Monday, Nov. 3, 08	Michal Jasiczak (University of Poznan & Institute of Mathematics Polish Academy of Sciences) A priori estimates for the $\overline{\partial}$-equation on K\"{a}hler manifolds: There is a growing interest in analysis on singular spaces. While there are many ways one can understand the term 'singular space' in different branches of mathematics, in complex analysis it usually refers to a singular analytic set. The regular part of such a set is a complex manifold, which sometimes can be equipped with a complete K\"{a}hler metric (e.g. the Ohsawa metric). Motivated by this fact we intend to analyze influence of geometry of a complete K\"{a}hler manifold on regularity of the $\overline{\partial}$-problem. More specifically, in order to obtain a priori H\"{o}lder estimates for the $\overline{\partial}$-problem on such a manifold, we will have a look on methods introduced in real analysis by S. Campanato, C. B. Morrey, E. DeGiorgi and J. Nash. This will lead us to an interesting non-linear PDE. As a corollary, we will also obtain a 'weak version' of the Schwarz Lemma for holomorphic functions and harmonic forms on a complete K\"{a}hler manifold.
Monday, Nov. 17, 08	Henry de Thelin (University Paris 11, France) Dynamics of meromorphic maps on compact Kahler manifolds: We study the dynamics of a meromorphic map on a compact Kahler manifold. We give a criterion that allows us to produce a measure of max- imal entropy and we apply this criterion for a family of generic birational maps of CP^k. This is a joint work with Gabriel Vigny.
Tuesday, Dec. 2, 08 3-4 pm (4088 EH)	Erlend Fornaess Wold (University of Oslo) Polynomial Convexity and Totally real manifolds: To generalize the Andersen-Lempert theory of automorphisms of C^n to the non-compact setting, it will be important to understand polynomial convexity in this case. We will discuss generic properties of totally real manifolds in C^n.