Zu Chongzhi Mathematics Research Seminar
Date and Time (China standard time): Wednesday, November 15, 9:00-10:00 am
Zoom: 926 0787 7463; Passcode: dkumath
Title: Partial convexity conditions and the $\overline{\partial}$-problem
Speaker: Debraj Chakrabarti
Bio: Dr. Debraj Chakrabarti is a professor from Central Michigan University. He obtained his PhD from University of Wisconsin, Madison in 2006. After his graduation, Dr. Chakrabarti worked at Indian Institute of Technology and Tata Institute of Fundamental Research. He joined Central Michigan University as an assistant professor in 2013 and has been there since then. Dr. Chakrabarti ’s research lies in the field of several complex variables. His main current research interests are the Cauchy-Riemann equations and $L^p$-Bergman spaces.
Abstract:
It is well-known that on a Stein manifold the $\overline{\partial}$-problem can be solved in each degree $(p,q)$ where $q\geq 1$, or in other words the Dolbeault cohomology vanishes in these degrees. There has been great interest in finding sufficient conditions on complex manifolds which ensure that the Dobeault cohomology in degree $(p,q)$ is finite dimensional or vanishes. Andreotti-Grauert introduced the notions of $q$-convex/$q$-complete manifolds, which generalize Steinness. For manifolds with boundary, Hormander and Folland-Kohn introduced the condition now called $Z(q)$ which ensures finite-dimensionality of the cohomology in degree $q$ as well as $\frac{1}{2}$ estimates for the $\overline{\partial}$-Neumann operator. These conditions ($q$-convexity/completeness and $Z(q)$) are biholomorphically invariant characteristics of the underlying complex manifold.
In the context of Hermitian manifolds, a different type of sufficient condition implies that the $L^2$-cohomology in degree $(p,q)$-vanishes. Here one assumes that the sum of any $q$-eigenvalues is positive, and this also leads to the vanishing of the $L^2$-cohomology via the Bochner-Kohn-Morrey formula. These conditions are not biholomorphically invariant (they depend on the choice of the metric).
