Xiaojun Huang (Rutgers University): ”Complex geodesics and complex Monge-Ampère equations with boundary singularities”
Abstract. This is a joint work with Xueping Wang from USTC. We study complex geodesics and complex Monge-Ampère equations on bounded strongly linearly convex domains in Cn. More specifically, we prove the uniqueness of complex geodesics in such domains with prescribed boundary value and direction, when the boundaries of these domains have minimal regularity. The existence of such a complex geodesics was proved by the first author in the 1990s, but the uniqueness was left open. Using this uniqueness result and a uniform C1,1/2-estimate of complex geodesics and their dual mappings, we solve a homogeneous complex Monge-Ampère equation with prescribed boundary singularity, which was first considered by Bracci-Patrizio-Trapani on bounded strongly convex domains in Cn with smooth boundary in their two important papers. The fundamental solution of the homogeneous Monge-Ampère equation was considered in the early 80’s in a fundamental paper of Lempert.
George Marinescu (University of Cologne): ”On the singularities of the Bergman projections for lower energy forms on complex manifolds with boundary”
Abstract. We determine the boundary behavior of the spectral kernel of the d-bar-Neumann Laplacian of a domain with smooth boundary near points where the Levi form is non- degenerate. As a consequence we show that the Bergman projection admits an asymptotic expansion under a certain closed range condition for d-bar in L2. This is a joint work with Chin-Yu Hsiao.
Ngaiming Mok (University of Hong Kong):”Functional Transcendence on Quotients of Bounded Symmetric Domains”
Abstract. Finite-volume quotients of bounded symmetric domains Ω, which are naturally quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic Geometry, Arithmetic Geometry and Number Theory, and an important topic revolves around functional transcendence in relation to universal covering maps of such varieties. While a lot has already been achieved in the case of Shimura varieties (such as the moduli space Ag of principally polarized Abelian varieties) by means of methods of Diophantine Geometry, Model Theory, Hodge Theory and Complex Differential Geometry, techniques for the general case of not necessarily arithmetic quotients Ω/Γ =: XΓ have just begun to be developed. For instance, Ax-type problems for subvarieties of products of arbitrary compact Riemann surfaces of genus ≥ 2 have hitherto been intractable by existing methods. We will explain a differential geometric approach leading to characterization results for totally geodesic subvarieties of XΓ for the universal covering map π : Ω → XΓ. Especially, we will explain how uniformization theorems for bi- algebraic varieties can be proven by analytic methods involving the Poincaré-Lelong equation in the cocompact case (joint work with S.-T. Chan), generalizing in the cocompact case earlier results of Ullmo-Yafaev (2011) in the case of arithmetic quotients. More generally, we will consider the Zariski closures of images of algebraic sets under the universal covering map π : Ω → XΓ. In the arithmetic case, Klingler-Ullmo-Yafaev (2016) resolved the hyperbolic Ax-Lindemann Conjecture in the affirmative ascertaining that such Zariski closures are weakly special (equivalently totally geodesic). I will explain how the arithmeticity condition can be dropped at least in the cocompact case by a completely different proof using foliation theory, Chow schemes, partial Cayley transforms and Kähler geometry.
Mihai Paun (Bayreuth University):”An extension result for twisted canonical forms”
Abstract. We will report on a recent joint work with J. Cao. We obtain an extension criterion for the canonical forms defined on an infinitesimal neighborhood of the central fiber of a family of Kähler manifolds. We will present some aspects of the proof of our main result, as well as its motivations.
Ben Weinkove (Northwestern University): ”Degenerating PDEs on complex surfaces”
Abstract. I will give an overview of the parabolic complex Monge-Amp`ere equation in the setting of complex surfaces, and the geometric behavior of the solutions as the equations degenerate. I will also discuss some results of X.S. Shen and W. Liu on the constant scalar curvature equation. My talk will emphasize open problems and new directions.