Aarhus–Gothenburg seminar

Monday June 9, in Pascal

9:30–10:20: Yanbo Fang, Aarhus: Behavior of KE metric around isolated lc singularity: baby examples

Abstract: On a Stein neighbourhood of an isolated log canonical singularity, Datar-Fu-Song have constructed negative Kähler-Einstein metrics (within certain class), and they conjecture that some asymptotic cuspidal metric behaviors can be captured in terms of algebro-geometric data of the singularity. Together with Spotti, we investigate some preliminary examples including Tsuchihashi cusp singularities and certain isolated hypersurface singularities, focusing on metric collapsing limits in a "generic" sense. This project continues the PhD thesis work of L.Engberg.

11:00–11:50 Antonio Trusiani: Regularity of solutions to complex Monge-Ampère equations in big cohomology classes

Abstract: Yau’s Theorem on the existence of smooth solutions to complex Monge-Ampère equations in the Kähler setting has solved Calabi’s Conjecture and opened countless avenues of research. Among them, the study of complex Monge-Ampère equations in big cohomology classes has been very active and has lead to several important geometric applications.

I will prove that solutions to complex Monge-Ampère equations in big cohomology classes are smooth on the ample loci, also producing global Laplacian estimates. At the end of the talk I will discuss applications such as the solution to the analogue of Calabi’s Conjecture in the big setting.

13:30–14:20 Ludvig Svensson, Chalmers: A calculus for computing finite parts of divergent complex geometric integrals

Abstract: We consider divergent integrals of certain forms on a reduced pure-dimensional complex space X. The forms under consideration are singular along a subvariety defined by the zero set of a holomorphic section s of a holomorphic vector bundle E->X. Equipping E with a smooth Hermitian metric allows us to define a finite part of the divergent integral as the action of a certain current extension of the form. I will talk about a current calculus that can be used to compute finite parts for a special class of forms. Furthermore, I will discuss how it is, in principle, possible to reduce the computation of the finite part for a general form to this class.

15:00–15:50 Martin de Borbon, Loughborough: Polyhedral Kahler metrics on CP^n

Abstract: A polyhedral Kahler (PK) metric on a complex manifold is a polyhedral metric, compatible with the topology, that is Kahler on its regular part. I will talk about a joint project with Dmitri Panov in which we 'classify' PK merics on CP^n with cone singularities along hyperplane arrangements (in the case that the cone angles are less than 2pi). By classify, I mean we give necessary and sufficient conditions on the cone angles that guarantee existence of such metrics. Our proof of existence uses the Kobayashi-Hitchin correspondence for parabolic bundles. The equations on the cone angles are linear and quadratic and mean zero first and second Chern classes of a certain parabolic bundle on a log resolution of the arrangement. Arrangements admitting such metrics are quite rigid and difficult to find (unless n=1). So far, the only examples I know come from complex reflection arrangements. The talk will be based on my papers with Panov: "Polyhedral Kahler cone metrics on C^n with cone singularities along a hyperplane arrangement", "A Miyaoka-Yau inequality for hyperplane arrangements in CP^n", and a forthcoming paper "PK metrics on CP^n" (to appear soon).

Tuesday June 10, in Euler

9:30–10:20 Giuseppe Barbaro, Aarhus: Non-Kähler Calabi-Yau metrics

Abstract: Pluriclosed metrics with vanishing Bismut-Ricci form, called Bismut-Hermitian-Einstein (BHE), provide a crucial example of non-Kähler Calabi-Yau geometries and are therefore relevant to mathematical physics and uniformization problems in complex geometry. They are natural candidates for ‘canonical metrics’ in complex non-Kähler geometry, arising as fixed points of the pluriclosed flow and being related to Yau’s Problem concerning compact Hermitian manifolds with holonomy reduced to subgroups of U(n). A remarkable result of Gauduchon-Ivanov states that the only complex non-Kähler BHE surfaces, are quotients of the standard Bismut-flat Hopf surface. We further extend this strong rigidity to BHE 3-folds. Furthermore, using a stability result for the pluriclosed flow, we deduce the non-existence of non-trivial homogeneous BHE metrics. These results in low dimensions and in the homogeneous setting together with the examples we have motivate the hypothesis that all BHE manifolds have parallel Bismut torsion (PBT). We therefore provide a complete classification of BHE metrics with PBT by specializing our more general classification of pluriclosed manifolds with PBT.

11:00–11:50 Annamaria Ortu, Chalmers: Stability and moment maps

Abstract: A guiding problem in Kähler geometry is to find an equivalence between the existence of a given special metric and an algebro-geometric stability condition. In this context, a natural step towards proving the equivalence is to show that, given a manifold that admits a special metric, stable deformations of the manifold still admit a special metric. I will explain a way to address this type of problem by reducing it to a finite-dimensional problem and using the theory of moment maps.