Abstracts
Takeo Ohsawa : Recent topics in the L² Dolbeault cohomology on pseudoconvex manifolds
To the memory of Professor Pierre Dolbeault
In the theory of functions of complex variables, Riemann (1857) first tried to introduce a coordinate system in the set of algebraic functions of one variable. Periods of integrals of Riemann surfaces are components of such a coordinate. By generalizing this notion of periods, Griffiths established a theory of period mappings for families of compact Kähler manifolds. Based on the theory of Griffiths, T. Fujita discovered a convexity property of this generalized period mapping in terms of the curvature of the direct image of relative canonical bundles. This is in the context of Hodge theory where the L² harmonic forms are analyzed.
On the other hand, after the foundation of several complex variables was established by K. Oka and H. Cartan, the L² method was developed to obtain effective results in complex analysis and geometry, based on Dolbeault's isomorphism theorem. An extension theorem with L² growth condition was also obtained in this context. It was applied to study singularities of plurisubharmonic functions, to prove the invariance of plurigenera for families of projective varieties, and to study the multiplier ideal sheaves. Recently, J. Cao proved in a general context that an L² extension theorem implies the semipositivity of the direct image of relative canonical bundles. Vice versa, B. Berndtsson and L. Lempert showed that such a curvature property implies an L² extension theorem in an optimal form.
Since this progress is closely related to the solution of a long-standing conjecture of N. Suita by Z. Błocki and by Q.-A. Guan and X.-Y. Zhou, it might be a good occasion to give an overview of this development going back to the origin of the ideas of function theory on Riemann surfaces and complex manifolds of higher dimension. Some of the well known basic results will be also reviewed in the setting of the L² theory.
Taeyong Ahn : Dynamics of Henon mappings and applications to complex geometry
In this talk, some basic dynamical properties of Henon mappings will be first introduced. Then, we consider some applications of the dynamics of the Henon mappings. The first example is the Fatou-Bieberbach domains and related facts. Next, we introduce Brody curves and injective Brody curves and then discuss some non-trivial examples of Brody curves and injective ones.
Tsz On Mario Chan : A vanishing theorem for Cousin-quasi-tori
Cousin-quasi-tori are non-compact analogue of compact complex tori which have no non-constant holomorphic functions on them and are weakly pseudoconvex. Without harmonic theory at the disposal, I will show how a vanishing theorem for the cohomology of holomorphic line bundles on Cousin-quasi-tori is proved by solving $\dbar$-equations using the classical $L^2$ method and a Runge-type approximation. The result generalises Mumford's Index Theorem on compact complex tori and gives a finer vanishing result than that given by the theorem of Andreotti and Grauert.
Young-Jun Choi : Positivity of fiberwise Kahler-Einstein metrics
Let $p:X\rightarrow Y$ be a \emph{family of K\"ahler manifold}, i.e., a surjective holomorphic mapping between complex manifolds such that every fiber $X_y:=p^{-1}(y)$ is a K\"ahler manifold. If every fiber admits a K\"ahler-Einstein metric, then it induces a \emph{fiberwise K\"ahler-Einstein metric}, which is defined by a real $(1,1)$-form on $X$ which is K\"ahler-Einstein when it is restricted on each fiber. In this talk, we discuss the positivity (on the total space $X$) of fiberwise K\"ahler-Einstein metrics on families of bounded strongly pseudoconvex domains, canonically polarized compact K\"ahler manifolds and Calabi-Yau manifolds.
Robert Xin Dong : Two proofs for boundary asymptotics of the relative Bergman kernel metric for elliptic curves
On each complex manifold, the Bergman kernel is a canonical volume form determined by the complex structure. As the complex structure changes, we study the asymptotic behaviors of the relative Bergman kernel metric for a family of compact Riemann surfaces near the boundary of their moduli space. For the genus one case, the asymptotic expansion is determined up to the fourth term, which shows that the Levi form of the relative Bergman kernel metric has hyperbolic growth at the node. Two proofs relying and not relying on the elliptic functions will be provided, respectively. If time permits, I will elaborate on the genus two case as well.
Pham Hoang Hiep : Singularities of plurisubharmonic functions
In this talk, we will give a short survey about plurisubharmonic functions and invariants of their singularities. We also show how to apply the L^2-extension theorem of Ohsawa and Takegoshi to study invariants of their singularities.
Genki Hosono : Approximations and examples of singular Hermitian metrics on vector bundles
We study singular Hermitian metrics on vector bundles.There are two main results in this talk. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some kind of positively curved singular Hermitian metrics whose standard approximations are not Nakano-semipositive.
The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank 2 bundles on elliptic curves.
Takayuki Koike : On some analogues of Ueda theory and their applications
Let $C$ be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of $C$ when $C$ is non-singular or is a rational curve with a node. We propose analogues of Ueda's theory in two manners: Ueda's theory for the case where $C$ admits nodes and a codimension two analogue of Ueda's theory. As an application, we study singular Hermitian metrics with semi-positive curvature on the anti-canonical bundle of the blow-up of the projective plane at 9 points, and the blow-up of the three dimensional projective space at 8 points.
Long Li : On the Reductivity of Automorphism group of conic Kaehler-Einstein manifolds
Matsushima proved the automorphsim group is reductive on Kaehler-Einstein(KE) manifolds, and this result has been used as a base in
Bando-Mabuchi's work on uniqueness of KE metrics when c_1(X)>0. Recently, Berndtsson(and so on) generalized this uniqueness result to conic KE metrics, by his previous work on subharmonicity of log-Bergman kernels. However, today we will follow Bando-Mabuchi's old path to investigate this uniqueness problem, and use Homander's L^2 techniques to recover Matsushima's result in conic setting. And hopefully, this will give another proof of uniqueness of conic KE metrics.
Alexander Rashkovskii :
1. Pluricomplex Green functions
I will discuss several types of pluricomplex Green functions and their applications in pluripotential theory.
2. Extremal cases for log canonical threshold
We show that a recent result of Demailly and Pham Hoang Hiep implies a description of plurisubharmonic functions with given Monge-Ampere mass and smallest possible log canonical threshold. It also gives a new proof of a result of Qi'an Guan and Xiangyu Zhou on plurisubharmonic functions with given log canonical threshold and smallest possible Lelong number.