Tien Cuong Dinh (Singapore)

- Title : Unique Ergodicity for foliations on compact Kaehler surfaces

- Abstract : Let F be a holomorphic foliation by Riemann surfaces on a compact Kaehler surface. Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed (1,1)-current. Then there exists a unique (up to a multiplicative constant) positive  ddc-closed (1,1)-current directed by F. This is a very strong ergodic property of F. Our proof uses an extension of the theory of densities to a class of non-ddc-closed currents. A complete description of the cone of directed positive ddc-closed (1,1)-currents is also given when F admits directed positive closed currents. The talk is based on a recent joint work with Viet-Anh Nguyen and Nessim Sibony.

Sungmin Yoo (KIAS)

- Title: Variation of Kahler-Einstein metrics on strictly pseudoconvex domains in Kahler manifolds

- Abstract: The postivity of the variation of Kahler-Einstein for a family of canonically polarized compact Kahler manifolds was proved by Schumacher. In case of noncompact manifolds case, Choi proved that the variation of Kahler-Einstein metrics for a family of strictly pseudoconvex domains in C^n is positive, when the variation is given by the canonical coordinate projection map.
In this talk, we will study the variation of Kahler-Einstein metrics for a family of strictly pseudoconvex domains in Kahler manifolds, when the variation is given by a holomorphic surjective map between two Kahler manifolds. We will also discuss the extension of the variation across singular fibers, using the argument of Paun. This is joint work with Young-Jun Choi from Pusan National University.

Takayuki Koike (Osaka City U.)

- Title: On a neighborhood of an elliptic curve and a gluing construction of K3 surfaces

- Abstract: In this talk, we construct K3 surfaces by gluing two rational surfaces given by blowing-up the projective plane at "general" nine points. For such K3 surfaces, one can concretely calculate the period maps. By observing the result of this calculation, one can show that such K3 surfaces constitute a large family which includes non-projective K3 surfaces as general elements. This gluing construction is based on Arnol'd's theorem on the existence of nice neighborhoods of an elliptic curve embedded in a surface whose normal bundle satisfies Diophantine-type condition in the Picard variety. This is a joint work with T. Uehara. We also explain our motivation from the study on the complex analytic structure of a neighborhood of an elliptic curve embedded in a surface, mainly based on Ueda theory.

Frank Loray (Rennes)

- Title : Neighborhoods of elliptic curves

- Abstract : In this talk, we consider an elliptic curve C with an embedding into a smooth complex surface S and we want to understand the structure of the germ of surface (S,C) near the image of the curve. The problem of classifying such germs of neighborhoods up to analytic equivalence has been investigated by Grauert when C.C<0, Ilyashenko when C.C>0, and Arnold when C.C=0 but the normal bundle is not torsion. With O. Thom and F. Touzet, we recently completed the formal classification of such germs of neighborhood in the torsion case. In this talk, I will focus on the analytic classification when the normal undle is trivial, which is a work in progress with F. Touzet and S. M. Voronin.