Complex manifolds, dynamics and birational geometry:
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November 10-14, 2014,
Laboratory of Algebraic Geometry, Moscow

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Bruno Klingler (Jussieu)
The hyperbolic Ax-Lindemann-Weierstrass conjecture

The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional algebraic independence statement for the uniformizing map of an arithmetic variety. In this talk I will describe the conjecture, its role and its proof (joint work with E.Ullmo and A. Yafaev).

Shin-ichi Matsumura (Kagoshima University)
Injectivity theorems with multiplier ideal sheaves and their applications

In this talk, I give an injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities. This result can be seen as a generalization of various injectivity and vanishing theorems.

The proof is based on a combination of the theory of harmonic integrals and the L^2-method for the dbar-equation. To treat transcendental singularities, after regularizing a given singular metric, we study the asymptotic behavior of the harmonic forms with respect to a family of the regularized metrics. Moreover we obtain L^2-estimates of solutions of the dbar-equation by using the Cech complex.

As applications of this injectivity theorem, I give some extension theorems for holomorphic sections of pluri-log-canonical bundle from subvarieties to the ambient space. Moreover, by combining techniques of the minimal model program, we obtain some results for semi-ampleness related to the abundance conjecture in birational geometry.

This talk is based on the preprint in arXiv:1308.2033v2 and a joint work with Y. Gongyo in arXiv:1406.6132v1.

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Laboratory of Algebraic Geometry and its Applications