### Complex manifolds, dynamics and birational geometry:

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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).
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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