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|