An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities

Abstract

The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain L 2 -estimates of solutions of the ∂-equation by using the de Rham-Weil isomorphism between the ∂cohomology and the Čech cohomology.

Injectivity theorems with multiplier ideal sheaves for higher direct images under K"ahler morphisms

“…In [Mat13] and [FM16], by combining the theory of harmonic integrals with the L 2method for the ∂-equation, we succeeded to obtain the above results in the absolute case (see also [GM13] and [FM16] for applications). The proof of the main results is based on transcendental methods developed in [Mat13], [FM16], and [Take95].…”

“…In [Mat13] and [FM16], by combining the theory of harmonic integrals with the L 2method for the ∂-equation, we succeeded to obtain the above results in the absolute case (see also [GM13] and [FM16] for applications). The proof of the main results is based on transcendental methods developed in [Mat13], [FM16], and [Take95]. One of the advantages of our method is that we can prove the main results for Kähler morphisms (not only projective morphisms) and singular metrics with non-algebraic singularities.…”

“…At the end of this section, we briefly explain the sketch of the proof of Theorems 1.2, comparing with the absolute case established in [Mat13]. The proof of Theorem 1.3 is essentially the same as in Theorem 1.2.…”

“…Particularly, the following theorem is one of the most useful generalizations of Kollár's injectivity theorem for deformations of projective varieties. On the other hand, also from the analytic viewpoint, we can approach to Kollár's result (for example, see [Eno90], [Fuj12], [Fuj13a], [FM16], [Mat13], [Mat14], [Ohs04], [Take95], and [Take97]). This paper contributes to the study of the injectivity theorem and its applications from the analytic viewpoint.…”

“…The following theorems, which are the main results of this paper, can be seen as a generalization of Theorem 1.1 and Takegoshi's result to pseudo-effective line bundles. Moreover Theorem 1.2 and Theorem 1.3 include various injectivity theorems, for example, [Eno90], [FM16], [Fuj12], [Fuj13a], [GM13], [Kol86a], [Mat13], [Mat14], [Take95], [Take97], and so on.…”