Vanishing theorems for projective morphisms between complex analytic spaces

We discuss vanishing theorems for projective morphisms between complex analytics spaces and some related results. They will play a crucial role in the minimal model theory for projective morphisms of complex analytic spaces. Roughly speaking, we establish an ultimate generalization of Koll\'ar's package from the minimal model theoretic viewpoint.Comment: 21 pages, v2: very minor revision, v3: a minor revision which makes it compatible with arXiv:2209.06382 [math.AG] and arXiv:2209.11401 [math.AG], v4: a very minor revision, v5: a minor revision, v6: very minor revisions, v7: very minor revisions following referee's comment

Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Koll'ar's injectivity theorem for adjoint bundles twisted by a suitable multiplier ideal sheaf. As applications, we generalize Koll'ar's vanishing theorem, Koll'ar's torsion-freeness, generic vanishing theorem, and so on, for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use the theory of harmonic integrals on noncompact K"ahler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems, which seems to be of independent interest.Comment: 34 pages, v2: minor revisions, problem on Bertini-type theorem was added, reference list was update