Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness, Kollár’s vanishing theorem, and a generic vanishing theorem 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 L 2 L^{2} -harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.

On second non-HLC degree of closed symplecitc manifold

In this note, we show that for a closed almost-Kähler manifold [Formula: see text] with the almost complex structure [Formula: see text] satisfies [Formula: see text] the space of de Rham harmonic forms is contained in the space of symplectic-Bott–Chern harmonic forms. In particular, suppose that [Formula: see text] is four-dimensional, if the self-dual Betti number [Formula: see text], then we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott–Chern harmonic forms.

A Poisson transform adapted to the Rumin complex

Let [Formula: see text] be a semisimple Lie group with finite center, [Formula: see text] a maximal compact subgroup, and [Formula: see text] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use [Formula: see text]-invariant differential forms on [Formula: see text] to construct [Formula: see text]-equivariant Poisson transforms mapping differential forms on [Formula: see text] to differential forms on [Formula: see text]. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on [Formula: see text] to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that [Formula: see text]. Thus, [Formula: see text] is [Formula: see text] with its natural CR structure and the relevant BGG complex is the Rumin complex, while [Formula: see text] is complex hyperbolic space of complex dimension [Formula: see text]. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

Differential operators on almost-Hermitian manifolds and harmonic forms

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.