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.

Extension of cohomology classes and holomorphic sections defined on subvarieties

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general L 2 L^2 extension theorem obtained by Demailly.

Restriction formula and subadditivity property related to multiplier ideal sheaves

We give a restriction formula on jumping numbers which is a reformulation of Demailly–Ein–Lazarsfeld’s important restriction formula for multiplier ideal sheaves and a generalization of Demailly–Kollár’s important restriction formula on complex singularity exponents, and then we establish necessary conditions for the extremal case in the reformulated formula; we pose the subadditivity property on the complex singularity exponents of plurisubharmonic functions which is a generalization of Demailly–Kollár’s fundamental subadditivity property, and then we establish necessary conditions for the extremal case in the generalization. We also obtain two sharp relations on jumping numbers, introduce a new invariant of plurisubharmonic singularities and get its decreasing property for consecutive differences.

Subadditivity of generalized Kodaira dimensions and extension theorems

In this paper, we introduce the notion of generalized Kodaira dimension with multiplier ideal sheaves, and prove the subadditivity of these generalized Kodaira dimensions for certain Kähler fibrations, which was originally obtained for Kodaira dimensions of algebraic fiber spaces by Kawamata and Viehweg. Our method is analytic and based on some new results in recent years. The crucial step in our proof is to prove an [Formula: see text] extension theorem for twisted pluricanonical sections on compact Kähler manifolds. Moreover, we also discuss the relation between two previous optimal [Formula: see text] extension theorems with singular weights on weakly pseudoconvex Kähler manifolds.

The metric geometry of singularity types

Let X be a compact Kähler manifold. Given a big cohomology class {\{\theta\}}, there is a natural equivalence relation on the space of θ-psh functions giving rise to {\mathcal{S}(X,\theta)}, the space of singularity types of potentials. We introduce a natural pseudo-metric {d_{\mathcal{S}}} on {\mathcal{S}(X,\theta)} that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge–Ampère equations with varying singularity type converge as governed by the {d_{\mathcal{S}}}-topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

ON COMPLEX HOMOGENEOUS SINGULARITIES

We study the singularity at the origin of $\mathbb{C}^{n+1}$ of an arbitrary homogeneous polynomial in $n+1$ variables with complex coefficients, by investigating the monodromy characteristic polynomials $\unicode[STIX]{x1D6E5}_{l}(t)$ as well as the relation between the monodromy zeta function and the Hodge spectrum of the singularity. In the case $n=2$, we give a description of $\unicode[STIX]{x1D6E5}_{C}(t)=\unicode[STIX]{x1D6E5}_{1}(t)$ in terms of the multiplier ideal.

On the singularity type of full mass currents in big cohomology classes

Let $X$ be a compact Kähler manifold and $\{\unicode[STIX]{x1D703}\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $\unicode[STIX]{x1D703}$-plurisubharmonic functions with full mass are the same as those of a current with minimal singularities. Second, given another big and nef class $\{\unicode[STIX]{x1D702}\}$, we show the inclusion ${\mathcal{E}}(X,\unicode[STIX]{x1D702})\cap \operatorname{PSH}(X,\unicode[STIX]{x1D703})\subset {\mathcal{E}}(X,\unicode[STIX]{x1D703})$. Third, we characterize big classes whose full mass currents are ‘additive’. Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.