arXiv research

The paper proves injectivity and vanishing theorems on compact Kahler manifolds.

We prove the classical Nakano vanishing theorem with Hörmander $L^2$-estimates on a compact Kähler manifold using Siu's so called $\partial\dbar$-Bochner-Kodaira method, thereby avoiding the Kähler identities completely. We then introduce singular hermitian metrics on holomorphic vector bundles, and proceed to prove a …

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact Kähler manifold $X$, $S^kE\…

We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type v…

We show vanishing theorems of $L^2$-cohomology groups of Kodaira-Nakano type on complete Hessian manifolds. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}(Ω)$ on a regular convex cone $Ω$ with the Cheng-Yau metric for $p>q$.

The study bounds growth of Hodge numbers and computes $L^2$-Betti numbers for irregular varieties.

Alternative metric defined on vector bundles, proving vanishing theorem.

We prove that the $L^2$ metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle ten…

For proper surjective holomorphic maps from K"ahler manifolds to analytic spaces, we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by Nakano semi-positive vector bundles by means of the higher direct image sheaves, by using the theory of harmonic integrals developed by Takegoshi…

Let M be a compact locally conformal hyperkaehler manifold. We prove a version of Kodaira-Nakano vanishing theorem for M. This is used to show that M admits no holomorphic differential forms, and the cohomology of the structure sheaf $H^i(O_M)$ vanishes for i>1. We also prove that the first Betti number of M is 1. This…

The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.

The paper defines positivity for singular metrics on vector bundles and proves related theorems.

The paper proves extension theorems for complex manifolds with Levi $q$-concave domains.

Solves Lempert's question on Nakano semi-positivity preservation.

Local vanishing theorems for complex spaces with smooth boundaries.

Generalizes Nakano-positivity to Hilbert space fields.

New curvature assumptions prove Nakano positivity for complex vector bundles.

Characterizes curvature positivity for Riemannian metrics on flat vector bundles.

Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as tra…

Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.

Proves curvature positivity of invariant direct images in complex geometry.

Generalized Nakano positivity for certain singular cases.

The paper proves vanishing theorems for complex line bundles using a new adiabatic limit approach.

Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.

Estimates cohomology dimensions for Nakano q-semipositive line bundles.

New characterizations of curvature operators for specific forms via L2-estimates.

The paper studies curvature properties of vector bundles and their applications to quasi-Fuchsian space.

Study complex structures with perturbed differential operators to compute curvature-like operators and obtain vanishing results.

This paper is a sequel to \cite{Berndtsson}. In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kähler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Naka…

Researchers find spectral gaps in quantum flag manifolds using twisted operators.

The paper characterizes positivity of holomorphic vector bundles via $L^p$-estimates and extensions.

The paper explores curvature positivity on Kähler and quasi-Kähler flag manifolds.

Paper proves direct image sheaf positivity for certain Kähler fibrations.

The paper defines new types of positivity and proves properties of Schur forms for vector bundles.

Analyzes Saito vanishing theorem using $L^{2}$ methods.

In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, ω_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positi…

Study extends complex sections on non-holomorphic objects on Kähler manifolds.

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.

We give several generalizations of the Kodaira vanishing and embedding theorems for Kähler manifolds to the case where the relevent line bundle has a small region of negative curvature. To prove the vanishing theorems we adapt techniques of Elworthy-Rosenberg for vanishing theorems in Riemannian geometry. For the embed…

We introduce and study a notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and P{ă}un. We define what it means for such a metric to be curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the cuvature $Θ^h$ as a matrix of currents. We …

Analyzes semi-characteristics on specific manifolds, proving a vanishing theorem.

Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.

The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.

We prove the logarithmic divergence of equivariant analytic torsion for one-parameter degenerations of projective algebraic manifolds, when the coefficient vector bundle is given by a Nakano semi-positive vector bundle twisted by the relative canonical bundle.

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and $M\ne SO_0(2,2)/SO(2)\tm SO(2).$ Let E be any vector bundle over M, Then any E-valued $L^2$ harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps fro…

Vanishing theorem on CR manifolds with non-negative curvature.

We prove a vanishing theorem for the twisted de Rham cohomology of a compact manifold.