arXiv research

Generalized Nakano positivity for certain singular cases.

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

Paper proves direct image sheaf positivity for certain Kähler fibrations.

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

Alternative metric defined on vector bundles, proving vanishing theorem.

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

Generalizes Nakano-positivity to Hilbert space fields.

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 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.

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.

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.

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…

Proves curvature positivity of invariant direct images in complex geometry.

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

Characterizes curvature positivity for Riemannian metrics on flat vector bundles.

New curvature assumptions prove Nakano positivity for complex vector bundles.

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

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 study the cohomology with high tensor powers of Nakano $q$-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the o…

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 …

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 …

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…

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

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

The paper proves positivity of characteristic forms for certain vector bundles.

Analytic torsion behavior studied for degenerating manifolds with equivariant bundles.

For one-parameter degenerations of compact Kähler manifolds, we determine the asymptotic behavior of the first Chern form of the direct image of a Nakano semi-positive vector bundle twisted by the relative canonical bundle, when the direct image is equipped with the L2-metric.

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…

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…

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

New characterization of Riemannian metric positivity and $L^2$ estimates for $d$ operator.

New system solves curvature for ample vector bundles, proving Griffiths conjecture.

The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we con…

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

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…

The paper studies curvature properties of direct image bundles.

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds of dimension $n$ and a relatively ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images.…

We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the $L^2$-Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem e.g., var…

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

Given an effectively parameterized family $f:X\to S$ of canonically polarized manifolds, the Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $K_{X/S}$. We use a global elliptic equation to show that this metric is strictly positive everywhere and give estimates. The dire…

The paper improves $L^2$-estimates for Dirac-Dolbeault operators on complex manifolds.

Proves positive mass theorem for AF spin manifolds with conical singularities.

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

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 $p>q$.

Motivated by our conjecture of an earlier work predicting the degeneration at the second page of the Frölicher spectral sequence of any compact complex manifold supporting an SKT metric $ω$ (i.e. such that $\partial\bar\partialω=0$), we prove degeneration at $E_2$ whenever the manifold admits a Hermitian metric whose t…

The paper examines positivity properties of singular Hermitian metrics.