arXiv research

Generalized Nakano positivity for certain singular cases.

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 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.

New curvature assumptions prove Nakano positivity for complex vector bundles.

Characterizes curvature positivity for Riemannian metrics on flat vector bundles.

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

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

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

Paper proves direct image sheaf positivity for certain Kähler fibrations.

Alternative metric defined on vector bundles, proving vanishing theorem.

We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the optimal $L^p$-estimate condition, the multiple coarse $L^p$-estimate condition, th…

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

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

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.…

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…

The paper explores curvature positivity on Kähler and quasi-Kähler flag 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 studies curvature properties of direct image bundles.

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