arXiv research

Constructs a convex Finsler metric on vector bundles under specific conditions.

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.

New curvature assumptions prove Nakano positivity for complex vector bundles.

For any complex vector bundle $E^k$ of rank $k$ over a manifold $M^m$ with Chern classes $c_i \in H^{2i}(M^m,\Z)$ and any non-negative integers $l_1, >..., l_k$ we show the existence of a positive number $N(k,m)$ and the existence of a complex vector bundle $\hat E^k$ over $M^m$ whose Chern classes are $ N(k,m) \cdot l…

The paper extends positivity results from vector bundles to Kobayashi positive ones.

We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least th…

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

The paper extends Schwarz's lemma to RC-positivity and complex manifolds.

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

The paper finds many negatively curved Kähler metrics on complex manifolds.

The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E\times\R^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any sim…

We use a sheaf-theoretic approach to obtain a blow-up formula for Dolbeault cohomology groups with values in the holomorphic vector bundle over a compact complex manifold. As applications, we present several positive (or negative) examples associated to the vanishing theorems of Girbau, Kawamata-Viehweg and Green-Lazar…

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\…

Study on positivity properties of vector bundle Monge-Ampère equation.

We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.

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…

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

The paper proves positivity of third Chern form for certain vector bundles.

Characterizes curvature positivity for Riemannian metrics on flat vector bundles.

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

Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.

The paper proves conditions for vector bundles to be Kobayashi and Griffiths positive.

We generalize a construction of Hitchin to prove that, given any compact Kähler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a Kähler metric with positive holomorphic sectional curvature.

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

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any $n\in\mathbb{N}_{0}$ and…

Noncommutative Kähler structures were recently introduced by the second author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. I…

In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex proj…

Alternative metric defined on vector bundles, proving vanishing theorem.

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vecto…

Study shows some biquotients have non-biquotient tangent bundles.

New product manifolds can have non-negative curvature.

Study stabilizes second-order systems to first-order dynamics.

Study on Kähler manifolds with non-positive mixed curvature and its implications.

The paper proves a pointwise Gysin formula for vector bundles and applies it to show positivity of polynomials.

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

In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

Two remarks on curvature properties of Kähler manifolds.

Proves generic surjectivity of vector bundles via degeneration.

The paper proves conditions for Kähler-Einstein metrics on certain bundles.

The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.

Proves curvature positivity of invariant direct images in complex geometry.

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…

New findings on biquotient bundles lacking inverses in various dimensions.

Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contac…

The study of Chern numbers on vector bundles uses combinatorial methods to establish bounds and ordering.

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

This is a survey of results on positivity of vector bundles, inspired by the Brunn-Minkowski and Prékopa theorems. Applications to complex analysis, Kähler geometry and algebraic geometry are also discussed.