arXiv research

Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.

Paper proves positivity of Chern-Weil forms for certain vector bundles.

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

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

The paper examines positivity properties of singular Hermitian metrics.

The Griffiths conjecture asserts that every ample vector bundle $E$ over a compact complex manifold $S$ admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ to induce a Griffiths …

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

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

Alternative metric defined on vector bundles, proving vanishing theorem.

Unique solution found for Demailly's equation on stable bundles.

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

Proves a conjecture about Riemann surfaces using PDEs.

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

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

In this paper we study a particular version of the Hermitian curvature flow (HCF) over a compact complex Hermitian manifold $(M,g,J)$. We prove that if the initial metric has Griffiths positive (non-negative) Chern curvature $Ω$, then this property is preserved along the flow. On a manifold with Griffiths non-negative …

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

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

Solved Demailly systems for Vortex bundles on manifolds.

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

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

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…

Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.

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

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge--Ampère type, whose correspondi…

Metrics are semipositively curved if they meet a specific asymptotic condition.

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result to find some homotopy groups of the Brill-Noether loci for a generic curve.

Study on algebraic curves' invariants and vanishing criteria.

Study of weakly Kähler hyperbolic manifolds, proving Lang and Green-Griffiths conjectures.

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.

We exhibit examples of projective varieties with degenerate Gauss mappings and determine numerical invariants of such varieties. Our examples provide counter-examples to an asserted structure theorem of Griffiths and Harris (Ann. Sci. ENS 1979).

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 holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a c…

Griffiths' first obstruction formula for vector bundles is derived.

We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Gri…

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…

In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern-Weil form can be represented by a given form ? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Gr…

New stability criteria for vector bundles linked to Hermite-Einstein geometry.

The paper studies curvature properties of direct image bundles.

Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.

In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eige…

In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This dis…

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

Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $\mathbb{X}^{n-1} \subset \mathbb{P}^n(\mathbb{C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the `celestial' horizon lies near $d \geqslant 2n…

Generalizes jet differential bounds and proves asymptotic Serre duality.

The infinitesimal period relation (also known as Griffiths' transversality) is the system of partial differential equations constraining variations of Hodge structure. This paper presents a study of the characteristic cohomology associated with that system of pde.

I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.

We discuss the Morse estimates for the curvature of several metrics on Semple weighted projective bundle over a projective variety. Following Demailly works on holomorphic Morse inequalities we show an analogue of his results along the Green-Griffiths conjecture for invariant jets.