arXiv research

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

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

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

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 conjecture about Riemann surfaces using PDEs.

Study on Hermitian manifolds with curvature, finding geometric properties.

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 …

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

Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.

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

Solved Demailly systems for Vortex bundles on manifolds.

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.

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

Griffiths extremal metrics solve complex Finsler equations and quantify Kähler geometry.

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

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

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

We shall show that $q$-semipositivity of the vector bundle $E$ over a Kähler total space $\mathcal X$ implies the Griffiths-semipositivity of the $q$-th direct image of $\mathcal O(K_{\mathcal X/B}\otimes E)$. As an application, we shall give a negative-curvature criterion for the generalized Weil-Petersson metric on t…

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

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

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

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

Alternative metric defined on vector bundles, proving vanishing theorem.

Study on algebraic curves' invariants and vanishing criteria.

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

The paper examines positivity properties of singular Hermitian metrics.

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 holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…

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

Griffiths' first obstruction formula for vector bundles is derived.

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 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…

Paper proves direct image sheaf positivity for certain Kähler fibrations.

Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.

Holomorphic foliations on complex manifolds without invariant complete intersections.

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations …

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

The paper studies curvature properties of direct image bundles.

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

Generalizes jet differential bounds and proves asymptotic Serre duality.

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 …

Improved bounds for algebraic degeneracy and hyperbolicity of hypersurfaces.

Study of curves in Lie sphere geometry using moving frames and variational principles.

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 investigate the fundamental group of Griffiths' space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word b…

We calculate the singular homology and Čech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda's proof that the first singular homology groups of these spaces are isomorphic.

The dual variety X* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X* is a hypersurface in the dual space (P^N)*. If dim X* < N - 1, then the variety X is called dually degenerate. The authors refine these definitions for a varie…