arXiv research

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

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

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

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…

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

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

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

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 …

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

The paper examines positivity properties of singular Hermitian metrics.

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 …

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).

Proves a conjecture about Riemann surfaces using PDEs.

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

Griffiths' first obstruction formula for vector bundles is derived.

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

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

Solved Demailly systems for Vortex bundles on manifolds.

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

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

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…

Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.

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

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

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…

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

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.

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…

We study the variational problem for $N$-parallel curves on a Finslerian surface by means of Exterior Differential Systems using Griffiths' method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler-Lagrange equations for this type of variational problem…

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.

Extends fibering theorems for 3-manifolds.

Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first defi…

We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual the Schubert VHS form a basis of the invariant characteristic cohomol…

In this paper we prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 4. This is accomplished by introducing a smaller yet equivalent line…

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…

We study a Laplacian operator related to the characteristic cohomology of a smooth manifold endowed with a distribution. We prove that this Laplacian does not behave very well: it is not hypoelliptic in general and does not respect the bigrading on forms in a complex setting. We also discuss the consequences of these n…

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…

A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in …

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…