arXiv research

New proof shows holomorphic sectional curvature fully determines curvature tensor.

The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

Classifies Kähler metrics with constant holomorphic curvature.

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…

The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.

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.

Directly proves Wu's theorem on negative curvature metrics.

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…

The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.

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

We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant $δ$ depending on the ratio such that $\cosα\ge…

The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomor…

Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.

The paper proves a Schwarz lemma for mappings between specific types of manifolds.

Study curvature of blown-up manifold, finds negative holomorphic sectional curvature.

The study proves the non-existence of certain Kähler metrics with specific curvature properties.

We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold…

The main result of this note is that, for each $n\in \{1,2,3,\ldots\}$, there exists a Hodge metric on the $n$-th Hirzebruch surface whose positive holomorphic sectional curvature is $\frac{1}{(1+2n)^2}$-pinched. The type of metric under consideration was first studied by Hitchin in this context. In order to address th…

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identical…

Sharp lower bound for curvature in Kähler manifolds.

The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.

In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invar…

On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional…

Survey on two non-Kähler geometry conjectures.

Study proves properties of compact Hermitian surfaces with specific curvature conditions.

The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kaehler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizi…

We study the conditions under which a Kählerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for $(T^*M,G,J)$ to be a Kählerian manifold, is expres…

Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant θ-holomorphic sectional curvature for a number θ in (0,π/2) then M is of constant sectional curvature or a Kähler manifold of constant holomorphic sectional curvature. Th…

We study Kaehlerian manifolds with Norden metric $g$ and develop the theory of their holomorphic hypersurfaces with constant totally real sectional curvatures. We prove a classification theorem for the holomorphic hypersurfaces of $(\mathbb{R}^{2n+2}, g, J)$ with constant totally real sectional curvatures.

We prove that any $4$-dimensional almost-Kähler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is Kähler.

The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.

The study characterizes symmetries in Kaehler manifolds.

Study estimates the first eigenvalue on Kähler manifolds with specific curvature conditions.

The paper proves a structure theorem for compact Kähler manifolds with semi-positive holomorphic sectional curvature.

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.

The Wu-Yau theorem is verified for negative curvature, and new examples of Kähler-Einstein metrics are found.

The paper explores applications of Gauduchon metrics in complex geometry.

Establishes metrics with positive curvature on projective line bundles.

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio $λ$ of the maximum and the minimum of the holomorphic sectional curvatures $< 2$, then there exists a positive constant $δ>\frac{29(λ-1)}{\sqrt{(48-24λ)^{2}+(29λ-29…

It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.

The aim of this paper is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature.

Let $(M,ω)$ be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of $M$ is of general type. Moreover, we can extend the theorem to the…

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…

Curvature on Kähler toric manifolds

Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact Kähler manifolds. In this short note, we give anothe proof of their theorems by using the Kähler-Ricci flow.