Related papers: Complex manifolds with negative curvature operator

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…

The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P_q), a potential theoretic condition on $(0,q)$-forms that generalizes Catlin's property (P_q) to CR…

We record two remarks. First, for a compact K\"ahler manifold with semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration is exactly the number of non-truly-flat directions. Second, for compact K\"ahler…

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a…

Let M be a compact complex surface which admits a Kaehler metric whose scalar curvature has integral zero; and suppose the fundamental group of M does not contain an Abelian subgroup of finite index. Then if M is blown up at sufficiently…

We prove that compact K\"ahler manifolds whose sectional curvatures are close to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and…

In this paper, we announce the following results: Let M be a Kaehler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler-Ricci…

In this paper, we prove some rigidity theorems for compact Bach-flat $n$-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 1.4 have the additional properties of being sharp.

Recent work on the relation between a special class of K\"ahler manifolds with positive first Chern class and critical N$=$2 string vacua with c$=$9 is reviewed and extended. (Based on a talk presented at the International Workshop on…

On a compact K\"ahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a K\"ahler metric with semi-negative holomorphic sectional curvature.…

In this paper, we establish a K\"ahlerian or projectivity criterion for a class of compact Hermitian surfaces with non-positive second Chern-Ricci curvature.

In this paper, we present a unified flow approach to prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree. When the conformal class of its Hermitian metric contains a balanced metric, we…

We adapt Bost's algebraicity characterization to the situation of a germ in a compact K\"ahler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-P\u{a}un and of Druel to foliations on compact K\"ahler…

In this paper, we first prove that a compact K\"ahler manifold is projective if it satisfies certain quasi-positive curvature conditions, including quasi-positive $S_2^\perp,\, S_2^+,\,\mbox{Ric}_3^\perp, \,\mbox{Ric}_3^+$ or…

By studying a complex Monge-Amp\`ere equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact K\"ahler manifold $N^n$ with $\Ric_k< 0$ for some integer $k$ with $1<k<n$, and the…

Given any integer $n\geq 2$, we construct a compact K\"ahler-Einstein manifold of dimension n of negative sectional curvature which is not covered by the ball.

In a recent work, Kai Tang conjectured that any compact Hermitian manifold with non-zero constant mixed curvature must be K\"ahler. He confirmed the conjecture in complex dimension $2$ and for Chern K\"ahler-like manifolds in general…

Let $(X,\omega)$ be a compact K\"{a}hler manifold with a K\"{a}hler form $\omega$ of complex dimension $n$, and $V\subset X$ is a compact complex submanifold of positive dimension $k<n$. Suppose that $V$ can be embedded in $X$ as a zero…

The property of admitting an astheno-K\"ahler metric is not stable under the action of small deformations of the complex structure of a compact complex manifold. In this paper, we prove necessary cohomological conditions for the existence…

We conjecture that certain curvature invariants of compact hyperk\"ahler manifolds are positive/negative. We prove the conjecture in complex dimension four, give an "experimental proof" in higher dimensions, and verify it for all known…