arXiv research

The paper shows how certain complex projective varieties can be broken down into simpler types.

Study compact Kähler manifolds with pseudo-effective tangent bundles.

Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.

In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$ to a flat projective manifold $Y$ such that its general fiber is rationally conn…

We study the distribution of the common zero sets of $m$-tuples of holomorphic sections of powers of $m$ singular Hermitian pseudo-effective line bundles on a compact Kähler manifold. As an application, we obtain sufficient conditions which ensure that the wedge product of the curvature currents of these line bundles c…

Study on $m$-positivity in Kähler manifolds with new Monge-Ampère-type equation.

Study shows non-polyhedral structure in moduli spaces for n≥8.

We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge…

We describe the positive cone and the pseudo-effective cone of a non-Kählerian surface. We use these results for two types of applications: - Describe the set $σ(X)$ of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-Kähhlerian surface, and decide whether the assignment $…

The paper shows how compact Kähler manifolds with a special bundle can be broken down into simpler parts.

Solves open problems on curved projective varieties.

Let $(X,g)$ be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure $J$ compatible with $g$, then the canonical bundle $K_X$ is not pseudo-effective and the Kodaira dimension $κ(X,J)=-\infty$. We also introduce the complex Yamabe number $λ_c(X)$ for compact …

In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp. negative) scalar curvature if and only if $K_X$ (resp. $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, …

The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the h…

The study constructs universal invariants for non-Archimedean metrics on projective varieties.

The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a rela…

Let $φ:\Bbb C^n\to X$ a holomorphic map to an $n$-dimensional connected compact complex manifold $X$. We establish links between the positivity properties of the canonical bundle of $X$ and the rate of growth of $φ$ which extend results of Kodaira and Kobayashi-Ochiai. For example: if the average degree of $φ$ on balls…

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex mani…

In this paper, with the aim of establishing a structure theorem for a compact Kähler manifold $X$ with semi-positive holomorphic sectional curvature, we study a morphism $φ: X \to Y$ to a compact Kähler manifold $Y$ with pseudo-effective canonical bundle. We prove that the morphism $φ$ is always smooth (that is, a subm…

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

The paper proves injectivity and vanishing theorems on compact Kahler manifolds.

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. …

This paper divides into two parts. Let $(X,ω)$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $ω$ satisfies the assumption that $\partial\overline{\partial}ω^k=0$ for all $k$, we generalize the volume of the cohomology class in the Kähler setting to the Hermitian setting, and prove that the volume is…

The paper proves a logarithmic partial derivative lemma and applies it to several geometric problems.

The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.