arXiv research

Study Miyaoka-Yau inequality for certain projective manifolds.

Paper proves structure of compact Kähler 3-folds with specific bundles.

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

The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.

Study on Kähler manifolds with non-positive mixed curvature and its implications.

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

Miyaoka-Yau inequality proven for smooth minimal models.

Researchers prove existence of cscK metrics on smooth minimal models.

Study on a weaker curvature condition for Kähler manifolds.

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …

Alternative proof of semipositivity and nefness for K-semistable log-Fano pairs.

We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the pro…

The paper studies foliations on smooth projective varieties and their properties.

The study of projective varieties with nef anticanonical divisors and log terminal singularities.

Explains differences and similarities of strictly nef and ample vector bundles.

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…

Study positive characteristic Fano 4-folds with nef tangent bundles.

In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety $X^n$ $(n>4)$ has strictly nef $Λ^2 TX$, then it is isomorphic to $\mathbb{P}^n$ or quadric $\mathbb{Q}^n$. We also prove that on elliptic curves, strict…

It is proved by Kawamata that the canonical bundle of a projective manifold is semi-ample if it is big and nef. We give an analytic proof using the Ricci flow, degeneration of Riemannian manifolds and $L^2$-theory. Combined with our earlier results, we construct unique singular Kahler-Einstein metrics with a global Rie…

Solves open problems on curved projective varieties.

The paper proves structures for complex projective varieties with certain tangent bundle properties.

We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.

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 proves inequalities for orbifold second Chern classes in Fujiki's class.

We generalize the maximal time existence of Kähler-Ricci flow in Tian-Zhang and Song-Tian to conical case. Furthermore, if the twisted canonical bundle $K_{M}+(1-β)[D]$ is big or big and nef, we can expect more on the limit behaviors of such conical Kähler-Ricci flow. Moreover, the results still hold for simple normal …

Study projective klt pairs with nef anti-canonical divisor and their properties.

In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ an…

Projective manifolds with specific bundles are isomorphic to simpler spaces.

Fundamental groups of certain Kähler orbifolds have polynomial growth.

The paper studies algebraic fibre spaces with specific properties and proves key results about their structure.

Study on volumes and scalar curvature in complex geometry, proving new bounds and conditions.

Study the Albanese map for Kähler manifolds with nef anticanonical bundle.

The paper classifies certain singular projective varieties with specific properties.

In this paper, we study smooth complex projective varieties $X$ such that some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If $T_X$ is strictly nef, then $X$ isomorphic to the projective space $\…

The paper establishes inequalities for Chern classes and numbers on polarized manifolds and nef vector bundles.

We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex $3$-folds of the form $P(T^*X)$ whose tangent bundles are nef. Moreover, we show that if $X$ is a Fano manifold such t…

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

Study compact Kähler manifolds with pseudo-effective tangent bundles.

We consider the Kähler Ricci flow on a smooth minimal model of general type, we show that if the Ricci curvature is uniformly bounded below along the Kähler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci curvature lower bound assumption, the Gromov-Hausdorff limit of the…

Study of tangent bundle positivity on complex projective varieties.

Let X --> B be a holomorphic submersion between compact Kahler manifolds of any dimension, whose fibres and base have no non-zero holomorphic vector fields and whose fibres all admit constant scalar curvature Kahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar cu…

Two types of nonvanishing results are presented for compact Kähler varieties.

In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divis…

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

Estimates Lelong numbers of Monge-Ampère products for Kähler manifolds.

Paper proves Hermitian-Yang-Mills metrics for stable Kähler bundles.

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…

Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cota…