arXiv research

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

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.

Projective manifolds with specific bundles are isomorphic to simpler spaces.

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…

Study compact Kähler manifolds with pseudo-effective tangent bundles.

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 $\…

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…

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…

Study of tangent bundle positivity on complex projective varieties.

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…

The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.

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

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…

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

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…

Solves open problems on curved projective varieties.

In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, w…

Fundamental groups of certain Kähler orbifolds have polynomial growth.

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

Study Miyaoka-Yau inequality for certain projective manifolds.

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

The study of Chern numbers on vector bundles uses combinatorial methods to establish bounds and ordering.

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

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

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

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

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

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

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

Researchers prove existence of cscK metrics on smooth minimal models.

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

Study on a weaker curvature condition for Kähler manifolds.

Miyaoka-Yau inequality proven for smooth minimal models.

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.

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

In this note, we continue the investigation of a projective Kähler manifold $M$ of semi-negative holomorphic sectional curvature $H$. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent {\it truly flat} directions of $H$ in the tangent spaces. We prove th…

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…

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…

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, …

Let $X$ be a smooth variety and let $L$ be an ample line bundle on $X$. If $π^{alg}_{1}(X)$ is large, we show that the Seshadri constant $ε(p^{*}L)$ can be made arbitrarily large by passing to a finite étale cover $p:X'\rightarrow X$. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an …

The paper establishes lower bounds on Yang-Mills functionals for fibrations.

Study on triviality of tangent and generalized tangent bundles of manifolds.

The article investigates conditions for isomorphism of singular tangent bundles.

Extends differential geometry concepts to manifolds with super tangent bundles.

Researchers compute the cohomology of an elliptic tangent bundle.