arXiv research

Study Miyaoka-Yau inequality for certain projective manifolds.

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

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

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

Study on a weaker curvature condition for Kähler manifolds.

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

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

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 …

Explains differences and similarities of strictly nef and ample vector 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…

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…

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…

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…

Projective manifolds with specific bundles are isomorphic to simpler spaces.

Fundamental groups of certain Kähler orbifolds have polynomial growth.

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…

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

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

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…

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

Study numerically flat bundles on Fujiki manifolds using algebraic groups.

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

Study compact Kähler manifolds with pseudo-effective tangent bundles.

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

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

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

Study of tangent bundle positivity on complex projective varieties.

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

Researchers prove existence of cscK metrics on smooth minimal models.

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal Q}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have be…

We study the algebraic dimension a(X) of a compact hyperkaehler manfold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact Kaehler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0,n and 2n are possible. In case of middle dimension,…

Study Fano fibrations and Kähler-Einstein metrics on their bases.

Miyaoka-Yau inequality proven for smooth minimal models.

Log-conformal projective pairs restrict to simple geometric structures.

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

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

Let $\mathcal{O}(D)$ be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety $X$. Given a continuous toric metric $\|\cdot\|$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,φ_{\bar{D}})$ where $φ_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D…

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…

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

The paper proves a version of the SYZ conjecture for hyperkahler manifolds.

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 Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…