arXiv research

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

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…

Solves Teissier problem for nef classes on compact Kähler manifolds.

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 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.

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

Study b-divisors on Kähler manifolds linking them to currents.

New proof of $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes.

Fundamental groups of certain Kähler orbifolds have polynomial growth.

Study Miyaoka-Yau inequality for certain projective manifolds.

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

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

Projective manifolds with specific bundles are isomorphic to simpler spaces.

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

Study of tangent bundle positivity on complex projective varieties.

The paper proves conditions for solutions of $J$-equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.

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

We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.

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

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

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

Researchers prove existence of cscK metrics on smooth minimal models.

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

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…

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

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and $θ$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,θ)$, making it a complete geodesic metric space.

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

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

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 on a weaker curvature condition for Kähler manifolds.

The paper classifies certain singular projective varieties with specific properties.

We consider two applications of the strata of differentials of the second kind (all residues equal to zero) with fixed multiplicities of zeros and poles: Positivity: In genus $g=0$ we show any associated divisorial projection to $\overline{\mathcal{M}}_{0,n}$ is $F$-nef and hence conjectured to be nef. We compute the c…

Study confirms conjecture on Hermitian manifolds with bounded mass.

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

Study compact Kähler manifolds with pseudo-effective tangent bundles.

Miyaoka-Yau inequality proven for smooth minimal models.

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

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

Let M be a compact, holomorphically symplectic Kahler manifold, and $η$ a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of $η$ is parabolic, that is, its top power vanishes. We prove that all Lelong sets of $η$ are coisotropic. When M is generic, this is used to show that all Le…

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

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

Taubes established fundamental properties of $J-$holomorphic subvarieties in dimension 4 in \cite{T1}. In this paper, we further investigate properties of reducible $J-$holomorphic subvarieties. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is $J-$nef. For a spherical class…

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…

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…

Let $\{α\}$ and $\{β\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional Kähler manifold $X$ such that the difference of intersection numbers $\{α\}^n - n\,\{α\}^{n-1}.\,\{β\}$ is positive. We solve in a number of special but rather inclusive cases the quantitative part of Demailly's Transce…