arXiv research

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

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

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

The paper classifies certain singular projective varieties with specific properties.

In this paper, we study a projective klt pair $(X, Δ)$ with the nef anti-log canonical divisor $-(K_X+Δ)$ and its maximally rationally connected fibration $ψ: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+Δ)$ on $X$ coincides with that of the anti-log canonical div…

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

The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.

Establishes Kobayashi-Hitchin correspondence for nef and big classes.

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

In this paper, we show that the canonical divisor of a smooth toroidal compactification of a complex hyperbolic manifold must be nef if the dimension is greater or equal to three. Moreover, if $n\geq 3$ we show that the numerical dimension of the canonical divisor of a smooth $n$-dimensional compactification is always …

The Miyaoka-Yau inequality is proven for certain singular varieties with big canonical or anticanonical divisors.

The paper classifies minimal projective varieties satisfying a specific equality.

Solves open problems on curved projective varieties.

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 …

Fix a K3 lattice $Λ$ of rank two and $L\inΛ$ a big and nef divisor that is positive enough. We prove that the generic $Λ$-polarised K3 surface has an integral nodal rational curve in the linear system $|L|$, in particular strengthening previous work of the first named author. The technique is by degeneration, and also …

The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We e…

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…

The note proves positive currents induced by VKE with mixed singularities.

Study on Kähler-Einstein metrics on quasi-projective manifolds.

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

We study the algebraic dimension of twistor spaces of positive type over $4\bbfP^2$. We show that such a twistor space is Moishezon if and only if its anticanonical class is not nef. More precisely, we show the equivalence of being Moishezon with the existence of a smooth rational curve having negative intersection num…

Criterion for projectivisation on klt spaces, characterizing quotients and stability.

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.

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.

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.

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.

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…

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.