arXiv research

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

Proves polynomial injectivity of Fubini-Study map for ample line bundles.

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

The paper studies volumes of direct images for high tensor powers of ample bundles.

Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense o…

Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.

We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties.

Shows CM line bundles are ample on K-stable varieties.

The paper studies positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.

Constructs currents and heights on K3 surfaces.

It is conjectured that the moduli b-divisor of the Kawamata-Kodaira canonical bundle formula associated to a klt-trivial fibration $(X,B)\to Z$ is semi-ample. In this paper, we show the semi-ampleness of an arbitrarily small perturbation of the moduli b-divisor by a fixed appropriate divisor which roughly speaking come…

Continuous metrics on ample bundles lie in infinite-dimensional cones.

Proves projectivity and ampleness of a Kähler manifold using complex Monge-Ampère equation.

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact Kähler manifold $X$, $S^kE\…

On a compact Kähler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the Kähler-Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration.

We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at…

One of the key requirements for incorporating machine learning into the drug discovery process is complete reproducibility and traceability of the model building and evaluation process. With this in mind, we have developed an end-to-end modular and extensible software pipeline for building and sharing machine learning …

The paper solves a 25-year-old problem about maximal growth distributions on manifolds.

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. …

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, …

The paper proves conditions for vector bundles to be Kobayashi and Griffiths positive.

In this paper, we introduce the associated geodesic-Einstein flow for a relatively ample line bundle $L$ over the total space $\mathcal{X}$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $(\mathcal{X}, L)$ is nonlinear semistable if the {associated} Donaldson …

Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety…

Let (X,D) be a klt pair. Assuming either K_X+D big or -(K_X+D) ample, and that the coefficients of D are greater than 1/2, we show that the Kähler-Einstein metric attached to (X,D) -whenever it exists- has cone singularities along D on the log-smooth locus of the pair intersected with the ample locus of K_X+D (in the n…

Characterizes projective submanifolds in high dimensions.

Estimates curvature for long-time continuity method solutions.

The unit ball is characterized by a Kähler-Einstein potential.

Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

Consider a divisor D with simple normal crossings in a compact Kähler manifold X. We show in this article that a Kähler metric in an arbitrary class, with constant scalar curvature and cusp singularities along the divisor is unique in this class when K[D] is ample. This we do by generalizing Chen's construction of appr…

We prove that the $L^2$ metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle ten…

The paper proves boundedness of envelopes in complex manifolds.

Classifies surfaces with T-singularities and ample canonical class.

Let $S$ be a smooth projective variety and $Δ$ a simple normal crossing $\mathbb{Q}$-divisor with coefficients in $(0,1]$. For any ample $\mathbb{Q}$-line bundle $L$ over $S$, we denote by $\mathscr{E}(L)$ the extension sheaf of the orbifold tangent sheaf $T_S(-\log(Δ))$ by the structure sheaf $\mathcal{O}_S$ with the …

Constructs projective moduli spaces for Calabi-Yau pairs.

Study Kähler-Ricci flow on rational homogeneous varieties using algebraic geometry and representation theory.

The paper extends positivity results from vector bundles to Kobayashi positive ones.

Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavio…

Let $π:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $π_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for la…

Let X be a smooth complex projective variety of dimension d. It is classical that ample line bundles on X satisfy many beautiful geometric, cohomological, and numerical properties that render their behavior particularly tractable. By contrast, examples due to Cutkosky and others have led to the common impression that t…

In this short note, we prove the existence of constant scalar curvature Kähler metrics on compact Kähler manifolds with semi-ample canonical bundles.

Complex projective varieties are quotients of polydiscs under specific group actions.

Study of monodromy and vanishing cycles for complete intersection curves.

The paper classifies minimal projective varieties satisfying a specific equality.

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds of dimension $n$ and a relatively ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images.…

The proof of the comparison principlein [EGZ11] is not complete. We provide here an alternative proof, valid in the ample locus of any big cohomology class, and discuss the resulting modifications.

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

Sufficient condition for log-continuity of complex Monge-Ampère solutions.