arXiv research

New system solves curvature for ample vector bundles, proving Griffiths conjecture.

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

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.

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

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

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\…

Proves complex Finsler bundles with positive curvature are ample and biholomorphic to projective space.

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

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

Compact Hermitian manifolds with quasi-negative curvature have ample canonical line bundles.

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

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

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 extends positivity results from vector bundles to Kobayashi positive ones.

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…

The paper studies geodesic-Einstein metrics on line bundles over fibration.

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…

Study on Kähler-Ricci flow's infinite-time singularities.

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

Characterizes projective submanifolds in high dimensions.

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…

For Fano homogeneous toric bundles, we obtain a formula of the greatest lower bound on Ricci curvature. We also give a criteria for the ampleness of a kind of line bundles over general homogeneous toric bundles.

Estimates curvature for long-time continuity method solutions.

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

Study local curvature of Kähler-Ricci flow on semi-ample manifolds.

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

The paper proves a CM line bundle is ample on K-moduli spaces.

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

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…

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

Study compact Kähler manifolds with nonpositive holomorphic sectional curvature and their canonical bundles.

Study of monodromy and vanishing cycles for complete intersection curves.

In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

In this paper, we study the relations between the log term of the Szegö kernel of the unit circle bundle of the dual line bundle of an ample line bundle over a compact Kählermanifold. We proved a local rigidity theorem. The result is related to the classical Ramadanov Conjecture.

The paper establishes conditions for optimal sampling configurations on complex manifolds.

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 $X$ be a smooth projective complex variety with an ample line bundle $L$, and let $D$ be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on $X-D$ and $μ_L$-stable parabolic $λ$-flat bundles with trivial characteristic numbers on $(X,D)$. Especially, …

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

The spectrum of the Laplace-Dolbeault operator for any line bundle with parallel curvature on a flat complex torus is computed. The Ray-Singer analytic torsion is then deduced, generalizing thus Bost's result for ample line bundles and Ray-Singer's ones for flat bundles, of which we a geometric interpretation is given.

The paper proves positivity of characteristic forms for certain vector bundles.

We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the…

The paper establishes inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.

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…

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

Study on positivity of CM line bundles on moduli space of klt good minimal models with κ=1.

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 …

Study stability thresholds of big line bundles, proving bounds and generalizing results.