arXiv research

Study shows mass distribution of random holomorphic sections follows a central limit theorem.

Study shows normal distribution in divisor counts of random sections on complex manifolds.

The paper equidistributes zeros of random polynomials and sections on manifolds.

In this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space $X$. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of r…

The paper estimates variance of random sections on complex manifolds.

The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.

Study of random sections on complex spaces converging to equilibrium metrics.

The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.

Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.

We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.

Surveying random sections on Kähler manifolds, leading to metrics.

Study Bergman kernels and zero distributions of random sections on Kähler manifolds.

We study a tower of normal coverings over a compact Kähler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furth…

The study examines spaces of holomorphic sections vanishing along subvarieties in complex spaces.

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

Geometric quantization results for Riemann surfaces with semi-positive line bundles.

New proof shows holomorphic sectional curvature fully determines curvature tensor.

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.

Classifies Kähler metrics with constant holomorphic curvature.

Let $X$ be a compact normal complex space of dimension $n$, and $L$ be a holomorphic line bundle on $X$. Suppose $Σ=(Σ_1,\ldots,Σ_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $τ=(τ_1,\ldots,τ_\ell)$ is an $\ell$-tuple of positive real numbers, and consider the space $H^0_0 (X, L^p)$…

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

The study bounds dimensions and proves existence of holomorphic sections on Kähler Ricci shrinkers.

The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.

We generalize a construction of Hitchin to prove that, given any compact Kähler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a Kähler metric with positive holomorphic sectional curvature.

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

Directly proves Wu's theorem on negative curvature metrics.

The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…

The paper develops quantitative estimates for holomorphic sections over bounded domains.

We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with a single complex point has $C^{2,α}$-close sections such that any (non…

We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…

The paper proves a Schwarz lemma for mappings between specific types of manifolds.

Linear statistics of random zero sets are integrals of smooth differential forms over the zero set and as such are smooth analogues of the volume of the random zero set inside a fixed domain. We derive an asymptotic expansion for the variance of linear statistics of the zero divisors of random holomorphic sections of p…

The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomor…

The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.

The study identifies holomorphic sections on jet spaces of the Riemann sphere.

We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si…

Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.

Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.

The main result of this note is that, for each $n\in \{1,2,3,\ldots\}$, there exists a Hodge metric on the $n$-th Hirzebruch surface whose positive holomorphic sectional curvature is $\frac{1}{(1+2n)^2}$-pinched. The type of metric under consideration was first studied by Hitchin in this context. In order to address th…

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identical…

The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.

In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invar…

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant $δ$ depending on the ratio such that $\cosα\ge…

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 proves the non-existence of certain Kähler metrics with specific curvature properties.

The study characterizes symmetries in Kaehler manifolds.