arXiv research

Extends residue theory to flags of holomorphic distributions.

The study generalizes a specific geometric correspondence to higher dimensions.

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

We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular hol…

Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. Svensson [7] has shown that this formula simplifies to a Bochner type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. Here, …

The paper studies distributions on surfaces and their connection to twistor spaces.

We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study th…

The study classifies and analyzes two-dimensional holomorphic distributions on a four-dimensional projective space.

This paper extends geometric structure theory to infinite type structures.

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

New insights into $(2,3,5)$-distributions via Legendrian curves.

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

Study cone structures on contact manifolds to understand their geometric properties.

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…

We show a residues formula for maps generically transversal to regular holomorphic distributions.

In this paper, the theory of functions of one complex variable is explored to study linearly full unramified holomorphic two-spheres with constant curvature in $G(2,n)$ satisfying that the generated harmonic sequence degenerates at position $2$. Firstly, we determine the value distribution of the curvature and give the…

Let $M$ be a super Riemann surface with holomorphic distribution $\mathcal{D}$ and $N$ a symplectic manifold with compatible almost complex structure $J$. We call a map $Φ\colon M\to N$ a super $J$-holomorphic curve if its differential maps the almost complex structure on $\mathcal{D}$ to $J$. Such a super $J$-holomorp…

Study CR-statistical submanifolds in holomorphic statistical spaces.

The article proves the existence of horizontal immersions into fat distributions and contact structures.

We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distr…

We study anti-holomorphic semi-invariant submersions from Kählerian manifolds onto Riemannian manifolds. We prove that all distributions which are involved in the definition of the submersion are integrable. We also prove that the O'Neill's tensor $\mathcal{T}$ vanishes on the invariant vertical distribution. We give n…

Study of flows on 7D manifolds with holomorphic properties.

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

A holomorphic Engel structure determines a flag of distributions $\mathcal{W}\subset \mathcal{D}\subset \mathcal{E}$. We construct examples of Engel structures on $\mathbf{C}^4$ such that each of these distributions is hyperbolic in the sense that it has no tangent copies of $\mathbf{C}$. We also construct two infinite…

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

Study of flows on complex manifolds with holomorphic properties.

This paper studies ruled real hypersurfaces in indefinite complex projective space.

The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kaehler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizi…

In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This dis…

We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.

Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prov…

Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.

We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.

The singular set of a foliation is always connected under certain conditions.

This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.

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 $H^0_0 (X, L^p)$…

Study horizontal discs in fat distributions, proving their existence.

We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+ε)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $ε\in {0,1}$ and $m \geq 2$. In particular, given such a distribution $\mathcal{N}$, say, we obtain algebraic conditions on the Weyl tensor and t…

Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.

We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples o…

In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi str…

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

Develops theory of d-holomorphic connections on Klein surfaces.

Survey on holomorphic structures on complex manifolds.

In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection $\bar{\nabla}$ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…

The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic las…

Holomorphic Lie algebroid connections on Riemann surfaces are characterized.

Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Cech …