Stable bundles on Hopf manifolds

Classification of holomorphic Pfaff systems on Hopf manifolds

We classify holomorphic Pfaff systems (possibly non locally decomposable) on certain Hopf manifolds. As consequence, we prove some integrability results. We also prove that any holomorphic distribution on a general (non-resonance) Hopf manifold is integrable.

关于Hopf面上不规则束集合的渺小性

On the meagerness of the set of irregular bundles on Hopf surfaces

研究Hopf面上稳定束的模空间中不规则束的渺小性，通过证明其在模空间中为渺小集，揭示其在拓扑结构中的稀疏性。

例外霍普夫映射、权移位及其在向量丛上的应用

Exotic Hopf maps, weight shifting and applications to vector bundles

本研究借助动机同伦论构造整数环上霍普夫映射同纬映射的显式多项式代表元，并由此得到整数环上三维射影空间的Jouanolou装置上的显式秩2向量丛。

正交丛的各向同性截面的Hopf指数

A Hopf index for isotropic sections of orthogonal bundles

将Hopf指数推广到带有二次型的丛的各向同性截面，得到八个类似物，应用于余截面局部化虚拟循环和DT^4虚拟循环。

Automorphisms of Hopf manifolds

We prove that the automorphism groups of Hopf manifolds are Jordan.

On Gauss-Bonnet and Poincaré-Hopf type theorems for complex $\partial$-manifolds

We prove a Gauss-Bonnet and Poincaré-Hopf type theorems for complex $\partial$-manifold $\tilde{X} = X - D$, where $X$ is a complex compact manifold and $D$ is a reduced divisor. We will consider the cases such that $D$ has isolated singularities and also if $D$ has a (not necessarily irreducible) decomposition $D=D_1\cup D_2$ such that $D_1$, $D_2$ have isolated singularities and $C=D_1\cap D_2$ is a codimension $2$ variety with isolated singularities.

Holomorphic bundles on diagonal Hopf manifolds

Let $A$ be a diagonal linear operator on $\C^n$, with all eigenvalues satisfying $0<|α_i|<1$, and $M = (\C^n\backslash 0)/<A>$ the corresponding Hopf manifold. We show that any stable holomorphic bundle on $M$ can be lifted to a $G$-equivariant coherent sheaf on $\C^n$, where $G=(\C^*)^l$ is a Lie group acting on $\C^n$ and containing $A$. This is used to show that all stable bundles on $M$ are filtrable, that is, admit a filtration by a sequence $F_i$ of coherent sheaves, with all subquotients $F_i/F_{i-1}$ of rank 1.

Classification of holomorphic foliations on Hopf manifolds

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 holomorphic distributions on Hopf manifolds.

A decomposition theorem for projective manifolds with nef anticanonical bundle

Let X be a simply connected projective manifold with nef anticanonical bundle. We prove that X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC fibration of any projective manifold with nef anticanonical bundle.

Stable bundles on non-Kahler elliptic surfaces

In this paper, we study the moduli spaces $\mathcal{M}_{δ,c_2}$ of stable rank-2 vector bundles on non-K\" ahler elliptic surfaces, thus giving a classification these bundles; in the case of Hopf and Kodaira surfaces, these moduli spaces admit the structure of an algebraically completely integrable Hamiltonian system.

Embedding of LCK manifolds with potential into Hopf manifolds using Riesz-Schauder theorem

An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theorem and Montel theorem. We give an alternative argument for complex surfaces, deducing embedding theorem from the Spherical Shell Conjecture.

关于半充足向量丛与可平行化紧复流形

On semiample vector bundles and parallelizable compact complex manifolds

该研究利用全纯对称微分刻画可平行化紧复流形及其商空间，证明了不同半充足性余切丛下小平维数为0的紧复流形的结构，还得到相关基本群性质并提出猜想。

Ricci-flat Deformations of Holomorphic Vector Bundles

In this paper we give a criterion for a deformation of a hermitian vector bundle to be Ricci-flat. As an application we show that on a Kähler manifold, every deformation of a vector bundle can be made Ricci-flat whereas on some Hopf manifolds, the non-existence of a Ricci-flat deformation is related to non-trivial vector bundles on the universal cover ${\mathbb C}^n\setminus\{0\}$. On a surface with $b_1(X)\not=0$ filtrable Ricci-rigid vector bundles prove to be very special. We apply this to Inoue and Hopf surfaces.

Tropical Hopf manifolds and contracting germs

Classical Hopf manifolds are compact complex manifolds whose universal covering is $\mathbb{C}^d \setminus \{0\}$. We investigate the tropical analogues of Hopf manifolds, and relate their geometry to tropical contracting germs. To do this we develop a procedure called monomialization which transforms non-degenerate tropical germs into morphisms, up to tropical modification. A link is provided between tropical Hopf manifolds and the analytification of Hopf manifolds over a non-archimedean field. We conclude by computing the tropical Picard group and $(p,q)$-homology groups.

On singular variants of the Singer-Hopf Conjecture

We propose singular variants of the Singer-Hopf conjecture, formulated in terms of the Euler-Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove the conjecture under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle.

高阶向量包在霍夫表面上的研究

Higher rank bundles on Hopf surfaces

研究高阶向量包在霍夫表面上的性质，证明存在具有特定$c_2$值的稳定包，并引入非常不可约包的概念。

非凯勒椭圆曲面上的（余）Higgs丛

(Co)-Higgs bundles on Non-Kähler Elliptic Surfaces

本文研究了非凯勒椭圆曲面上的Higgs与co-Higgs丛，给出了非平凡稳定Higgs丛和co-Higgs丛存在的底空间亏格条件及拓扑光滑性条件，并完整描述了秩2情形下的非平凡co-Higgs丛。

$L^2$-extension theorems for jet sections of nef holomorphic vector bundles on compact Kähler manifolds and rational homogeneous manifolds, I

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 tangent bundles are rational homogeneous spaces.

Hopf surfaces in locally conformally Kahler manifolds with potential

An LCK manifold with potential is a compact quotient M of a Kahler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on $X$ by holomorphic homotheties and maps f to a function proportional to f. It is known that M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman LCK manifold with potential contains a complex surface with normalization biholomorphic to a Hopf surface H. Moreover, H can be chosen non-diagonal, hence, also not admitting a Vaisman structure.

Curvature restrictions on a manifold with a flat Higgs bundle

We shall prove a semi-negative curvature property for a manifold with a flat admissible Higgs bundle.

Projective manifolds whose tangent bundle contains a strictly nef subsheaf

Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group $π_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space.

Flat Vector Bundles over Parallelizable Manifolds

We study flat vector bundles over complex parallelizable manifolds.

Manifolds with nef cotangent bundle

Generalising a classical theorem by Ueno, we prove structure results for manifolds with nef or semiample cotangent bundle.

Tangent bundle of a manifold of K$3^{[2]}$-type is rigid

We prove that the tangent bundle of a manifold of K$3^{[2]}$-type is rigid.

Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds

We construct covariant $q$-deformed holomorphic structures for all finitely-generated relative Hopf modules over the irreducible quantum flag manifolds endowed with their Heckenberger--Kolb calculi. In the classical limit these reduce to modules of sections of holomorphic homogeneous vector bundles over irreducible flag manifolds. For the case of simple relative Hopf modules, we show that this covariant holomorphic structure is unique. This generalises earlier work of Majid, Khalkhali, Landi, and van Suijlekom for line modules of the Podleś sphere, and subsequent work of Khalkhali and Moatadel…

Toric Fano manifolds with nef tangent bundles

In this note we prove that any toric Fano manifold with nef tangent bundle is a product of projective spaces. In particular, it implies that Campana-Peternell conjecture hold for toric manifolds.

Fano manifolds and stability of tangent bundles

We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.

Projective manifolds with splitting tangent bundles, I

We investigate the universal cover of projective threefolds whose tangent bundle is a direct sum of subbundles in case the Kodaira dimension is not 1 and 2. We also prove results on Fano manifolds with splitting tangent bundles in any dimension.

Fano manifolds of coindex three admitting nef tangent bundle

We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.

Complex manifolds with generating tangent bundles

Let $M$ be a close complex manifold and $TM$ its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then $M$ is a complex homogeneous manifold. Our proof depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces.

Extremal metrics on toric manifolds and homogeneous toric bundles

In this paper, we prove the Yau-Tian-Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles.

Cohomologies of harmonic bundles on quasi-compact Kaehler manifolds

In this note, we survey our recent work concerning cohomologies of harmonic bundles on quasi-compact Kaehler manifolds.

Cohomology of holomorphic line bundles and Hodge symmetry on Oeljeklaus-Toma manifolds

We prove the Hodge symmetry type result on the Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in the direct sum of holomorphic line bundles. Consequently, we show the vanishing and non-vanishing of Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in holomorphic line bundles.

Comodule Structures, Equivariant Hopf Structures, and Generalized Schubert Polynomials

In this article, the comodule structure of Chow rings of Flag manifolds $\operatorname{CH}(G/B)$ is described by Schubert cells. Its equivariant version gives rise to a Hopf structure of the equivariant cohomology of flag manifolds $H^*_B(G/B)$. We get two identities of generalized Schubert polynomials as explanations of the geometric facts.