Some Results on Holomorphic Vector Bundles Over General Hopf Manifolds

1 Introduction In recent years, holomorphic vector bundles on non-algebraic manifolds received increased attention. The main problem is to determine which continuous complex vector bundle carries a holomorphic structure or furthermore a holomorphic filtrable structure. D. Mall studied vector bundles on the primary Hopf manifolds。     

Some Results of Holomorphic Vector Bundles on Non-primary Hopf Manifolds

In recent years,there are a lot of work on holomorphic vector bundles on non-algebraic     

一般Hopf曲面上向量丛的结构

得到了非主Hopf曲面上连续复向量丛全纯结构的存在性及可滤性问题的充要条件.     

一般Hopf曲面上向量丛的结构

得到了非主Hopf曲面上连续复向量丛全纯结构的存在性及可滤性问题的充要条件.     

Effective Bounds on Holomorphic Mappings into Complex Hyperbolic Manifolds

We give an explicit upper bound on the number of non-constantholomorphic maps from a quasi-projective manifold into a complexhyperbolic manifold of finite volume. This gives an effectiveversion of the results of Sunada and Noguchi.     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group.E is a holomorphic vectorbundle of rank r with trivial pull-back to W=C~n-{0}.We prove the existence of a non-vanishingsection of L(?)E for some line bundle on X and study the vector bundles filtration structure of E.These generalize the results of D.Mall about structure theorem of such a vector bundle E.     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group. E is a holomorphic vector bundle of rank r with trivial pull-back to W = ℂ ⁿ –{0}. We prove the existence of a non-vanishing section of L⊗E for some line bundle on X and study the vector bundles filtration structure of E. These generalize the results of D. Mall about structure theorem of such a vector bundle E. The research was supported by 973 Project Foundation of China and the Outstanding Youth Science Grant of NSFC (grant no. 19825105)     

Holomorphic Symplectic Manifolds and Lagrangian Fibrations

We introduce the geometrical nature of fibre space structures of an irreducible symplectic manifold and holomorphic Lagrangian fibrations.     

Functions Holomorphic along Holomorphic Vector Fields

The main result of the paper is the following generalization of Forelli’s theorem (Math. Scand. 41:358–364, 1977): Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function φ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary. K.T. Kim and G. Schmalz were supported by the Scientific visits to Korea program of the AAS and KOSEF. E. Poletsky was supported by NSF Grant DMS-0500880. G. Schmalz gratefully acknowledges support and hospitality of the Max-Planck-Institut für Mathematik Bonn.     

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C^*-algebras.Mathematics Subject Classifications (2000): primary 32Q60; secondary 53C15, 58B12.     

Flat Riemannian Manifolds as Torus Bundles

Vasquez showed that for any finite group G there exists a number n(G) such that for every flat Riemannian manifold M with holonomy group G there exists a fiber bundle T → M → N, where T is a flat torus and N is a flat manifold of dimension less than or equal to n(G). We show that n(H) ≤ n(G) if H Δ leftG or G = N ? H and use this result to describe groups with the Vasquez number equal to 2 or 3.     

Globalization of Holomorphic Actions on Principal Bundles

Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P → X. We show that if there is a holomorphic action of the complexification G^C of G on. X, this lifts to a holomorphic action of G^C on the bundle P → X. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification G^C of G acts holomorphically and transitively on G/H. We also show that the ends of a homogeneous complex manifold G/H with more than two ends essentially come from a space of the form S/Γ, where Γ is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and Γ being explicitly constructed in terms of G and H.     

Semi-stability for Holomorphic Bundles with Global Sections

In this paper, we establish a generalized Hitchin-Kobayashi correspondence between the τ-semi-stability and the existence of approximate τ-Hermitian-Einstein structure on holomorphic pair (E,Φ). As an application, we obtain the Bogomolov type inequality for τ-semi-stable holomorphic pair.     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kèahler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985(when the constant is zero or negative) and by Apostolov±Davidov±Muskarov in 1996(when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this a...