Rational Curves on Complex Manifolds

We give an overview of the study of rational curves on a complex manifold. Starting from Mori’s celebrated theorem on the extremal rays of the cone of curves, we describe some of its applications, and some of the tools used to produce rational curves in Algebraic Geometry and in Kähler Geometry.     

Laplacian on Complex Finsler Manifolds

In this paper, the Laplacian on the holomorphic tangent bundle T1,0M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using the Chern Finsler connection associated with (M,F). Utilizing the initiated “Bochner technique”, a vanishing theorem for vector fields on the holomorphic tangent bundle T1,0M is obtained.     

Complex Powers and Non-compact Manifolds

Abstract

We study the complex powers A ^z of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called “Guillemin algebras, ” whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyl's formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131–160]. A Guillemin algebra can be thought of as an algebra of “abstract pseudodifferential operators.” Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,…) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for A ^z , when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melrose's work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).     

Almost Complex Manifolds and Hyperbolicity

One of the sufficient conditions for a complex manifold to be (complete) hyperbolic (in the sense that its intrinsic pseudo-distance is a (complete) distance) is that it admits a (complete) Hermitian metric with holomorphic sectional curvature bounded above by a negative constant. The concept of hyperbolicity can be readily extended to almost complex manifolds. We will show that the above result for hyperbolicity can be generalized to the almost complex case. As an application, we prove that every point of an almost complex manifold has a complete hyperbolic neighborhood. In real dimension 4, this fact was established by Debalme and Ivashkovich [2] by a completely different method.     

复Finsler流形上的Hodge-Laplace算子

本文定义了强拟凸复Finsler流形上的Hodge-Laplace算子,并给出其水平部分的局部坐标表示．     

关于复Finsler流形的一个注记

设(M,G)为n维复Finsler流形,TM为M的全纯切丛,得到了TM上的Hermite度量hTM=G(-ij)(z,v)dzi(×)d(-z)j+G(-ij)(z,v)δvi(×)δ(-v)j为K(a)hler度量的充要条件是M为全纯曲率为0的Kahler流形,其中G(-ij)=(а)2G/(а)vi(а)(-v)j,1≤i,j≤n.推广了Cao-Wong的某些结果.     

A Remark on the Reduction of Cauchy–Riemann Manifolds

The concept of a moment map for an action of a real Lie group by CR–diffeomorphisms on a CR–manifold M of hypersurface type is introduced. It gives rise to a natural reduction procedure in the sense that we construct a CR–structure on an associated orbit manifold M₀ for the case that the group acts freely and properly on M.     

Monodromy Invariants – From Symplectic to Smooth Manifolds

Recently, together with Auroux and Donaldson, we have introduced some new invariants of four-dimensional symplectic manifolds. Building on the Moishezon–Teicher braid factorization techniques, we show how to compute fundamental groups of compliments to a ramification curve of generic projection. We also show that these fundamental groups are only homology invariants and outline the computations in some examples.Demonstrating the ubiquity of algebra, we go further and, using Braid factorization, we compute invariants of a derived category of representations of the quiver associated with the Fukaya–Seidel category of the vanishing cycles of a Lefschetz pencil and a structure of a symplectic four-dimensional manifold. This idea is suggested by the homological mirror symmetry conjecture of Kontsevich. We do not use it in our computations, although everything is explicit. We outline a procedure for finding homeomorphic, nonsymplectomorphic, four-dimensional symplectic manifolds with the same Saiberg–Witten invariants. This procedure defines invariants in the smooth category as well.     

Some Remarks on Almost and Stable Almost Complex Manifolds

In the first part we give necessary and sufficient conditions for the existence of a stable almost complex structure on a 10-manifold M with H₁(M;?) = 0 and no 2-torsion in H₁(M;?) for i = 2,3. Using the Classification Theorem of Donaldson we give a reformulation of the conditions for a 4-manifold to be almost complex in terms of Betti numbers and the dimension of the ±-eigenspaces of the intersection form. In the second part we give general conditions for an almost complex manifold to admit infinitely many almost complex structures and apply these to symplectic manifolds, to homogeneous spaces and to complete intersections.     

Collars in Complex and Quaternionic Hyperbolic Manifolds

We use the complex and quaternionic hyperbolic versions of Jørgensen's inequality to construct embedded collars about short, simple, closed geodesics in complex and quaternionic hyperbolic manifolds. In general, the width of these collars depend both on the length of the geodesic and on the rotational part of the group element uniformising it. For complex hyperbolic space we are able to use a lemma of Zagier to give an estimate based only on the length. We show that these canonical collars are disjoint from each other and from canonical cusps. We also calculate the volumes of these collars.     

Number Variance of Random Zeros on Complex Manifolds

We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set with smooth boundary is asymptotic to , where is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact K?hler manifold. Received: August 2006 Revision: March 2007 Accepted: April 2007     

Cohomologies of Generalized Complex Manifolds and Nilmanifolds

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fiber bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.     

从积不出来的积分到黎曼曲面理论

从介绍椭圆和双纽线弧长积分等积不出来的椭圆积分开始,给出数学家法尼亚诺和欧拉导出这些椭圆积分加法公式的计算过程,以及经典的阿贝尔定理的具体含义.讲解大数学家黎曼引入的黎曼曲面的重要概念,并且从几何的角度给出椭圆积分之所以积不出来的内在原因.介绍黎曼在黎曼曲面理论方面的重要工作及其对代数几何与现代数学发展的深远影响.     

A Note on the Wong-Rosay Theorem in Complex Manifolds

The authors prove a version, in utmost generality, of the Bun Wong-Rosay theorem on a complex manifold M. The essence of the result is that a domain Ω?M with non-compact automorphism group and boundary orbit accumulation point that is strongly pseudoconvex must be biholomorphic to the unit ball in C ⁿ .     

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.     

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.     

乘积复流形上的Szabó度量

设(M_1,α),(M_2,β)均为Hermitian流形,本文证明了积流形M_1×M_2上的复Szabó度量F_ε是Berwald度量,且当α,β为K(?)hler度量时,F_ε是强Kahler-Finsler度量,此外本文还给出了F_ε的全纯曲率的显式表达式．     

Global Volterra Series for Nonlinear Meromorphic Systems on Complex Manifolds

A global Volterra series is obtained for an meromorphic systemdefined on a complex manifold c. Local bilinearizations arepieced together by a fibre bundle giving rise to a ‘twisted’bilinear system on c.