关于6维近凯勒流形中二阶平行拉格朗日子流形的注记

证明6维严格近凯勒流形中的二阶平行拉格朗日子流形一定是全测地的,这推广了L.Vrancken等人文中的一个重要结果.特别地,得到了齐性近凯勒S³×S³中该类拉格朗日子流形的完全分类.     

代数函数域及模型的Kodaira维数

Iitaka 对特征零情形引进了代数簇的 Kodaira 维数的概念,由此发展的一套理论对代数几何中的双有理分类问题起到重要的作用(参见(3)和(7)).由罗昭华定义的(参见(6))任意特征代数函数域的 Kodaira 维数的概念是观察双有理问题的一个新的途径.在本文中,我们首先证明了罗意义下的 Kodaira维数当代数函数域进行某种特殊的扩张(即称为正则扩张)时是不变的.另外,我们定义了代数函数域之模型的 Kodaira 维数,并就此证明了关于代数簇的一个母纤维定理.     

近凯勒流形中一般子流形的淹没

本文讨论了近凯勒(nearly K(a|¨)hler)流形中一般子流形的淹没,证明了:如果π:M→B是近凯勒流形M中子流形M到殆埃尔米特(almost Hermitian)流形B的淹没,那么B是近凯勒流形.另外,本文给出了关于这种淹没分解理论的一些性质,并且研究了M和B的全纯截面曲率之间的关系.     

示性式的超渡(续完)

<正> 7.Grassmann流形与陈类 设为复N维向量酉空间中全体n维子空间所成的Grassmann流形,它是一个齐性空间     

Lie代数双极化与齐性抗仿凯勒流形的新进展

本文介绍Lie代数双极化与齐性仿凯勒流形的若干新进展,并提出了若干相关问题,指出该领域可能发展的若干方向。     

关于切丛和单位切球丛的度量的一个注记

本文在黎曼流形$(M,g)$的切丛$TM$ 上研究与参考文献[10]中平行的一类度量$G$以及相容的近复结构$J$.证明了切丛$TM$关于这些度量和相应的近复结构是局部共形近K\"{a}hler流形,并且把这些结构限制在单位切球丛上得到了切触度量结构的新例子.     

近切触流形的φ~*-解析向量场（英文）

本文引入了近切触流形(M,φ,ξ,η,g)中φ~*-解析向量场的概念,并研究了其性质.利用近切触流形的性质,证明了切触度量流形中的φ~*-解析向量场v是Killing向量场且φv不是φ*-解析的.特别地,如果近切触流形M是正规的,得到v与ξ平行且模长为常数.另外,证明了3维的切触度量流形不存在非零的φ~*-解析向量场.     

复Grassmann流形中的实迷向极小二维球面

本文研究了复Grassmann流形中极小曲面的两种迷向性质之间的关系.首先给出了实迷向的定义,之后用整体微分形式刻画实迷向性质,然后介绍Calabi的线丛上的联络理论作为主要计算方法,最终运用调和序列理论证明复迷向性质强于实迷向.作为应用,本文证明了复射影空间中的极小二维球面是实强迷向的.     

靶流形为齐次流形的弱次椭圆Q调和映射的正则性

该文证明了靶流形为齐次流形的弱次椭圆Q调和映射是内部正则的，这里Q是定义域的 齐次维数。这一结果推广了Hajlasz和Strzelecki的相应结果[2].作为推论得到了靶流形为齐次流形的p维p调和映射的正则性.     

3-对称Finsler流形（英文）

3-对称Finsler流形是3-对称黎曼流形的推广.本文给出了3-对称Finsler流形的定义,并将3-对称Finsler流形用齐性空间的形式表示.同时,本文还给出了在齐性空间上存在3-对称Finsler度量的条件,并讨论了3-对称Finsler流形与3-对称黎曼流形的关系.最后,本文给出了自然约化的3-对称Finsler流形的旗曲率和曲率张量.     

Kähler Metrics on the Projective Bundle of a Holomorphic Finsler Vector Bundle

Let (M, g) be a compact Kähler manifold and (E, F) be a holomorphic Finsler vector bundle of rank r ≥ 2 over M. In this paper, we prove that there exists a Kähler metric φ defined on the projective bundle P (E) of E, which comes naturally from g and F. Moreover, a necessary and sufficient condition for φ having positive scalar curvature is obtained, and a sufficient condition for φ having positive Ricci curvature is established.     

Kodaira dimensions and hyperbolicity of nonpositively curved compact Kähler manifolds

In this article, we prove that a compact Kähler manifold M ⁿ with real analytic metric and with nonpositive sectional curvature must have its Kodaira dimension, its Ricci rank and the codimension of its Euclidean de Rham factor all equal to each other. In particular, M ⁿ is of general type if and only if it is without flat de Rham factor. By using a result of Lu and Yau, we also prove that for a compact Kähler surface M ² with nonpositive sectional curvature, if M ² is of general type, then it is Kobayashi hyperbolic.     

Seiberg–Witten Invariants of Nonsimple Type and Einstein Metrics

We study the behavior of the moduli space of solutions to theSeiberg–Witten equations under a conformal change in the metric of aKähler surface (M,g). If the canonical line bundle K _M is ofpositive degree, we prove there is only one (up to gauge) solution tothe equations associated to any conformal metric to g. We use this, toconstruct examples of four dimensional manifolds withSpin ^c -structures, whose moduli spaces of solutions to theSeiberg–Witten equations, represent a nontrivial bordism class ofpositive dimension, i.e. the Spin ^c -structures are not inducedby almost complex structures. As an application, we show the existenceof infinitely many nonhomeomorphic compact oriented 4-manifolds withfree fundamental group and predetermined Euler characteristic andsignature that do not carry Einstein metrics.     

The Kodaira dimension of diffeomorphic Kähler threefolds

We provide infinitely many examples of pairs of diffeomorphic, non-simply-connected Kähler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of non-deformation equivalent, diffeomorphic Kähler threefolds.

     

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.     

Analysis and geometry of Floer theory of Landau-Ginzburg model on ℂn

This article studies the Floer theory of Landau-Ginzburg (LG) model on ℂⁿ: We perturb the Kähler form within a xed Kähler class to guarantee the transversal intersection of Lefschetz thimbles. The C⁰ estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.     

On Almost Hermitian 4-manifolds with J-invariant Ricci Tensor

We prove that every almost Hermitian 4-manifold with J-invariant Ricci tensor which is conformally flat or has harmonic curvature is either a space of constant curvature or a Kähler manifold. We also obtain analogous results on almost Kähler 4-manifolds.     

Flat nearly Kähler manifolds

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form $\zeta \in \Lambda^3 (\mathbb{C}^m)^*We classify flat strict nearly K?hler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K?hler factor of maximal dimension and a strict flat nearly K?hler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form . The first nontrivial example occurs in dimension 4m = 12.      

Compact Kähler manifolds with nonpositive bisectional curvature

Let (M ⁿ , g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N ^k with c ₁ <  0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.