复微分几何中的几个问题

We plan to study the following three topics in complex geometry: (1) a set of Chern number inequalities conjectured to be satisfied by compact Kahler manifolds with nonpositive sectional curvature. We propsoed a general conjecture in arbitrary dimension, based on what is known in the two-dimensional case. We would like to investigate the three-dimensional cases here. The geography of Chern numbers has been a difficult problem in complex geometry and algebraic geometry. We hope that our study in the three-dimensional cases would help to advance the knowledge in this direction. (2) The differntial-geometric approach towards the generalized Hartshorne conjecture. This is a classic problem in complex geometry. Our starting point is centered around the nonnegativity of the so-called quadratic bisectional curvature, a condition that has been receiving more and more attention recently. It has been proved that the majority (but not all) of the Kahler C-spaces with second betti number equal to 1 do satisfy this condition. Thus it becomes the first known geometric quantity whose nonnegativity is satisfied by some Kahler C-space which is not Hermitian symmetric (note that the nonnegativity of bisectional curvature or orthogonal bisectional curvature implies that the manifold must be a Hermtian symmetric space).(3) Real Kahler Euclidean submanifolds with small codimensions. Typically, the study of Euclidean submanifolds are in the real (namely, Riemannian) case, while the complex (Kahler) case is not well-studied at all. We find that there are actually lots of questions that one can ask about such submanifolds, and we raised a few here in this proposal.

我们希望研究复几何中的以下三个子方向：（一）非正曲率的紧凯勒流形的陈数所必须满足的一组不等式。我们根据二维时的情形提出了关于任意维数的猜测，并希望研究三维时的情形。陈数地理分布的问题一直是复几何和代数几何中的一个难题，我们希望通过对三维特殊情形的研究推进这方面的进展。（二）关于广义Hartshorne猜测的微分几何的探讨。这是复几何中的一个传统问题，我们的出发点是围绕近年来比较热门的一个几何条件，　即所谓的二次双截曲率非负的条件。已经发现的事实是，这个条件被大部分（但不是全部）齐性凯勒流形所满足，因而成为第一个能容纳非对称空间的条件。之前的双截曲率或垂直双截曲率的非负性都被证实只能用来刻画对称空间。（三）关于小余维的实凯勒子流形的结构性和凯勒延拓的研究。欧氏空间中的子流形几何学通常都是研究实的性情，复的情形研究很少。我们发现，这方面实际上有大量的问题可提可做，在这里我们提出了几个典型问题。

本课题属于基础数学中微分几何领域的研究，主要围绕着有关复流形的有关课题。 复流形是数学中普遍关注的研究对象，在代数几何，拓扑学，大范围分析，多复变， 以及数学物理中都被实质性地涉及。我们主要从微分几何的角度出发， 关注给定的复流形上能有什么样的最佳厄米度量存在。在本项基金的资助下，我们获得了三项研究成果， 其一是得到余维不超过4 的实凯勒子空间的结构定理， 将1990年代的结构性定理从余维1和2的情形推广到了余维3和4的情形； 其二是得到了实凯勒子空间的一个柱面定理； 其三是系统性地研究了两类特殊的厄米度量： 其黎曼曲率或陈曲率具有凯勒曲率的所有对称性。 我们分别称之为 “G-凯勒似的” 或 “凯勒似的”。当流形为紧时， 这两种情形都是平衡度量。这几项研究都具有原创性， 在实凯勒流形和非凯勒流形的研究上具有国际前沿性， 我们相信其将对这两个子方向的进一步发展起到推动作用。