代数几何与拓扑的交叉

The title characterizes well the type of research we plan to do, but is perhaps not very descriptive. To make it so, we have divided the proposal into three parts that are only modestly related: .(1) Topology and Hodge theory of Baily-Borel compactifications, We intend to analyse the homotopical and cohomological properties (including) mixed Hodge structures of Baily-Borel compacfitications. .(2) Configuration spaces, WZW-theory and polydifferentials. We want to identify the inner product on a conformal block (whose existence has been since long conjectured by physicists) in terms of toplogy and algebraigc geometry. .(3) Cohomological dimension of moduli spaces of curves. The goal is to determine the cohomological dimensions for constructible sheaves and for coherent sheaves on various moduli spaces parametrizing curves; for the moduli space of curves of genus g ≥ 2, M_g, we conjecture these numbers to be 4g − 5 resp. g − 2..(4) Moduli of hyperkaehler manifolds and automorphic forms. We want to construct algebra’s of automorphic forms that are suggested by the geometric invariant theory of hyperkaehler manifolds.

我们将该课题分成四个相对独立的部分:.(1)Baily-Borel紧化的拓扑和霍奇理论。我们将分析Baily-Borel紧化的同伦和上同调性质，以及其混合霍奇结构。.(2)构形空间，WZW理论和多微分。利用拓扑和代数几何的语言来刻画共形块（长期以来，物理学家猜想其存在）上的内积。.(3)曲线的模空间的上同调维数。我们的目标是决定各种曲线的模空间上的可构层和凝聚层的上同调维数；对亏格g大于或等于2的曲线的模空间Mg，我们猜想这些数分别为4g-5和g-2。.(4)超凯勒流形的模空间和自守形式。试构造超凯勒流形的几何不变量理论给出的自守形式的代数。

这项研究的目标是仿照有限阶分块的闭曲面微分同胚群的分类理论，以及带边紧曲面的拟-Anosov微分同胚的分类理论给出类似的结果。我期望这方面完整的理论应该解决Nielsen实现问题：M映射类群的有限子群是否可以提升成M上保持定向的有限群作用？这对于闭曲面是可行的，但是对于一般的4-流形则不行。我和Farb尝试研究对于特殊的4-流形，如K3曲面或有理曲面的底4-流形，Nielsen实现问题是否成立。对于K3曲面，我们得到了群是2阶的条件下，Nielsen实现问题的一个完整答案。例如，我们证明了广义Dehn扭曲不能提升到对合。这部分内容已经整理成文，很快便可以面世。最后，沿着Thurston的思路，我们还得到了微分同胚群的部分分类。