Combinatorial structures on Riemann surfaces and topological properties of the moduli space

我们考虑黎曼曲面与阿贝尔微分构造闪电对。闪电对是由某种组合数据确定的复平面上的分段线性回路。在对该方法的研究过程中，我们得到了以下结果：1. 1.我们从几何上定义了Haefliger（6，3）-纽结的解纽数，即在6维球体中光滑嵌入的3维球体，并确定了每个Haefliger（6，3）-纽结的解纽数。2.我们在S^2上的S^3-丛上构造了一个新的Einstein度量的无穷级数，它包含了无穷多个非齐次的Einstein度量.研究了射影空间和SL 3上的Beilinson和伯恩斯坦关于D^^-^<（m）>的局部化定理，证明了倾斜层是由Frobenius自同态取结构层象的对偶而得到的.给出了莫利定理的一个简短证明。

We consider Riemann surfaces with abelian differentials constructed from lightning pairs. A lightning pair is a piecewise linear loop in the complex plane determined by a certain kind of combinatorial data. During the study of this method we obtained the following results:1. 1. We gave geometrically a definition of unknotting numbers of a Haefliger (6,3)-knot, which means a smoothly embedded 3-dimensional sphere in the 6-dimensional sphere, and determined the unknotting number of each Haefliger (6,3)-knot.2. We constructed explicitly a new infinite series of Einstein metrics on the S^3-bundles over S^2, which containing infinite numbers of inhomogeneous ones.3. We investigated the localization theorem of Beilinson and Bernstein for D^^-^<(m)> on the projective spaces and SL_3 in positive character, and we showed that a tilting sheaf is obtained by taking the dual of the image of the structure sheaf by the Frobenius endomorphism.4. We presented a short proof of Morley's theorem.