Principal Simple-Group-Bundle over an Elliptic Surface

通过前一年的活动，证明了椭圆曲面上的主简单群束的研究是e型例外群中最有趣的例子，它与del-Pezzo曲面的q量子化研究密切相关。这里的参数q可以解释为在边界值q = 0处将原曲面恢复为极限时椭圆曲线的模量。q量子化就是这样一种表面变形。通过今年的研究，对于一般的q值，计算了变形曲面的基本几何不变量，如陈氏数、霍奇数等。这一结果将为澄清表面的混合Hodge结构，考虑其Torelli问题，以及在整个变形空间中表征c-量化提供设备。这一结果是通过观察量子化与Kummer曲面或abelian曲面具有一定的相似性（代数对应）来证明的，因此，在量子化的参数空间上可以构造一个相干的函数束，并且可以通过从空间中自然产生的某个热方程的解束来实现，这也是非常值得期待的。人们很容易回到gbs理论的最初目标。

Through the activity of the prvious year, it turned out that the study of principal simple-croup-bundle over- an elliptic surface is in the most interesting case of the excepional groups of the E-type, closely related with the study of the q-quantization of a del-Pezzo surface. The parameter q here might be explained to be such a moduli of elliptic curves as the original surface is recovered to be the limit at the boundary value q = 0. The q-quantization is such a deformation of the surface.Through the investigation of this year, for a general value of q, fundamental geometric invariants of the deformed surface such as the Chern numbers, the Hodge numbers etc. are calculated. This result will give devices for clarifying the mixed Hodge structure of the surface, considering then the Torelli problem to it, and characterising the c-quantisation: in the whole deformatin space. This result was shown by observing a ertain similarity (algebraic correspondence) of the quantization to a Kummer or an abelian surface, so it is also highly expectable that there can be constructed a coherent sheaf of theta-functions over the parameter space of the quantization and that this can be realized as the solution-sheaf of a certain heat equation naturally arising from the space Through, the affirmation of this expectation, one would easily turn back to the original objective in the GBS-theory.