Principal Simple-Group-Bundle over an Elliptic Surface

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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.