Algebraic and Geometric Study on the Structure of Moduli Spaces

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Moduli in geometry is a set of geometric objects endowed with the universal geometric structure. It is known that not only a moduli space itself is a rich geometric object but also it is often a useful tool in studying geometry. For example the moduli space of Hermit-Nesting connections reflects strongly the differential geometric nature of the base manifold. On the other hand, as the fact that a Hermit-Nesting connection is nothing but a stable vector bundle shows us, we realize that moduli spaces constructed independently in different fields sometimes coincide with each other. This has been promoting direction of studying the theory of moduli spaces from various viewpoints. In this project we have carried out our study on classifying spaces, moduli of various connections and moduli of vector bundles, in cooperation with the specialists of topology, differential geometry, number theory, algebraic geometry and commutative algebra in our department. We have got the following results.1.We completed the computation of Betty numbers of the moduli spaces of stable sheaves of rank 2 on the projective plane and furthermore we got a similar results on ruled surfaces.2.We could clarify an interesting relationship between parabolic stable vector bundles on the projective plane and instantaneous. Using this we could prove that the moduli spaces of instantaneous are connected.3.The standard of the moduli spaces of stable sheaves of rank 2 the on projective plane are dominated by those of the moduli spaces of parabolic stable vector bundles. They are related under a generalization of the elementary transformation of vector bundles.4.We could develop deep study of reflexive sheaves on surfaces with rational double points and their deformations.5.We applied our results on vector bundles to the theory of conformal field theory.