Study of Moduli and its Applications

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The results we have obtained are the following.1) The curve of the jumping lines of second kind of a stable vector bundle of rank 2 on the projective plane deeply reflects the structure of the vector bundle. We found a way to determine the multiplicity and the tangent cone at a singular pint of the curve.2) We showed a close relationship between an ordinary double point on a surface and the space of deformations of a reflexive sheaf on it.3) We generalized Bogomolov's inequality and found an interesting relationship with the moduli space of stable curves.4) Stable sheaves on a degenerating family of varieties were studied and gave a way to find the structure of the moduli space of stable sheaves on a nonsingular variety through a combination of simpler varieties.5) We constructed families of surfaces on which there is a polarization such that the dimension of the moduli space of stable vector bundles is bigger than expected even for very big second Chern classes.6) One of the subjects expected further development is the moduli space of complexes of sheaves. To get good moduli we have to give a proper definition of stable complex. Instead of this we introduced the notion of simple complex and constructed their moduli space in the category of algebraic spaces.7) One of the most important problem is the boundedness of semistable sheaves in positive characteristic cases. In fact, this has been the main target to be solved in this project. The boundedness for a surface was proved in 1973 and for the rank 2, 3 cases in 1980 by M.Maruyama. After 20 years of no progress, we could take a step forward, that is, succeeded in proving the rank 4 case. Unfortunately the method we took cannot be generalized to higher rank cases directly and we nay have to find another way to complete the proof for general cases.