Algebraic-geometrical and arithmetical study of a quotient space of a Riemannian symmetric space by an arithmetic group

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Quotinets of pseudo-hermitian symmetric sapces by certain arithmetic groups are realized often as a moduli space of various types of polarized compact Kahler manifolds, and through this fact their structures are clarified. From this point of view we have obtained the following results.1. (1) For a general complex tori, we have classifieid the possible types of its rational endomorphism rings, and the corresponding pseudo-hermitian symmetric space. (2) We have obtained the complete classification of finite automorphism groups of complex tori of dimension two, which reflects the structure of the singularity of our quotient space. Especially, we have obtained explicit description of the moduli space of complex tori with fixed abstract group as automorphism group, as a quotient of a pseudo-hermitian space by arithmetic group. (3) We have found that the spaces in (2) are closely related with certain root lattices.2. (1) As a general structure theorem for compact Kahler symplectic manifolds we have obtained an analogy of Lefschetz-Hodge decomposition theorem, and some remarkable property of a natural 2n-form on the second cohomology group. (2) As an application of (1) we have shown the smoothness of the local moduli space of sufh manifolds; this shows that their moduli space is essentially realized as an open subset of hermitian symmetric space of type IV.3. In general, using the notion of extremal metrics due to Calabi, we have introduced the notion of stability in analogy with that in algebraic geometry, and obtained an entirely new formulation of the moduli theory. There seems much to be developped in the future in this theory.