CoInvestigator(Kenkyūbuntansha) 
HOSHIRO Tosihiko Department of Science, Himeji Institute of Technology, Professor, 理学部, 教授 (40211544)
UMEDA Tomio Department of Science, Himeji Institute of Technology, Professor, 理学部, 教授 (20160319)
IWASAKI Chisato Department of Science, Himeji Institute of Technology, Professor, 理学部, 教授 (30028261)
HIRANO Katuhiro Department of Science, Himeji Institute of Technology, Lecturer, 理学部, 講師 (90316034)
FUJIWARA Takasi Department of Science, Himeji Institute of Technology, Assoc.Professor, 理学部, 助教授 (10202293)

Research Abstract 
Let (V, o) be an isolated singularity with complex dimension n, in a complex euclidean space (C^N/,o). Let M be the intersection of this V and the real hyperspere S^2N1_E(o), centered at the origin o with radius ε. Then, over M, a CR structure is induced from V, and this CR structure determines the isolated singularity V(Rossi's theorem). So, with this in mind and with Kuranishi equivalence for CR structures, the deformation theory of CR structures is established and the versal family of CR structures is constructed. Related to mathematical physics, it is found that: the SeibergWitten invariant is quite useful in studying the geometry of the moduli space (for example, CalabI Yau manifolds).Therefore it seems natural to try to obtain a similar result in isolated singularities. Let {(M,^<φ(t)>T''),t ∈ M} be the versal family of CR structures of (M,^<o>T''), constructed our former paper. In the construction of the versal family, we have to handle a second order diferential operator(so, the corresponding Laplace operator must be a 4thorder differential operator). For scalar valued differential forms, this phenomenon occurs in the middle dimension degree (in our case, n and n  1). In fact, the harmonic space of differential forms of the middle dimension degree is determined by fourth order partial differential equations, and its solution space has a particular subspace, which is determined by second order partial differential equations. While in algebraic geometry, for Al singularities and their moduli spaces, flat coordinates are found by K. Saito. My first motivation is that: there might be a relation with Saito flat coordinate and the above 4th order partial differential equations. This relation is studied in my recent research, in Al singularities and some HilzebruchJung singularities.
