DEFORMATIONS WITH SECTION: COTANGENT COHOMOLOGY, FLATNESS CONDITIONS, AND MODULAR SUBGERMS

摘要

We study modular subspaces corresponding to two deformation functors associated with an isolated singularity X_0: the functor Def_(X_0) of deformations of X_0 and the functor Def_(X_0)~s of deformations with section of X_0. After recalling some standard facts on the cotangent cohomology of analytic algebras and the general theory of deformations with section, we give several criteria for modularity in terms of the relative cotangent cohomology modules of a deformation. In particular, it is shown that the modular strata for the functors Def_(X_0) and Def_(X_0)~s of quasi-homogeneous complete intersection singularities coincide. Then flatness conditions for the first cotangent cohomology modules of the deformation functors under consideration are compared.