Unimodular invariants of totally real tori in C^n

Abstract

We study the global invariants of real analytic manifolds in the complex space with respect to the group of holomorphic unimodular transformations. We consider only totally real manifolds which admits a certain fibration over the circle. We find a complete set of invariants for totally real tori in C^n which are close to the standard torus. The invariants are obtained by an analogous classification of complex-valued analytic $n$-forms on the standard torus. We also study the realization of certain exact complex-valued analytic $n$-forms on the standard torus through non-critical totally real embeddings.