Complexes of differential forms associated with a normalized manifold over the algebra of dual numbers

Abstract

We construct some complexes of differential forms on a smooth manifold M_n^D over the algebra of dual numbers D on the base of a decomposition of the tensor product TM_n^D⊗_ℝD into the Whitney sum of two subbundles. It is shown that these complexes can be obtained as restrictions of some complexes of holomorphic (D-smooth) forms defined on the tangent bundle TM_n^D. For holomorphic fiber bundles over M_n^D, we introduce complexes of D-valued forms holomorphic along the fibers and express in terms of cohomology classes of such complexes the obstructions to existence of holomorphic connections in holomorphic principal bundles.