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There are two different ways of defining complex structures of the moduli space of irreducible Einstein-Hermitian connections (cf. [Don 1], [Don2]5 [U-Y]) ; i.e. a differential geometric way (cf. [I], [Ko], [L-O]) and an algebro-geometric way (cf. [Ma]). It has been unclear whether these two complex structures are isomorphic when they are non-reduced i.e. their structure sheaves have nilpotent elements. A main reason for this is that the deformation theory of Kuranishi type for vector bundles (e.g. [Ak]) has not been fully generalized so that we can hardly deal with non-reduced structures in differential geometric arguments. Main purposes of this paper are to give a complete generalisation of the deformation theory of Kuranishi type for vector bundles and to prove that the above two complex structures are isomorphic to each other. In §§1 and 2, we will give a generalisation of the local deformation theory of Kuranishi type for vector bundles. In §1, we will show the existence of semi-universal local family of holomorphic structures (Theorem 1) using Banach analytic space argument in [Dou]. By [Ko], Ch. VII or [L-O] together with the arguments of §1, the moduli space of simple holomorphic structures will be a (non-

Kuranishi family of vector bundles and algebraic description of the moduli space of Einstein-Hermitian connections