On flat families and complete families of analytic spaces onto complex manifold

Abstract

Let 7~: (X, X ) -~ (M, (9) be a holomorphic mapping of an analytic space (X, JC) onto a complex manifold (M, (9). For each t eM, the fibre X t : r c l ( t ) is an analytic set in X. Let m t c (gt be the ideal sheaf of germs of all holomorphic functions which vanish on the point t and (mt o re) the idealsubsheaf in generated by g o 7c, g ~ mr. Let us put ~ , . . = ~ / ( m t o ~z) l X . Then (X t , 2/f t) is an analytic space. Thus a holomorphic mapping lr: X --~ M gives a family of analytic spaces X~, t ~ M . A family is reduced if all Xt are reduced analytic spaces (or the analytic spaces which are defined by Cartan and Serre). A family ~: X ~ M is flat at x ~ X if TOrl~(2/f~, (9]mt)=O where l r ( x ) t . Grauert and Kerner give in [43 an equivalent condition to flatness. In this paper, we shall give an another equivalent condition to flatness, that is. rc is flat at x if and only if the sequence of homomorphisms t~, ..., ~,, of ~'~x into 24~x is a regular sequence where (t I . . . . , t,) are local coordinates in a neighborhood of ~z (x). We shall give that if zr is an open mapping, then rc is flat. It is showed in [6] that a reduced family zr: X ~ M is an open mapping if and only if ~ is flat. Let re: X --+ M be a flat reduced family and let Xo be the fibre at a point 0e M. Then there exists a natural homomorphism of the sheaf ~-* (M) into the sheaf ~Y~(Xo), where J-* (M) is the sheaf which the hotomorphic vector field on M is lifted to X and 3-;~(Xo) is the sheaf of the infinitesimal deformation of Xo. If the natural homomorphism ~ * ( M ) ~ ( X 0) is surjective. ~ is called complete at x s X . Kerner proved in [6] that ~z is complete at a regular point of X. In this paper, we shall give that taking X ' = ~z-~ ( M 3 instead of X o in the theorem proved by Kerner, a similar consequence holds, where M' is any submanifold of M.