Stability of certain holomorphic maps (Q1176349)

A holomorphic map \(f:X\to Y\) is said to be source-stable (resp. target- stable) if any small deformation of \(X\) (resp. \(Y\)) lifts to a deformation of the triple \((f,X,Y)\). In the present paper the author proves the following results on stability of holomorphic maps between compact reduced complex spaces. Theorem 1.1. Let \(f:X\to Y\) be an embedding with ideal sheaf \(I={\mathcal I}_{X,Y}\). Suppose no component of \(X\) is containing in the singular locus of \(Y\) and that \[ \text{Ext}^ 1_ X(I/I^ 2{\mathcal O}_ X)=0. \] Then \(f\) is target-stable. Theorem 2.1. Let \(f:X\to Y\) be a morphism with \[ f_ *{\mathcal O}_ X={\mathcal O}_ Y,\quad R^ 1f_ *{\mathcal O}_ X=0. \] The \(f\) is source- stable. Theorem 2.3. Let \(f:X\to Y\) be a surjective morphism étale in codimension 2 and flat in codimension 1 and assume moreover that \(X\) is locally \(S_ 3\). Then \(f\) is target-stable. For proofs the author uses his previous results on the theory of deformations of holomorphic maps [Lect. Notes Math. 1389, 246-253 (1989; Zbl 0708.14006)].