A Regularity Theorem for Deformations of a Compact Complex Surface

Let ƒ:M →D ⊑C ⁿ be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially to D, if the fibers of ƒ¦_D∖Z are either Hirzebruch surfaces (projective bundles overP ₁), Hopf surfaces (elliptic bundles overP ₁), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.