Deformation of varieties defined by sections in homogeneous vector bundles

Abstract

In the present paper we prove some generalizations of a theorem of K. Kodaira and D. Spencer. They showed in 1958 that any smooth, projective algebraic hypersurface which is not a K3 surface remains projective algebraic under small deformation. In a previous paper [13] we have shown that the analogous statement remains valid for projective algebraic complete intersections with singularities, too. Recently, there has been studied an interesting and rather different generalization by Borcea [2]. From his point of view the complex projective space is the most simple example in the class of homogeneous rational K/ihler manifolds of rank 1. Borcea generalizes the result of Kodaira and Spencer to smooth complete intersections in homogeneous manifolds of this class, which covers for instance all Grassmannians. All these results correspond to special cases of the following question: Consider a compact complex manifold Z together with a vector bundle E and a section of it, vanishing along a subvariety X of Z of codimension equal to rank E. In which case can every small deformation of X be obtained by varying the section in H~ E)? A sufficient criterion is the vanishing of the cohomology groups