Impossibility criterion of being an ample divisor*)

In [So 1], Sommese gave many examples of manifolds that cannot be ample divisors in any manifold.His theory works also to construct non-smoothable singularities (see [So 2]).In this note we give the following criterion: THEOREM.Let A be a manifold such that H¹(A, T[-L])=0 for any ample line bundle L on A , where T is the tangent bundle of A. Then A cannot be an ample divisor in any manifold unless A Pⁿ .As we shall see in 1, this result follows easily from a characterization theorem of projective spaces due to Mori-Sumihiro [MS].In 2, we show that various types of manifolds, including many of those in [So 1], satisfy the above criterion.In 3, similarly as in [So 2], we construct examples of non-smooth- able singularities. Notation, convention and terminology.Usually we employ the notation which is commonly used in algebraic geo- metry.We work in the category of C-schemes of finite type.In most cases everything is assumed to be proper over $Spec(C)$ .Point means a closed point.Variety is an irreducible, reduced scheme.Manifold is a non-singular variety.Vector bundles are confused with locally free sheaves.Line bundles are regarded as linear equivalence classes of divisors, and their tensor products are denoted additively.Now we list up some symbols.$[D]$ : The line bundle associated with a (Cartier) divisor D .F[L] : F⊗₀L , where F is a coherent sheaf and L is the invertible sheaf corresponding to a line bundle L .TM : The tangent bundle of a manifold M .EX : The pull back of a vector bundle E on Y by a morphism X→ Y .Sometimes we write simply E instead of EX , when there is no danger of confusion.