Relations of Normal Bundles and Flips of Small Contractions on Projective Manifolds

Abstract

Let X be a smooth projective variety over the complex number field. Let f : X → Y be a small contraction, and suppose that each irreducible component Eiof the exceptional locus E of f is a smooth subvariety. Assume that dim E ≤ ½ ( dim X + 1), and the normal bundle [Formula: see text]. Then each Ei≅ Pdim Eior Qdim Ei. Moreover, the flip f+: X+→ Y of f exists.