A complex manifold X of dimension n together with an ample vector bundle E on it will be called a generalized polarized variety. The adjoint bundle of the pair (X,E) is the line bundle KX + det(E). We study the positivity (the nefness or ampleness) of the adjoint bundle in the case r := rank(E) = (n - 2). If r ≥ (n - 1) this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski. If KX + detE is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map π : X → W from X onto a normal projective variety W with connected fiber and such that KX + det(E) = π*H, for some ample line bundle H on W. We describe those contractions for which dimF ≤ (r - 1). We extend this result to the case in which X has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which dimF = r for every fiber and π is birational. ©1997 American Mathematical Society.
Contractions on a manifold polarized by an ample vector bundle
MELLA, Massimiliano
1997
Abstract
A complex manifold X of dimension n together with an ample vector bundle E on it will be called a generalized polarized variety. The adjoint bundle of the pair (X,E) is the line bundle KX + det(E). We study the positivity (the nefness or ampleness) of the adjoint bundle in the case r := rank(E) = (n - 2). If r ≥ (n - 1) this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski. If KX + detE is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map π : X → W from X onto a normal projective variety W with connected fiber and such that KX + det(E) = π*H, for some ample line bundle H on W. We describe those contractions for which dimF ≤ (r - 1). We extend this result to the case in which X has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which dimF = r for every fiber and π is birational. ©1997 American Mathematical Society.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
