Ample Vector Bundles With Sections Vanishing on Projective Spaces or Quadrics

Abstract

Let ɛ be an ample vector bundle of rank r≥2 on a compact complex manifold X of dimension n≥r+1 having a section whose zero locus is a submanifold Z of the expected dimension n–r. Pairs (X, ɛ) as above are classified under the assumption that Z is either a projective space or a quadric.

Extending extremal contractions from an ample section

“…The other is a very strong result, a Weak Lefschetz type theorem for ample vector bundles, proved by Sommese in [17] and subsequently with slightly weaker assumptions in [13].…”

“…This implies, by [13,Theorem A], that X 0 is a projective space P n and E 0 decomposes as l r O P n ð1Þ, but this contradicts the ampleness of E. r Remark 4.2. Let us note that a blow-up of a projective space P nÀr along a linear space Y of codimension m d 3 cannot be an ample section of a line bundle or of a vector bundle which is a direct sum of line bundles; this follows from [7,Proposition 5.8].…”

“…Through the paper we will assume that Z is not minimal, we assume also that dim Z d 2; if dim Z ¼ 1 then Z F P 1 and this case is treated in [13], where the problem of special sections of an ample vector bundle was studied first.…”

Connections between the geometry of a projective variety and of an ample section

“…In [2] and [3] some examples were given where the inclusion NE(X) ⊃ NE(Z) is strict. Let us recall here [2, Example 4.10], which in turn generalizes an example of L. Bǎdescu (see also [18,Example 4.2]): consider the sequence…”

Ample vector bundles of small curve genera

“…Now when e is ample, we classified pairs XY e with gXY e 0 and 1 in [4] and [5], respectively, under the assumption Ã. The purpose of this paper is to show that gXY e^0 when e is simply supposed to be ample, and to classify polarized pairs XY e with gXY e % 1 without assuming the condition Ã. Needless to say, gXY e is nothing but the usual sectional genus in case n 2.…”