A vector bundle characterization of pn (original) (raw)

Ample Vector Bundles with Sections Vanishing on Special Varieties

International Journal of Mathematics, 1999

Let ε be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section s ∈ Γ(ε) whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X-r:= n - r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.

Note on Characterization of Projective Spaces

Communications in Algebra, 2007

We prove a numerical characterization of P n for varieties with at worst isolated local complete intersection quotient singularities. In dimension three, we prove such a numerical characterization of P 3 for normal Q-Gorenstein projective varieties. This result was first proved by Cho, Miyaoka and Shepherd-Barron [CMSB02] and later simplified by Kebekus [Ke01]. The main goal of this paper is to relax the assumption on smoothness. Theorem 1.2. Let X be a projective variety of dimension n ≥ 3 with at most isolated local complete intersection quotient (LCIQ) singularities. Assume that there is a K X-negative extremal ray R such that C •(−K X) ≥ n+1 for every curve [C] ∈ R. Then X ∼ = P n. Note that the numerical condition in Theorem 1.2 is weaker: we only require this condition only for curves in one extremal ray, instead of all curves. The next corollary follows immediately. Corollary 1.3. Let X be a projective variety of dimension n ≥ 3 with at most isolated LCIQ singularities. If C •(−K X) ≥ n+1 for all curves C ⊂ X, then X ∼ = P n. Combining with methods from the minimal model program (MMP), we obtain the following stronger result when dim X = 3.

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

Mathematische Nachrichten, 2006

Let X be a smooth complex projective variety and let Z = (s = 0) be a smooth submanifold which is the zero locus of a section of an ample vector bundle E of rank r with dim Z = dim X − r. We show with some examples that in general the Kleiman-Mori cones N E(Z) and N E(X) are different. We then give a necessary and sufficient condition for an extremal ray in N E(X) to be also extremal in N E(Z). We apply this result to the case r = 1 and Z a Fano manifold of high index; in particular we classify all X with an ample divisor which is a Mukai manifold of dimension ≥ 4. In the last section we prove a general result in case Z is a minimal variety with 0 ≤ κ(Z) < dim Z.

Tseng: Note on characterizations of projective space

2005

Abstract. We prove a numerical characterization of P n for varieties with at worst isolated local complete intersection quotient singularities. In dimension three, we prove such a numerical characterization of P 3 for normal Q-Gorenstein projective varieties. 1.

On an elementary transformation of vector bundles in P^n

arXiv: Algebraic Geometry, 2017

By considering the equivalence between the category of locally free sheaves and the category of algebraic vector bundles, we show how elementary transformations of vector bundles can be used to prove a case of the maximal rank hypothesis. We in turn show how this can be applied in the study of minimal free resolutions.

Complements of hyperplane sub-bundles in projective space bundles over the projective line

2011

We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle H of a projective space bundle of rank r-1 over the projective line depends only on the the r-fold self-intersection of H . In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub-bundle with given r-fold self-intersection. Our proof exploits the unexpected property that every such complement comes equipped with the structure of a non trivial torsor under a vector bundle on the affine line with a double origin.

Ampleness in complex homogeneous spaces and a second Lefschetz theorem

Pacific Journal of Mathematics, 1983

This paper investigates how ampleness of the normal bundle of a smooth subvariety Y of a complex homogeneous space Z = G/H influences the intersection of Y with other subvarieties of Z. We consider a class of homogeneous spaces, rigged spaces, that includes Grassmannians, quadrics and P r \P* (the compliment in P r of a linear subspace P*). A result of Corollary 4.5.2 is: Let Z be a rigged homogeneous space with group G. Let 7 be a compact smooth subvariety of Z possessing an ample normal bundle NY. (See [10] for the definition of ample.) Then the map φ y :P(iV*y)^p« determined by the (/-sections of TZ is generically 1-1 (see 2.2 for the definition of φ γ). Corollary 4.5.2 and Theorem 5.2 imply that if X and Y are both smooth and compact subvarieties of Z with ample normal bundles, then for all gGG, except for a closed codimension 2 subvariety of G, X Π g~\Y) is either a transverse intersection, or has precisely one singular point and it is non-degenerate quadratic.