Ample vector bundle pdf

There is also a notion of amplitude for vector bundles, which seems to have mostly been developed by hartshorne. Eof a vector bundle on x by an embedding of projective varieties i. The space x is called a moisezon olomorphic vector bundle e over x is said to be ample if finitely. For basic results about nef and ample vector bundles, as well as the big vector bundles and qtwisted vector bundles discussed below, see 25, chapter 6. We follow hartshornes notion of ample vector bundles and nef vector bundles, as in har and laz. Christopher hacon, remarks on seshadri constants of vector bundles.

The system is designed so that solutions provide hermitian metrics with positive curvature in the sense of griffiths and even in the dual nakano sense. Ishihara studied ample vector bundles of rank 2 on a hirzebruch surface. However, as we will see, the positivity of the whole vector bundle a is more delicate especially in positive characteristics. Let h be an ample line bundle on x, whose restriction h z to z is a very ample line bundle and assume that z, h z is a bordiga surface, i. Stable ample 2vector bundles on hirzebruch surfaces. On the other hand, the notion of stability has proved to be crucial in the construction of moduli spaces of vector bundles with given numerical invariants. Hermitian differential geometry and the theory of positive and ample holomorphic vector bundles author.

For full access to this pdf, sign in to an existing account, or purchase an annual subscription. Invariants of ample vector bundles on smooth projective varieties. In this paper, we prove a vanishing theorem for the dolbeault cohomology of a product of symmetric and skewsymmetric powers of an ample vector bundle, twisted by a suitable power of its determinant line bundle. New invariants of ample vector bundles over smooth projective. We prove a general vanishing theorem for the cohomology of products of symmetric and skewsymmetric powers of an ample vector. Moreover, we completely classify the cases where equality holds. The development parallels and for the most part reduces to the corresponding. Ample vector bundles with zero loci having a bielliptic curve. A compact complex manifold m is called a hodge manifold if there exists a positive line bundle l over m.

Let e be an ample vector bundle on a projective manifold x, with a section vanishing on a smooth subvariety z of the expected dimension, and let h be an ample line bundle on x inducing a very ample ample line bundle h z on z. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. Consequently, an ample line bundle can be defined as a line bundle such that some tensor power of it is very ample. Classification of rank2 ample and spanned vector bundles on surfaces whose zero loci consist of general points atsushi noma abstract. Numerous and frequentlyupdated resource results are available from this search. Positivity and vanishing theorems for ample vector bundles. What would be a common, simple example of a nef and big divisor that is not ample.

X be a smooth surface, which is the zero locus of a section of an ample vector bundle. We denote by kx the canonical bundle of a smooth variety x. Let e be an ample vector bundle of rank r on a complex projective manifold x such that there exists a section s 2 gammae whose zero locus z s 0 is a smooth submanifold of the expected. A vector bundle on a smooth projective variety, if it is generically generated by global sections, yields a rational map to a grassmannian, called ko. Hermitian differential geometry and the theory of positive. New invariants of ample vector bundles over smooth. Hartshornes book, chapter v, and reids chapters on surfaces. Ample vector bundles on curves nagoya mathematical journal. Then hkx will always denote cohomology with complex coe. The numerical properties of ample vector bundles are still poorly understood. An algebraic or analytic vector bundle with an ample sheaf of regular resp. On very ample vector bundles on curves international. Pdf chern numbers of ample vector bundles on toric.

Robin hartshorne defined a vector bundle f on a projective scheme x over a field to be ample if the line bundle on the space of hyperplanes in f is ample. Here we generalize this result to flag manifolds associated to a vector bundle eon a complex. In our earlier paper 4 we developed the basic sheaftheoretic and cohomological properties of ample vector bundles. Since lis ample, there exists an integer n 0 0 such that the sheaf vnl. Are there any common strategies for finding examples. We also show that tensor products of semiample vector bundles with k ample vector bundles in the sense of sommese are k ample. Ample vector bundles, chern classes, and numerical criteria.

Curvature of vector bundles 535 another example of the situation in theorem 1. Pv be the associated projective bundle, with projection p m and tautological line bundle. In the paper ps00 we suspected that the vector bundle e associated to a branched. In this section, we give some criteria for a vector bundle to be ample modulo d, a divisor with simple normal crossings. Pdf ample vector bundles with sections vanishing on. A smooth projective variety x is called a fano manifold if its anticanonical bundle. A vector bundle e on xis nef or ample if the line bundle o1 on pe is nef or ample, respectively. E is an ample vector bundle if and only if e carries a gri. In this paper, we discuss the stability of rank2 ample vector bundles. In this paper, we study the nakanopositivity and dualnakanopositivity of certain adjoint vector bundles associated to ample vector bundles. Here we generalize this result to flag manifolds associated to a vector bundle e on a complex projective manifold x. In this paper, we will introduce new invariants of generalized polarized manifolds x. Ample vector bundles on curves nagoya mathematical. Consider the following two conditions on a vector bundle e on y, a scheme of finite type over an algebraically closed field k.

Ampleness equivalence and dominance for vector bundles. In particular, we will see that any quotient bundle of an ample vector bundle is ample. Very ample vector bundles of smalldelta1genus in this section we characterize pairs x,e,withe a very ample vector bundle of rank at least two, whose delta1genus is small. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. A vector bundle e on xis quasiample if, for every nonconstant morphism c xfrom a complete nonsingular curve c, e is ample on c. In the case where e is a line bundle, the terminology strictly nef has been used 18. This is the definition taken in positivity in algebraic geometry ii, by robert lazarsfeld, and in example 6. The only two vector bundles with base space a circle and onedimensional.

Ample vector bundles of very smalldelta1genus in order to extend the notion of delta1genus to the vector bundle setting, we give the following definition. Let v be a vector bundle over a smooth projective variety m, let p. In 2 we give the definition of an ample vector bundle in terms of sheaves generated by global sections. Then we are going to introduce some invariants of x,e which. The main result of this article is a general vanishing theorem for the cohomology of tensorial representations of an ample vector bundle on a smooth complex projective variety.

David gieseker, stable vector bundles and the frobenius morphism. In this section we show that an indecomposable bundle on an elliptic curve is ample if and only if its degree is positive. The positivity of the chern classes c, of an ample vector bundle on an algebraic surface is studied. A vector bundle eon xis nef or ample if the line bundle o1 on pe is nef or ample, respectively. Hermitianyangmills approach to the conjecture of gri. Several properties of ample line bundles extend to ample vector bundles. Notably the inequality 0 ample vector bundles proved that e is ample if and only if equation is ample. Our main purpose is to show that there is no numerical criterion for the ampleness of a vector bundle, even on a nonsingular projective variety in characteristic. Introduction let x be a smooth projective variety over the complex number field. Recall the following basic notions which can be found in the hartshornes book. Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of hermitianyangmills type for the curvature tensor.

Ample vector bundles and bordiga surfaces lanteri 2007. For basic results about nef and ample vector bundles, as well as the big vector bundles and qtwisted vector bundles discussed below, see laz, chapter 6. As a consequence, if an existence result could be obtained for every ample vector bundle, the gri. In order to study some ample and spanned vector bundles of rank 2 on threefolds, ballico ba introduced another notion of sectional genus for an ample vector bundle e of rank n gamma 1 on x of. P 1, but by 6, theorem 6 this would imply that f 0 f 2,a contradiction. Ample vector bundles on compact complex spaces by yozo. Let x be a complex smooth projective variety of dimension n, and let ebe an ample vector bundle on x. An easy proof yields the analogous result for direct sums. Ample vector bundles and branched coverings, ii thomas peternell andrew j. Sommese november 5, 2002 introduction to a degree dcovering f. Ample vector bundles on algebraic surfaces article pdf available in proceedings of the american mathematical society 2 march 1969 with 20 reads how we measure reads.

Ample vector bundles with zero loci having a bielliptic. Iitaka fibrations for vector bundles international. Varieties of small degree with respect to codimension and ample vector bundles lanteri, antonio and novelli, carla, journal of the mathematical society of japan, 2008. Then we are going to introduce some invariants of x,e whichare considered to be a generalization of invariants of polarized manifolds we introduced before. On very ample vector bundles on curves by paltin ionescu and matei toma university of bucharest, str.

X yof projective manifolds there is naturally attached a holomorphic rank d. Intuitively this is the object we get by gluing at each point p. As a particular example, consider a smooth projective variety xand its cotangent bundle x. It is wellknown that a line bundle on a complete toric variety is nef or ample if and. Given a vector bundle e on x, we can consider various notions of positivity for e, such as ample, nef, and big. A bundle over a manifold is trivial if it is simply the cartesian product of. This chapter is devoted to the basic theory of ample and nef vector bundles.

Let x be an dimensional smooth projective variety over an algebraically closed field k of characteristic zero, and e an ample and spanned vector bundle of rank n on x. Pdf hermitianyangmills approach to the conjecture of. For a line bundle, it is wellknown that the ampleness of the bundle is equivalent to its gri. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology. New invariants of ample vector bundles over smooth projective varieties yoshiaki fukuma abstract. Positivity and vanishing theorems for ample vector bundles kefeng liu, xiaofeng sun, and xiaokui yang abstract in this paper, we study the nakanopositivity and dualnakanopositivity of certain adjoint vector bundles associated to ample vector bundles. Introduction let xbe a projective scheme over an algebraically closed. For the converse, we will separate cases according to the. View enhanced pdf access article on wiley online library.

The blowup of a variety x along a smooth subvariety y isdenoted by bly x. Ample vector bundles on compact complex spaces by yozo matsushima and wilhelm stoll r an irreducible, compact space x of pure dimension m. The goal of this paper is to study this positivity, and gives some arithmetic applications of it. Pdf ample vector bundles with sections vanishing along. Let e be an ample vector bundle of rank r on a complex projective manifold x such that there exists a section s \in \gamma\cal e. Sectional geometric genera for ample vector bundles ishihara, hironobu, kodai mathematical journal, 2004. Vanishing theorems for tensor powers of an ample vector bundle. Chern numbers of ample vector bundles on toric surfaces. This is then clearly an even locally trivial holomorphic bration and there is a naturally. Varieties of small degree with respect to codimension and ample vector bundles. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. As applications, we get new vanishing theorems about ample vector bundles. Positivity of hodge bundles of abelian varieties over some.

Ample and spanned vector bundles of sectional genera three. Yoshiaki fukuma may 2, 2011 abstract let x be a smooth projective variety of dimension n, let e be an ample vector bundle of rank r on x with 1. The primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve defined over an algebraically closed field k. Gieseker, d the positivity of the chern classes of an ample vector bundle. In 3 we show that this condition is equivalent to three. These generalize the corresponding wellknown results for ample line bundles. The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, grothendieck 6 having shown that for a rational curve every vector bundle is a direct sum of line. We first give numerical conditions for the existence of nonspecial such bundles. Introduction let x be a smooth projective variety over.

Ample vector bundles and introduction to mori theory. If e is ample, then every quotient invertible sheaf l of e is ample, so has degree 0, so e satisfies the conditions of the proposition. Let x be a smooth complex projective variety and let z. Many properties of ample vector bundles carry over to quasiample vector bundles, as the following. The main result of this paper is that tensor products of semiample vector bundles over compact complex manifolds are semiample. Let e be an ample vector bundle of rank r on a smooth projective. Novelli journal of algebra 323 2010 671697 definition 3. The of meromorphic functions on x has a finite transcendence over c with 0 s trx 5 tn.