In particular, the total space lof a line bundle is also a complex manifold of dimension one higher than that of x, with a morphism l. Projective spaces for finitedimensional vector spaces over general fields are consid ered. In particular, this con rms the vision and completes the results of. Here is another interesting vector bundle the so called tautological. Recall that the real projective n space rpn is the space of lines. Exx2m of vector spaces of rank r, smoothly parametrized by x 2 m. Ov 1 of the projective bundle pv x is strictly nef.
Recall that a complex analytic structure on a topological manifold m of dimension nis given by a. We also wrote it as a quotient of the space of rank k matrices in mat n kc by the action of gl kc on the right. Two line bundles on projective space let denote the space of 1d subspaces of, or projective space. Suppose that the line bundle l is a pth power of an ample bundle or, more generally, that l l. This showed that grck cn can be covered by n k sets, each of which is in bijection with c k. Hence 2 can be interpreted as showing that the only universal cohomology classes in the. U i c, satisfying g ipf ijpg jp for points p2u i\u j. Two vector bundles over the grasmann g kv are the tautological bundle t. When f is a line bundle, these have been of extensive interest, e. Let e and f be vector bundles on a smooth projective variety y. Intuitively, one should think about e as the collection. Assuming the image in the grassmannian is an algebraic subscheme y, we can use y and the restriction of the tautological bundle to represent the hilbert functor. Maximal chow constant and cohomologically constant fibrations, joint with kristin devleming.
In particular, the total space l of a line bundle is also a complex manifold of. Every line bundle has a trivial section, namely the zero section sending the points of xto. The question arises from geometry of algebraic curves from arbarello, cornalba, griffiths and harris. Characterizations of projective spaces and quadrics by strictly nef. Divisors and line bundles jwr wednesday october 23, 2001 8. Positivity of hodge bundles of abelian varieties over some. Well examine the example of real projective space, and show that its a compact abstract manifold by realizing it as a quotient space. Instead, we define rpn by induction on n simultaneously with its tautological bundle of 2element sets. The projectivization p v of a vector space v over a field k is defined to be the quotient of v.
Oct 01, 2007 a vector bundle e on y is ample if the tautological line bundle o p y e 1 is an ample line bundle on the projective bundle p y e. The tautological ring of the moduli space of curves m g is a subring rm g of the chow ring am g. The first line is the definition of the complex projective bundle. None of these bounds holds for general fano manifolds. On the geometry of hypersurfaces of low degrees in the.
In fact one direction is clear, since tensoring by a line bundle wont change the bres of the projective bundle, the transition functions of y 1 and y 2 are the same. A section of a line bundle is the data of maps g i. In particular, there is a basis for this cohomology. It consists of a collection of vector spaces m,m m parametrized by m. Thus shows that any riemann surface can be embedded in projective. In low degree, the generic k3 surface can be exhibited as a complete intersection in projective space. V, and denote by e the tautological rankk quotient bundle of vg. More precisely, this is called the tautological subbundle, and there is also a dual ndimensional bundle called the tautological quotient bundle. Appendix d maps from curves to projective space 564. The canonical way is due to mumford and takemoto who embed the. On projective space bundle with nef normalized tautological.
Namely, a vector bundle e over a scheme is ample resp. We follow hartshornes notion of ample vector bundles and nef vector bundles, as in har and laz, chap. Line bundles and maps to projective space paul hacking 32310 let x be a compact complex manifold and p. Apr 27, 2011 in this paper, we study the structure of projective space bundles whose relative anticanonical line bundle is nef. Real projective space has a natural line bundle over it, called the tautological bundle. We discussed the tautological bundle over a grassmannian as the subset of cn grck cn of the form fv. Show that, by orthogonally projecting v 0 onto all lines. I wrote something down in my masters thesis that i admittedly dont completely understand, a. Finally, we study when the projectivization of whitney sum of the tautological line bundle and the tangent bundle over real projective space is di. Recall that a section of a vector bundle v x is a map from x to v whose value at x belongs to the vector space vx. Sheaf of sections of a line bundle, and the correspondence with line bundles.
Stability of tautological vector bundles on hilbert. A subscheme of projective space is determined by its equations. In particular, this con rms the vision and completes the results of vermeire in 30 and. As the base case, we take rp1 to be the empty type. May 17, 2016 a projective manifold determines and affine manifold of one dimension higher, called the tautological line bundle. Our intention is to introduce the isomorphism classes of line bundles on a projective space in three different ways by projective geometry. Whether or not that projective manifold is convex is equivelent to whether or not there is a certian kind of metric on the affine manifold. Maps to projective space correspond to a vector space of sections of a line bundle. Assuming the image in the grassmannian is an algebraic subscheme, we can use this subscheme to represent the hilbert functor. A vector bundle is a family of vector spaces that is locally trivial, i. We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual hilbert scheme h of subschemes of length n of a smooth projective surface x.
Let x be a smooth projective variety and e a rank r vector bundle on x. It consists of a collection of vector spaces m,m m parametrized. Pdf hochschild cohomology of schemes with tilting object. Assume that the relative anticanonical line bundle.
Projective bundles and the tautological divisor class. Similarly, the cohomology of some important moduli spaces, like the quot scheme on p1 or the moduli space of stable vector bundles of rank rand degree dwith xed determinant over a curve, can be understood in terms of tautological classes constructed via a universal family or a universal bundle. Stiefelwhitney classes of a projective space bundle. The classical definition of rpn, as the quotient space identifying antipodal points of the nsphere, does not translate directly to homotopy type theory. The real projective spaces in homotopy type theory core.
The tautological ring of the moduli space of curves. Bounds for the anticanonical bundle of a homogeneous. We now want to find a canonical form for matrices in this set glr, ks, s. Recall that we may take to be the tautological, or hopf, bundle. Recall that we may take to be the tautological, or hopf, bundle x. Mitchell august 2001 1 introduction consider a real nplane bundle. Download citation on projective space bundle with nef normalized tautological divisor in this paper, we study the structure of projective space bundles whose relative anticanonical line.
Recall that l embeds x gp into the projective space ps v of hyperplanes of the socalled schur power s v, which is the space of global sections of l. For a given line bundle lon a smooth variety x, we will denote the set of global sections by h0x. An abstract manifold cameron krulewski, math 2 project i march 10, 2017 in this talk, we seek to generalize the concept of manifold and discuss abstract, or topological, manifolds. On projective space x pn, the line bundle o1 is the inverse to the tautological line bundle from example 1. The main theorem, detailed in chapter iv, shows that the rst secant variety to a projective variety embedded by a su ciently positive line bundle is a normal variety. Here bundle simply means a local product with the indicated. Here is another interesting vector bundle the so called. Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space w. A canonical treatment of line bundles over general projective spaces. The usual constructions to create new vector spaces apply a. Finally, we study when the projectivization of whitney sum of the tautological line bundle and the tangent bundle over real projective space is diffeomorphic to the product of two real projective spaces. Let v r be a point of gand choose a decomposition v v r v n r. The universal line bundle, o pv1 on the projective space pv will be obtained via the blowup. If v is onedimensional so that pv consists of a single point, then a section of.
In mathematics, the tautological bundle is a vector bundle occurring over a grassmannian in a. One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space. If m is almost complex, then dim m 2n and all tangent spaces are complex. There is a tautological line bundle s pe which restricts on each. Under the canonical inclusion of projective spaces kpn. We denote by g kv the grasmann manifold of kdimensional subspaces of v and by pv g 1v the projective space of v. Let xbe a complex vector bundle of dimension nand let p. Summing up, we can say that p n is a compact topological manifold of complex dimension n.
Let be the one dimensional antitautological bundle of th. Its pretty much the same as what happens for the tautological bundle over projective space. The complex projective space cpn is a complex manifold. The projectivization pv of a vector space v over a field k is defined to be the quotient of. Apr 19, 2017 the classical definition of rpn, as the quotient space identifying antipodal points of the nsphere, does not translate directly to homotopy type theory. As an application, we get a characterization of abelian varieties up to finite etale covering. We write down complete curves in the moduli space based on the following three constructions. As an application, we characterize the real projective bundles over 2dimensional small covers by interpreting the. Give a trivializing open cover fu gand change of coordinate functions fg gfor l. Riemann surfaces uwmadison department of mathematics. Projective bundles over small covers and topological. The line bundle o1 plays an important role in algebraic geometry see.
Note that there is always a zerosection given by g ip 0 for all i, p2u. A section of the tautological line bundle l on pv assigns to each line in v a particular element of that line. It can also be constructed algebraically by shea fying the graded module s1, where s cx 0. Introduction to algebraic geometry, class 21 contents. In the case of projective space the tautological bundle is known as the tautological line bundle. Chapter 4 cones over projective manifolds and their canonical. Secondary 14m15, 32m10 the homogeneous compact complex manifolds x that admit an equivariant embedding in projective space are precisely the quotients x gp where g is a semisimple complex lie group and p is a parabolic subgroup. This is an expository note describing the isomorphism classes of algebraic line bundles on projective spaces in an elementary way. Given a coherent sheaf f on projective space, the tautological sheaf is the bundle v2 pr 2. A direct proof that toric rank 2 bundles on projective space split. We have seen the definition and first examples of vector bundles. Then we would get an injection of the schemes with a. We have already seen that the grassmannian together with its tautological bundle represents the functor classifying subspaces of a vector space. The incidence variety consisting of pairs vector,line such that the vector lies in the 1d subspace, has a natural map to by projection onto the.
72 1395 1785 1048 985 146 1662 1378 1805 16 585 942 742 1198 502 698 831 110 99 1690 1808 1698 1204 952 1283 580 848 663 1143 1593 902 1841 1706 1234 824 396 1540