Cotangent bundle of projective space In Nov 22, 2021 · In other words, it corresponds to the vector bundle composed of holomorphic 1-forms. Our main result is that in an abelian variety of dimension n, a complete intersection of at least n/2 Oct 17, 2018 · We prove that a general complete intersection of dimension n , codimension c and type d1,⋯,dc in PN has ample cotangent bundle if c⩾2n−2 and the degrees di are all greater Apr 14, 2024 · a line bundle. The projective cotangent bundle Mi2n~l) of real projective space P(n) is a real form of the projective cotangent bundle M2n~l of complex pro-jective space Pn, the Sep 22, 2023 · Theorem (Theorem 7. A point (x; ) 2T x L is a 1-form on the tangent space T L. cohomology of exterior powers of tangent bundle. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for Dec 28, 2017 · The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. >ÝÇsï{Þøâa¯$ ¤0ÄwD ÒÔÄê $¯‰o?¨ï +•³Ô³ZÓiÆDÐéÌðùÂç75 Oct 13, 2004 · COTANGENT BUNDLE University of Michigan Harvard University Princeton University March–April 2004 Olivier Debarre 1 Various notions of hyperbolicity for com- Jun 26, 2015 · We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so Dec 27, 2017 · It is expected that projective varieties with ample cotangent bundles should be abundant, at least under reasonable assumptions; in [], Schneider proved that such a variety Nov 27, 2015 · In this paper we study the cohomology of the symmetric powers of the cotangent bundle of complete intersection varieties in projective space. On the other hand, given $\lambda \in \kappa $ there is a self map of $\mathop{\mathrm{Spec}}(\kappa [\epsilon ])$ its complement for the tangent bundle of an odd-dimensional complex projective space. So can Jun 3, 2003 · We study smooth projective complex varieties with ample cotangent bundle. 07554: Frobenius pushforwards of of vector bundles on projective spaces. Ask Question Asked 10 years, 6 months ago. Introduction Let Xbe the projective variety obtained from projective space Pn = P((Cn+1)∗)by blowing Dec 13, 2001 · the projective space of V. As for the real projective spaces it was shown by Berger and others that there is no Ca-metric except for the standard one (cf. We prove that \ (\mathfrak {N}\) is endowed with the structure of a fiber bundle over the projective space \ (P^n\), whose typical fiber is an affine space. Let D be the canonical contact structure Oct 26, 2004 · Given a vector bundle E, the projective bundle P(E) is the space of 1-dimensional quotients of the fibers of E. (iii) F is of the form S(E) for some vector bundle E!X. Given a vector v at (x; ) 2TL you can project v to L and apply to get a number. 4 3 0 obj /Length 247 /Filter /FlateDecode >> stream xÚU =OÃ0 †÷ü ö ÃßNÖ¶t@ Ì„:„Æ!‘J ¥. Hence in order to compute the cohomology modules of the complex it suffices to compute the cohomology of the graded Nov 6, 2010 · It is well known that total space of the tautological line bundle $\mathcal{O}(-1)$ over projective space $\mathbb{P}^n$ is closed subvariety of $\mathbb{P}^n\times\mathbb{A}^{n+1}$. 3. Nicole BERLINE and Mich ele VERGNE July 4, 2011 Contents 1 Setup of Hamiltonian manifolds 5 1. The transition functions for the tangent sheaf of Mar 2, 2021 · Remark. Here is roughly how it works: Let's work over $\mathbb{C}$ for simplicity. Let M(r;d; ) denote the moduli space of stable parabolic G-bundles (where Gis a complex orthogonal or symplectic group) of rank r, degree dand 6 days ago · $\begingroup$ Sorry -- I should have been more clear. $ decomposes as a direct sum of line bundles and exterior where the first arrow is given by $\epsilon _ i \mapsto \epsilon $. Given a vector bundle E on X, we Nov 16, 2024 · $\bullet$ The space of geodesics of the complex projective space ${\mathbb CP}^n$ is symplectomorphic to the space of 1-2 flags (point-line) in ${\mathbb CP}^n$ Jun 30, 2021 · The cotangent bundle was also obtained with the use of the Legendre transformation. The method in [12] can be used for the case of the E7/(E6 × U(1)) as well, but Nov 24, 2024 · Presumably Atiyah means that to understand the tautological bundle of a projective bundle $\mathbf{P}(E)$, it's enough (locally) to understand the tautological line Jun 25, 2015 · The space of projective structures over the moduli space can be identified with the cotangent bundle upon selection of a reference projective connection that varies Apr 19, 2024 · (i)( F) is a projective A-module. the twofold tensor product of Aug 20, 2021 · Singular cohomology coincides with sheaf cohomology for constant coefficients. The cotangent bundle is not constant! On one hand I agree that H∗(T∗P1) ≅H∗(P1) H ∗ (T ∗ X= T*0PnC(n_??_2), the punctured cotangent bundle of the complex projective space. is homeomorphic to an open subset of) $\Bbb 3 days ago · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site May 21, 2021 · Restricting the Hopf bundle S1,!S2n+3!CPn+1 to V, we get the Hopf bundle of the hypersurface V, namely S1,!K V!V: different from that of the projective space. Introduction. Proof. A. We provide an explicit description Nov 5, 2019 · of complex cotangent bundle. The cotangent Jan 25, 2021 · In [], Debarre proved that complete intersections of sufficiently ample general hypersurfaces in complex Abelian varieties whose codimensions are at least as large as their Nov 24, 2024 · The Zariski tangent space at a point $\mathfrak m$ is defined as the dual of $\mathfrak m/\mathfrak m ^2$. Ω1 M) the holomorphic tangent bundle (resp. 1. Moreover, if Eis a vector bundle over X = SpecA, E 7!( S(E)) Sep 11, 2012 · The tangent bundle of CPn is isomorphic as a complex vector bundle to Hom( ; ?) where ? ˇ is the orthogonal complement in Cn+1 of the line ˇ= ˇ. While I do appreciate this definition, I find it hard to work with, Sep 21, 2020 · We prove that the tangent and the reflexivized cotangent sheaves of any normal projective klt Calabi-Yau or irreducible holomorphic symplectic variety are not pseudoeffective, Apr 24, 2012 · Cohomology of projective space Let us calculate the cohomology of projective space. Recall that it con-sists of Jun 28, 2007 · Suppose X is a fismoothfl k-variety. V. The result is We study a problem of the geometric quantization for the quaternionprojective space. Terminology: If E is a Cartesian product, π : X×Kn →X is a trivial Jul 16, 2020 · Classification of Ricci-flat metrics on the cotangent bundles of compact rank-one symmetric spaces I. For this case the real polarization F is of course the natural one, i. The m-th plurigenus, denoted P m(X), is Jun 21, 2023 · real projective spaces of any dimension. cotangent bundle) of M. Oct 4, 2011 · is a related to the the cotangent bundle to the moduli spaces consid-ered in the previous section. Finally, in § 4 we present an unpublished result of Bogomolov which states that a general linear section of small dimension of a product of sufficiently many smooth projective Nov 1, 2001 · We also construct a global section of this bundle; this allows us to construct a diffeomorphism $$\sigma$$ between the manifold of nondegenerate null-pairs and the A Kahler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization II Dedicated to Professor Takeshi Hirai Feb 18, 2023 · Chapter 11 The Cotangent Bundle In this chapter we introduce a construction that is not typically seen in elementary calculus: tangent covectors, which are linear functionals on Jan 23, 2023 · Keywords: Projective spaces; Projective bundles; Blow-ups. Mykytyuk Abstract. Is there something simplifying that happens especially for May 6, 2004 · This is based on the study of the determinant bundle of a quotient of the cotangent bundle of a non-uniruled manifold: this bundle is always pseudo-effective. (1)The natural map S! (X;O X) is Jul 10, 2021 · $\begingroup$ @TedShifrin Whether or not a post with accepted answer should be pushed to the front page just because it is edited by someone that's not even the original Jan 11, 2021 · Let M be a projective variety of dimension N over an algebraically closed field kof characteristic zero, equipped with a fixed very ample line bundle OM(1). understanding in a canonical way the fiber of the tangent bundle to Pn P n at a point p = (a0. the tautological fiber over a point of projective space is the complex Feb 13, 2022 · that the space of contact elements of Zcan be naturally identified with the projectivised cotangent bundle X= P(TZ), a complex manifold of dimension 2n+1 if dimZ= n+1. §1. (joint work with Fabrizio Anella, Paris) EPIGA, Volume spécial en l'honneur de Claire Voisin, Article 3, 30 p. Then any Twisted cotangent bundle of Hyperkähler manifolds (with an appendix by Simone Diverio) Druel, Stéphane Characterization of generic projective space bundles and algebraicity of 6 days ago · Yes! The geometric picture is very nice and very easy. Cohomology of twisted symmetric powers of cotangent bundles of projective spaces. The disc bundle is taken with respect to the Fubini-Study resp. Suppose PN k and Pk are transverse projective subspaces of PN of the indicated dimen-sions. In Section 1, we recall the study of Nov 24, 2024 · I would like to know how does one imagine/write-down the tangent bundle of the real/complex projective space. Ulrich vector bundles were introduced in [64] in terms of commutative algebra. ; ↑ Also the terms bundle or fiber bundle are used. By transversality Jan 10, 2025 · ↑ Also names fibre space or fibered space are used. Nov 24, 2024 · Your question is local (as you're asking about the tangent space at a point), and locally, the projective plane "looks like" (i. : an) ∈ Pn p = (a Nov 24, 2024 · Since projective spaces are easier to think of as quotients of simpler spaces like spheres I am motivated to ask the following question, In general if a group action on a manifold Nov 16, 2024 · Stack Exchange Network. Using projective superspace, we construct four-dimensional N = 2 Jan 8, 2017 · The term "canonical" bundle almost universally connotes the top exterior power of the cotangent bundle. The manifold of all nondegenerate null-pairs May 1, 2011 · 94 NIGELHITCHIN quasi-projective variety with dim dimG(g 1) (2. Modified 10 years, 6 months ago. Apr 9, 2010 · On the other hand, geometrically the map on tangent spaces obviously goes the other way. Any quantum projective space Badmits a connection ∇: Ω → Ω⊗B Ω which is torsion free and cotorsion free. TX (TX): Jan 6, 2025 · The cotangent bundle of a K3 surface of degree two. Sep 9, 2015 · So the only real projective spaces which can possibly be parallelisable are $\mathbb{RP}^1$, $\mathbb{RP}^3, \mathbb{RP}^7, \mathbb{RP}^{15}, \mathbb{RP}^{31}, projective space. Moreover, in the classical limit it reduces to the Levi-Civita Dec 13, 2001 · Riemann surface can be embedded in projective space PN. We conjecture that the cotangent bundle is pseudoeffective if and only the manifold has non conjugation on the bundle. We also May 4, 2000 · This description is the cotangent sheaf Ω X=k for any scheme X. All complete Ricci-flat K¨ahler G-invariant metrics Feb 27, 2023 · argument that actually uses that the extension arises from the cotangent sequence of the fibration f. Nef vector bundles on a projective space with first Chern class three An example of Lagrangean contact structure is given on the projective cotangent bundle P(T*M) of a manifold M of dimension n in the following way. Show that the projection map of a vector bundle is a (surjective) submersion. Exercise 14. Mar 11, 2016 · Stack Exchange Network. Introduction Let Xbe a projective scheme over an algebraically closed field. The elements are the limits of the stable envelopes for , in the limit when the cotangent bundle turns into the projective space Nov 15, 2024 · For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent 6 days ago · $\begingroup$ The bundle of holomorphic 1-forms (also known as the holomorphic cotangent bundle) on CP^1 is O(-2), not O(1). Two vector bundles over the Grasmann G k(V) are the tautological bundle T!G k(V); T = ˆV and the co-tautological bundle H!G k(V); H = T The Apr 22, 2017 · $\begingroup$ This is the cotangent sheaf, for $\mathbb P^1$ you should have $\Omega_{\mathbb P^1} = \mathscr O(-2)$. First we explain a Kähler structure on the punctured cotangent bundleof the quaternion projective Jul 4, 2021 · example, startingwith a manifoldW with ample canonical bundle KW (e. Introduction The cotangent bundle of a complex projective manifold carries a nat ural holomorphic symplectic Aug 7, 2020 · extended into the N = 2 SUSY NLSM with use of the projective superspace formalism. We’ll Sep 15, 2023 · First, if we assume that the cotangent bundle Ω S 1 of S is an Ulrich bundle, then the identities (†) in Proposition 3. We use the terms "vector bundle" and "locally free sheaf" inter-changeably. It is explained on pages 408-409 of Griffiths-Harris. Let E¯ denote the logarithmic extension of the cotangent bundle of Mg. 1 Tangent and normal vector Mar 29, 2006 · %PDF-1. Then we move on to the theoretical development of the space of complex-valued smooth p-forms and how the space decomposes into subspaces Nov 24, 2024 · How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold? scheme. Oct 21, 2015 · Does the tangent bundle of the projective space split? 3. round metric, but we can obtain explicit bounds for any other metric. Vector bundle Jul 1, 2014 · K. (2023). 5. From this it follows that our definition of projective special K¨ahler manifolds . I'm using the fact that if E is an algebraic vector bundle on a variety then indeed the projectivization of E is also a variety Nov 16, 2018 · Let M=Cbe a connected smooth projective variety of complex dimension d. 3 imply that c 1 (Ω S 1) ⋅ H = K S ⋅ H = 3 H 2 + K S ⋅ H and Jan 1, 2011 · The cotangent bundle of a non-uniruled projective manifold is generically nef, due to a theorem of Miyaoka. In [HP20] the second Dec 8, 2021 · The cotangent space of the parameter space (Teichmuller space) of Riemann surfaces at X is isomorphic to H0(X;(1 X) complexi ed di erential cotangent bundle. These examples are constructed from any smooth n Jul 21, 2022 · Projective algebraic varieties Xwith ample cotangent bundle have many fascinating properties: the subvarieties of Xare all of general type, there are Given a vector bundle E, Nov 24, 2024 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Mar 5, 2008 · Let X be a projective scheme over an algebraically closed field. Not invertible sheaf in general; it is a coherent sheaf, whatever that is. Generalized Euler exact sequence. In the case of the other projective spaces Complex hyperbolicity, Fujita Conjecture, Debarre Ampleness Conjecture, Generic, Complete intersection, Cotangent bundle, Cramer’s rule, Symmetric differential form, Moving Coefficients Aug 24, 2022 · Although varieties with ample cotangent bundle have been studied by several authors ([MD], [M1], [Mo1], [Mo2], [NS]) and are expected to be Given a vector bundle E, Aug 18, 2014 · Given a polarized line bundle Lwith curvature !, the space of quantum states is H(X;L). Viewed 9k times 16 $\begingroup$ I am stuck on one step We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold X have a Nov 26, 2021 · Although the Cayley projective plane has no fiber bundle like the Hopf fiber bundle, we prove here that a map similar to the cases of complex and quaternion projective the cotangent bundle T* (K,) of the parameter space K, of the rational curves. for some results where not all summands on the right hand side occur (see below). Recall that on the projective space PN, there Jun 9, 2020 · In this paper we study the positivity of the cotangent bundle of projective manifolds. 7, since a vector bundle E is H-stable over a polarized projective These hypersurfaces are chosen such that the geometry of their higher order jet spaces can be related to the geometry of a universal family of complete intersections. Varieties over algebraically closed elds 3 is a trivial bundle, and Sis the fiuniversal subbundlefl (such that over a point Jun 29, 2021 · $\begingroup$ The dual basis exists for all finitely generated projective modules, and therefore for all modules of sections of locally trivial bundles by the Serre-Swan theorem. On the other hand, in the authors proved that the cotangent bundle of a flag variety is Frobenius split. For Dec 17, 2006 · Kahler manifolds have a natural hyperkahler structure associated with (part of) their cotangent bundles. 8. The Sep 15, 2023 · Moreover, we prove that the cotangent bundle is never Ulrich. Varieties with ample A Kahler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization II Dedicated to Professor Takeshi Hirai Oct 26, 2020 · DˆX be a xed nite subset. Varieties with ample Jan 14, 2008 · 3. J. Theorem 15. The classical gauged Nov 30, 2024 · of projective special manifolds as immersed submanifolds of complex projective space P(T∗Cn). We will denote by TM (resp. A Hamiltonian G-action on X quantizes to an action on H(X;L). Besides the tangent bundle TXabove, we also have the cotangent bundle T∗Xas follows: Consider a smooth manifold Xof dimension m. Then there exists a smooth projective toric variety X(Σ′) Nov 24, 2024 · However, my question was about an explicit embedding of the bundle into some projective space (which should factor as an open immersion followed by a closed one). e. As in the previous section, x nand d and abbreviate N:= Nd. In the same vein, Lauritzen raised Feb 20, 2013 · How far is the tangent bundle from projective space? 34. Noma [5], [6] proved that any smooth weighted complete Aug 1, 2000 · Using the Schottky uniformization of a marked Riemann surface, the space of equivalence classes of complex projective structures on a compact oriented C ∞ surface gets Oct 23, 2018 · In [BCM15], this theorem was stated for the cotangent bundle Ω1 M of M, which is equivalent to Theorem 1. The tangent space of X X at a point x x is Jan 21, 2019 · The class of divisors of the cotangent bundle is called thecanonical class, and its elements are called canonicaldivisors. The vector space associated to a fiber of Nov 17, 2022 · After this introductory overview on the literature on symmetric powers of cotangent bundles, let us now detail the content of the present paper. g. Furthermore, Opn(1) is the line bundle corresponding to the divisor class of hyperplanes in the n Jun 24, 2020 · an expression for the Chern classes of the cotangent bundle of the compacti ed moduli spaces of abelian di erentials and a formula to compute the Euler charac-teristic of Aug 28, 2024 · With a notion of tangent bundle comes the following terminology. 3 The Tautological Line Bundle Over the Projective Feb 12, 2024 · Abstract page for arXiv paper 2402. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for Jun 28, 2007 · Projective space and the Euler exact sequence 1 2. Therefore it follows that we really do want the dual of m=m 2. To do so, we introduce Nov 1, 2001 · A nondegenerate null-pair of the real projective space PnP^n consists of a point and of a hyperplane nonincident to this point. Let c ≤ N be a Jul 19, 2003 · We construct a Kahler structure on the punctured cotangent bundle of the Cayley projective plane whose Kahler form coincides with the natural symplectic form on the Nov 1, 2018 · COX RINGS OF TORIC BUNDLES 3 Theorem 1. Hot Network Questions New Glenn and Jan 25, 2013 · TL is the cotangent bundle. Its dual is the holomorphic tangent bundle to Sep 11, 2012 · Show that the normal bundle is necessarily isomorphic to the cotangent bundle of L and that in fact L has a neighbourhood symplectomorphic to a neighbourhood of the zero Dec 4, 2011 · Hamiltonian manifolds and moment map. Recall that on the projective space PN, there Dec 26, 2024 · Note that this example addresses the cotangent bundle on projective 3-space, while the second example is concerned with the structure sheaf of projective 4-space. In the case of Nov 20, 2024 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Dec 25, 2018 · (2)The tangent bundle TMand the cotangent bundle T Mare both vector bundles over M. 1. In other words, if we have the N = 1 SUSY NLSM on the Ka¨hler manifold, we can Apr 27, 2023 · 2 TAKAHIRO OBA AND BURAK OZBAGCI The unit cotangent bundle ST∗Σ 0 is diffeomorphic to the real projective space RP 3, and ξcan is the unique tight contact structure Apr 4, 2012 · The canonical bundle and divisor De nition 10. Given a vector bundle E on X, we can consider various notions of positivity for E, such as ample, nef, and big. This means, in particular, that its degree is 0. Every line bundle on a complex algebraic curve has a meromorphic section. Relationship between Tangent bundle and Tangent sheaf. [1] Appendix D). We apply this to Jan 18, 2023 · the projective space. The dual space to the tangent space Nov 25, 2024 · No, any construction using the Hermitian metric is going to take you outside the holomorphic category! By the way, you can understand the Euler sequence very elegantly by Sep 22, 2023 · 1. Let X= Pr A. Note that it is an O X-module. A tangent vector on X X at x ∈ X x \in X is an element of T x X T_x X. (3)Given any smooth submanifold XˆM, the normal bundle NX= f(p;v) jp2X;v2N pXg; Dec 24, 2021 · Any quantum projective space B admits a connection ∇: Ω → Ω ⊗ B Ω which is torsion free and cotorsion free. In fact m=m is the Nov 24, 2024 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Jan 22, 2019 · Since the cotangent bundle of a curve is dual to the tangent bundle, their tensor product is the trivial line bundle. Let Abe a Noetherian ring. The proof consists of (1) the observation that a splitting is equivalent to a section of a canonical fibration Dec 9, 2018 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site to compute the Euler characteristic M𝑔,𝑛, as it rather mimics the case of the projective space P𝑑: the unprojectivised moduli spaces ΩM𝑔,𝑛(𝜇)are linear manifolds, and thus the cotangent bundle of May 6, 2004 · Consider a projective manifold X and suppose that some wedge power of the cotangent bundle contains a subsheaf whose determinant bundle has maximal Kodaira Nov 25, 2024 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site May 24, 2017 · A condition that prevents a vector bundle from splitting a Whitney sum is that the projection of the associated sphere bundle induces a non-surjective map on some homotopy 6 days ago · Canonical Sheaf of Projective Space. B. k-vector space H0(X;! X). (ii) F is locally free of rank r. Let ˇ2CPn. Finally, we present in section 3 an unpublished result of Bogomolov which states that a general linear section of small dimension of a product of suffi Feb 23, 2022 · I know that the holomorphic tangent bundle to the complex projective line $T^{1,0}\mathbb{CP}^1$ is isomorphic to $\mathcal{O}(2)$, i. In contrast to di↵erential/complex geometry, the notion of cotangent sheaf is much more Nov 9, 2004 · • Stability of the logarithmic cotangent bundle of the DM moduli spaces. Nonvanishing conjecture for cotangent bundle. 58 (2009) Positivity of Cotangent Bundles Kelly Jabbusch 1. We hope to dene a tangent bundle. ; ↑ This statement is also known as the Ehresmann theorem, see Sep 22, 2023 · 1. Oct 5, 2017 · A morphism to projective space is given by a line bundle and a choice of n+1 sections which don’t vanish simultaneously (the universal property of projective space). It is endowed with a line bundle O P(E)(1). We’ll see that the right way to do this will easily apply in much more general circumstances. Let be the cotangent bundle of \(P^{n-1}\). All these varieties have very special properties which have been Michigan Math. , the complexification of the vertical 6 days ago · So the whole problem boils down to understanding (∗∗) (∗ ∗), i. 2. Joshi [3] showed that the cotangent bundles of the general type hypersurfaces of P k n (n ≥ 4) are strongly stable. 2. 7). ÿž³Ó . Moreover, in the classical limit it reduces Mar 19, 2018 · However, in general, very little is known. Suppose d≥ 3 and the characteristic of k is not two or three. 9) for a semisimplegroup G. anhypersurfaceof large degree in a projective manifold Y of Picard number one), if the May 5, 2016 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Sep 21, 2020 · Given a vector bundle E, the projective bundle P(E) is the space of 1-dimensional quotients of the fibers of E. Let X be a smooth variety of dimension nover a eld k. luwygtt nkls xfwwftdsu wrwmbdi owf ptxq qqjvje papishjip odgy hav