View source for Vector bundle, algebraic

A morphism of varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963901.png" /> which locally (in the [[Zariski topology|Zariski topology]]) has the structure of a projection of a direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963902.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963903.png" />, such that the glueing preserves the linear structure of the vector space. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963904.png" /> is said to be the fibre space (bundle space), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963905.png" /> is the base and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963906.png" /> is the rank or dimension of the bundle. The morphisms of an algebraic vector bundle are defined in the same manner as in topology. A more general definition, which is suitable for any scheme, involves the concept of a [[Sheaf|sheaf]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963907.png" /> be a locally free sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963908.png" />-modules of finite (constant) rank; then the affine morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v0963909.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639010.png" /> is a sheaf of symmetric algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639011.png" />, is said to be the vector bundle associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639012.png" />. This terminology is sometimes also retained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639013.png" /> is an arbitrary quasi-coherent sheaf. The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639014.png" /> can be uniquely reconstructed from the algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639015.png" />, and the category of algebraic vector bundles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639016.png" /> is dual to the category of locally free sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639017.png" />-modules. Moreover, for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639018.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639019.png" /> the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639020.png" />-morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639021.png" /> bijectively corresponds to the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639022.png" />-module homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639024.png" /> is a structure morphism of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639025.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639026.png" />. In particular, the sheaf of germs of cross-sections of the algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639027.png" /> is identified with the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639028.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639029.png" />. The algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639030.png" /> is said to be the trivial vector bundle of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639032.png" />. The set of all algebraic vector bundles of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639033.png" /> on the scheme is in one-to-one correspondence with the cohomology set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639035.png" /> is the sheaf of automorphisms of the trivial vector bundle of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639036.png" />. Algebraic vector bundles of rank 1 are said to be line bundles; they correspond to invertible sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639037.png" />-modules and are closely connected with divisors (cf. [[Divisor|Divisor]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639038.png" />; the set of line bundles with the tensor product operation forms a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639039.png" /> (cf. [[Picard group|Picard group]]).

As in topology, the direct sum, tensor product, dual bundle, symmetric and exterior power, induced algebraic vector bundle, etc., are defined for algebraic vector bundles. In the case of an algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639040.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639041.png" />, the line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639042.png" /> is said to be the determinant bundle. To an algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639043.png" /> one can associate the projective bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639044.png" />, just like to a vector space one can associate a projective space (see [[Projective scheme|Projective scheme]]).

Examples of non-trivial algebraic vector bundles include canonical algebraic vector bundles on a [[Grassmann manifold|Grassmann manifold]]; in particular, there exists a canonical line bundle on the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639045.png" /> which corresponds to the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639046.png" />. If the algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639047.png" /> on the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639048.png" /> is a subbundle of a trivial algebraic vector bundle, such an imbedding will define a morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639049.png" /> to the corresponding Grassmann manifold, the canonical algebraic vector bundle on the Grassmann manifold being used to induce this morphism. Line bundles which determine an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639051.png" /> are said to be very ample (see [[Ample vector bundle|Ample vector bundle]]).

Other examples of algebraic vector bundles include the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639052.png" /> on a smooth variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639053.png" /> and bundles constructed from it by different operations (see [[Tangent bundle|Tangent bundle]]; [[Canonical class|Canonical class]]; [[Normal bundle|Normal bundle]]).

An algebraic vector bundle on a variety defined over the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639054.png" /> may be regarded both as an analytic and as a topological (in the complex topology) algebraic vector bundle. Analytic and algebraic vector bundles are equivalent on a complete algebraic variety (see [[Comparison theorem (algebraic geometry)|Comparison theorem (algebraic geometry)]]; [[Vector bundle, analytic|Vector bundle, analytic]]). Topological vector bundles do not always allow an algebraic structure, and even when they do, such a structure is usually not unique. If an algebraic vector bundle is regarded as topological, topological methods may be used; in particular, the Chern classes (cf. [[Chern class|Chern class]]) of algebraic vector bundles may be introduced. There also exists an abstract definition of Chern classes which involves the [[K-functor|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639055.png" />-functor]] or one of the variants of [[Etale cohomology|étale cohomology]].

The properties of an algebraic vector bundle will depend on whether its base is a complete or an affine scheme. If the base is affine, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639056.png" />, algebraic vector bundles correspond to projective modules of finite type over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639057.png" /> (cf. [[Projective module|Projective module]]). If the rank of the algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639058.png" /> is higher than the dimension of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639060.png" /> may be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639061.png" />, where 1 is the one-dimensional trivial bundle. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639062.png" /> is usually not uniquely defined. Moreover, if the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639063.png" /> is higher than the dimension of the base and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639064.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639065.png" /> [[#References|[4]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639066.png" /> is a non-singular one-dimensional scheme (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639067.png" /> is a [[Dedekind ring|Dedekind ring]]), any algebraic vector bundle is the direct sum of a trivial and a line bundle. This also applies to algebraic vector bundles on a non-singular affine surface over an algebraically closed field which is birationally equivalent to a ruled surface.

==The case of a projective base.==
The study of line bundles on projective varieties is a classical problem in algebraic geometry (cf. [[Picard group|Picard group]]; [[Picard scheme|Picard scheme]]). The study of algebraic vector bundles of higher ranks began in 1957, when it was shown by A. Grothendieck that algebraic vector bundles on the projective line are direct sums of line bundles. M. Atiyah classified algebraic vector bundles on an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639068.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639069.png" /> denotes the set of algebraic vector bundles of non-decomposable (into a direct sum) algebraic vector bundles of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639070.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639071.png" /> ( "degree"  is to be understood as the degree of the determinant of the bundle), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639072.png" /> is identical with the points of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639073.png" /> itself [[#References|[3]]].

The concept of stable algebraic vector bundles proved useful in the study of algebraic vector bundles on curves. For a given algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639074.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639075.png" /> be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639076.png" />; the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639077.png" /> is then said to be stable (or semi-stable) if for any subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639078.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639079.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639080.png" />). A stable bundle is simple (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639081.png" />) and, in particular, not decomposable. An algebraic vector bundle of degree 0 on an algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639082.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639083.png" /> is stable if and only if it is associated with an irreducible unitary representation of the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639084.png" /> [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639085.png" /> be the set of all semi-stable algebraic vector bundles of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639086.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639087.png" /> which are direct sums of stable algebraic vector bundles, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639088.png" /> be the subset of stable algebraic vector bundles. If the genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639089.png" /> of a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639090.png" /> is higher than 1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639091.png" /> has the natural structure of a normal projective variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639092.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639093.png" /> is an open smooth subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639094.png" /> [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639096.png" /> are coprime, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639097.png" /> and is therefore smooth. The moduli space of semi-stable algebraic vector bundles has been studied extensively. It is known, in fact, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639098.png" /> is a component of the Picard scheme for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v09639099.png" />; the fibres of the determinant mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390100.png" /> are unirational varieties; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390102.png" /> are coprime, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390103.png" /> uniquely determines the original curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390104.png" />. Since the universal family of algebraic vector bundles does not always exist over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390106.png" /> is not a representing object of a suitable functor [[#References|[1]]]. Most of these results were obtained for the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390107.png" />, even though many of them are also valid for an arbitrary algebraically closed field. Certain special facts are known for algebraic vector bundles on algebraic surfaces and projective spaces [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Narasimhan,   C.S. Seshadri,   "Stable and unitary vector bundles on a compact Riemann surface"  ''Ann. of Math.'' , '''82'''  (1965)  pp. 540–567</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Tyurin,   "On the classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390108.png" />-dimensional vector bundles over an algebraic curve of arbitrary genus"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30'''  (1966)  pp. 1353–1366  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.F. Atiyah,   "Vector bundles over an elliptic curve"  ''Proc. London Math. Soc. (3)'' , '''7'''  (1957)  pp. 414–452</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Bass,   "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390109.png" />-theory" , Benjamin  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.V. Dolgachev,   V.A. Iskovskikh,   "Geometry of algebraic varieties"  ''J. Soviet Math.'' , '''5''' :  6  (1976)  pp. 803–864  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''12'''  (1974)  pp. 77–170</TD></TR></table>



====Comments====
In recent work of S. Donaldson (cf. [[#References|[a2]]]–[[#References|[a3]]]) the moduli space of stable rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390110.png" /> vector bundles on a compact complex algebraic surface was shown to be isomorphic to the moduli space of instanton unitary vector bundles on the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390111.png" />-dimensional smooth manifold. An instanton vector bundle on a Riemannian manifold is a differential bundle together with a connection whose curvature form satisfies a certain system of non-linear differential equations. The moduli space of such instantons is independent of the choice of a Riemannian metric and is a new invariant of smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390112.png" />-manifolds. The theory of algebraic vector bundles allows one to compute these invariants in some cases. In this way there were given the first examples of homeomorphic but not diffeomorphic compact simply-connected smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096390/v096390113.png" />-manifolds.

Many new ideas in the theory of algebraic vector bundles on algebraic curves and projective spaces were inspired by theoretical physics (twistor theory, Yang–Mills theory and string theory).

====References====
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