User:Marsupilamov/Vector bundles in algebraic geometry

In mathematics, an algebraic vector bundle is a vector bundle for which all the transition maps are algebraic functions. All $SU(2)$-instantons over the sphere $S^{4}$ are algebraic vector bundles.

Definition

In sheaf theory, a field of mathematics, a sheaf of ${\mathcal {O}}_{X}$-modules ${\mathcal {F}}$ on a ringed space $X$ is called locally free if for each point $p\in X$, there is an open neighborhood $U$ of $p$ such that ${\mathcal {F}}|_{U}$ is free as an ${\mathcal {O}}_{X}|_{U}$-module. This implies that ${\mathcal {F}}_{p}$, the stalk of ${\mathcal {F}}$ at $p$, is free as a $({\mathcal {O}}_{X})_{p}$-module for all $p$. The converse is true if ${\mathcal {F}}$ is moreover coherent. If ${\mathcal {F}}_{p}$ is of finite rank $n$ for every $p\in X$, then ${\mathcal {F}}$ is said to be of rank $n.$

On algebraic curves

In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces. which is the classical approach, or as locally free sheaves on algebraic curves C in a more general, algebraic setting (which can for example admit singular points).

Some foundational results on classification were known in the 1950s. The result of Alexander Grothendieck, that holomorphic vector bundles on the Riemann sphere are sums of line bundles, is now often called the Birkhoff–Grothendieck theorem, since it is implicit in much earlier work of G. D. Birkhoff on the Riemann–Hilbert problem.

Michael Atiyah gave the classification of vector bundles on elliptic curves.

The Riemann–Roch theorem for vector bundles was proved in 1938 by André Weil, before the 'vector bundle' concept had really any official status. In fact, though, associated ruled surfaces were classical objects. See Hirzebruch–Riemann–Roch theorem for his result. He was in fact seeking a generalization of the Jacobian variety, by passing from holomorphic line bundles to higher rank. This idea would prove fruitful, in terms of moduli spaces of vector bundles. following on the work in the 1960s on geometric invariant theory.

See also

• Swan's theorem
• Coherent sheaf, a more general, or less restrictive, notion of sheaf
• Algebraic K-theory, a theory studying certain equivalence classes of algebraic vector bundles
• Projective module, an algebraic counterpart

• Holomorphic vector bundle, on a projective smooth algebraic variety, both notions coincide due to the GAGA principle

• Vector bundle the topological notion

References

• Sections 0.5.3 and 0.5.4 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math., 79 (1957), 121–138
• M. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. VII (1957), 414–52, in Collected Works vol. I

External links

• "Locally free". PlanetMath.
• The weblog Rigorous Trivialities on Locally free sheaves and vector bundles