Projective bundle theorem

Projective bundle theorem
Name	Projective bundle theorem
Field	Algebraic topology; Algebraic geometry; K-theory
Introduced	1950s–1970s
Keywords	Projective bundle; Chern classes; Grassmannian; Thom isomorphism

Projective bundle theorem

The Projective bundle theorem describes the cohomology or K-theory of a projective bundle associated to a vector bundle over a base space, giving an explicit presentation in terms of the base's invariants and a canonical hyperplane class. It connects ideas from Hermann Weyl-style characteristic classes, Jean-Pierre Serre-inspired sheaf cohomology, and Atiyah–Singer index theorem-adjacent K-theory, and underpins computations in enumerative geometry on schemes such as Grassmannians and Flag varietys. The theorem appears in multiple guises across the work of Raoul Bott, Michael Atiyah, Armand Borel, and Alexander Grothendieck.

Statement

Let E → B be a complex vector bundle of rank r over a compact Hausdorff space B or a scheme over a base S. Form the projective bundle P(E) whose fiber is the projective space associated to the fibers of E. The Projective bundle theorem asserts that the cohomology ring H*(P(E)) (or the Chow ring A*(P(E)), or the Grothendieck group K^0(P(E))) is a free module of rank r over the corresponding cohomology ring H*(B) (resp. A*(B), K^0(B)), generated by 1, ξ, ξ^2, …, ξ^{r-1}, where ξ is the first Chern class of the tautological line bundle O(1) on P(E). In particular, there is a relation given by the total Chern class c(E) and ξ: ξ^r + c_1(E) ξ^{r-1} + … + c_r(E) = 0 in H*(P(E)). Variants replace H* with singular cohomology, de Rham cohomology, étale cohomology, or topological K-theory, and replace Chern classes with corresponding characteristic classes used by Emmy Noether-influenced algebraic geometers.

Historical context and motivation

The theorem evolved from classical results on projective bundles in the work of Évariste Galois-era algebraists and was formalized in the mid-20th century alongside the development of characteristic class theory by Marston Morse, Shiing-Shen Chern, and Jean Leray. The formulation in algebraic geometry owes much to the foundations laid by Alexander Grothendieck in the Éléments de géométrie algébrique program and the development of intersection theory by William Fulton. Topological proofs feature techniques from the schools of Raoul Bott and Michael Atiyah, while algebraic and étale cohomology approaches draw on methods by Alexander Grothendieck, Jean-Pierre Serre, and later expositions influenced by Pierre Deligne. Motivations include computations on Grassmannians, enumerative problems addressed by David Hilbert-style systems, and formal structures needed in the proof of the Riemann–Roch theorem and its variants by Ferdinand Georg Frobenius-related algebraists.

Proofs and variants

Proofs come in several flavors. Topological proofs use the Leray–Hirsch theorem as developed in the work of Jean Leray and connections to the Thom isomorphism as studied by René Thom and Tadao Oda. Algebraic proofs use Grothendieck’s formalism of projective bundles and the splitting principle as deployed by Alexander Grothendieck and clarified in texts influenced by Nicholas Katz and Luc Illusie. K-theoretic formulations were advanced by Michael Atiyah and Friedrich Hirzebruch in the context of topological K-theory and generalized cohomology theories such as complex cobordism studied by David Quillen. Étale cohomology variants use methods from Pierre Deligne and play a role in arithmetic geometry involving schemes over fields considered by André Weil and Jean-Pierre Serre. Further variants address real projective bundles, quaternionic projective bundles studied in the work of Raoul Bott and Shoshichi Kobayashi, and motivic cohomology approaches connected to Vladimir Voevodsky.

Applications

The theorem is a standard tool in computations on Grassmannians, Flag varietys, and moduli spaces such as moduli of vector bundles on curves studied by David Mumford and Michael Atiyah. It is used in proofs of the Grothendieck–Riemann–Roch theorem instantiations, in calculations of Chern numbers that appear in the work of Hermann Weyl and Isaac Newton-era enumerative traditions, and in index computations related to the Atiyah–Singer index theorem. The Projective bundle theorem facilitates intersection theoretic calculations in William Fulton’s intersection theory framework, informs structural results about characteristic classes used by Shiing-Shen Chern, and supports computations in algebraic K-theory originating in the work of Daniel Quillen and John Milnor.

Examples

Classic examples include P(O_B^{\oplus r}) ≅ B × P^{r-1}, where the theorem recovers the Künneth-style decomposition used in the work of Élie Cartan and Hermann Weyl. For the tautological rank r bundle over a Grassmannian Gr(k,n) one obtains explicit presentations of H*(Gr(k,n)) consistent with Schubert calculus developed by Hermann Schubert and formalized by André Weil and Alexander Grothendieck. In algebraic geometry, applying the theorem to the projectivized cotangent bundle of a smooth projective variety yields relations used by Kunihiko Kodaira in classification theory and by Arnaud Beauville in the study of holomorphic symplectic varieties.

Generalizations and related theorems

Generalizations include Projective bundle theorems in generalized cohomology theories such as complex cobordism and elliptic cohomology studied by Michael Hopkins and Haynes Miller, and motivic versions developed by Vladimir Voevodsky. The theorem is related to the Leray–Hirsch theorem of Jean Leray, the Splitting principle often invoked in texts by Milnor and Stasheff, and to Grothendieck’s formal properties of projective morphisms found in the Éléments de géométrie algébrique program. Further connections appear with the Thom isomorphism of René Thom, the Riemann–Roch framework of Grothedieck-inspired authors, and Schubert calculus traditions of Hermann Schubert.

Category:Algebraic topology Category:Algebraic geometry Category:K-theory