In mathematics, particularly in the field of differential geometry and algebraic topology, Chern classes are important invariants associated with complex vector bundles. They provide a way to study the curvature and topology of these bundles, contributing to many areas such as characteristic classes, complex differential geometry, and algebraic geometry. To introduce Chern classes, let's break down the concepts: 1. **Complex Vector Bundles**: A complex vector bundle is a collection of complex vector spaces parametrized smoothly by a base manifold.
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For a complex vector bundle of rank , its Chern classes are integral cohomology classesThey are natural under pullback and obey the Whitney sum formula for Chern classes. For a complex line bundle, is the Euler class of its underlying oriented real two-plane bundle; generally is the Euler class of the underlying real bundle.[ref]
Characteristic Classes by John Milnor and James D. Stasheff. Chapter 14, definition and product theorem
For example, if on Complex projective space, the Euler sequence gives , truncated above degree . For an algebraic vector bundle on a smooth variety, algebraic Chern classes instead belong to the Chow groups; the cycle class map recovers the topological classes over the complex numbers.