Chern classes

Chern classes
Name	Chern classes
Caption	Characteristic classes for complex vector bundles
Field	Algebraic topology, Differential geometry
Introduced	1940s
Introduced by	Shiing-Shen Chern

Chern classes Chern classes are topological invariants associated with complex vector bundles that first arose in the work of Shiing-Shen Chern, influencing Henri Cartan, André Weil, Jean-Pierre Serre, John Milnor, and Raoul Bott. They play central roles in the development of Algebraic Topology, Differential Geometry, Complex Manifolds, Index Theorem, and Sheaf Theory and connect to invariants studied by Atiyah–Singer Index Theorem, Lefschetz Fixed-Point Theorem, Riemann–Roch theorem, and Hirzebruch–Riemann–Roch.

Introduction

Chern classes attach to each complex vector bundle over a topological space or smooth manifold a sequence of cohomology classes that behave naturally under pullback, direct sum, and tensor operations; this framework was pioneered by Shiing-Shen Chern and further developed by Hirzebruch, Jean Leray, Raoul Bott, Michael Atiyah, and Isadore Singer. The theory provides algebraic invariants used in classification problems encountered by researchers at Institute for Advanced Study, Princeton University, University of Chicago, and Harvard University and has consequences for the study of Calabi–Yau manifolds, Kähler manifolds, and the geometry of Complex Projective Space. Chern classes interact with characteristic classes like the Stiefel–Whitney class and Pontryagin class, and with categorical constructions in Grothendieck's algebraic geometry circle around Alexander Grothendieck and Jean-Pierre Serre.

Definitions and Construction

One standard construction defines total Chern class via transition functions of a complex vector bundle using Čech cohomology and classifying maps to the classifying space BU(n), leveraging results by J. H. C. Whitehead, Bott Periodicity discovered by Raoul Bott and Michael Atiyah; the topological definition yields classes in integral cohomology groups H^{2k}(X;Z) for base spaces X such as CW complexes and Differentiable manifolds. In the differential-geometric approach, Chern–Weil theory, developed by Shiing-Shen Chern and André Weil, produces Chern classes from curvature forms of a connection on a Hermitian bundle, connecting to the Chern–Simons form introduced by Shiing-Shen Chern and James Simons. Algebraic geometers, following Alexander Grothendieck and Oscar Zariski, define Chern classes in the Chow ring via locally free sheaves and implement them in intersection theory used by William Fulton and Jean-Pierre Serre.

Properties and Characteristic Relations

Chern classes satisfy naturality, Whitney sum formula, and normalization axioms codified in the work of Hirzebruch and Grothendieck; for vector bundles E and F over a base X, the total Chern class obeys c(E ⊕ F) = c(E) ∪ c(F), paralleling multiplicative laws found in K-theory by Michael Atiyah and Friedrich Hirzebruch. The first Chern class classifies complex line bundles via an isomorphism with H^2(X;Z) used in studies by Leray and Serre, and c_1 links to Picard group computations central to Mumford's work on moduli of curves. Relations to Pontryagin classes appear through realification maps investigated by John Milnor and James Stasheff, and integrality results tie into index formulas from the Atiyah–Singer Index Theorem and signature theorems of Hirzebruch.

Examples and Computations

For the tautological line bundle over Complex Projective Space CP^{n}, the total Chern class is given by 1 + α for generator α ∈ H^2(CP^{n};Z), a calculation appearing in texts by Hatcher, Milnor–Stasheff, and Bott–Tu. Tangent bundle Chern classes of complex projective varieties provide concrete invariants used by Enriques and Kodaira in classification of surfaces and by Yau in Calabi–Yau existence results; explicit computations for Flag manifolds and Grassmannians exploit Schubert calculus developed by Schubert and formalized by Ehresmann and Lascoux–Schützenberger. For complex line bundles over a torus studied by André Weil and Hermann Weyl, first Chern classes correspond to integer cohomology classes determined by periods appearing in work by Poincaré and Riemann.

Applications in Geometry and Topology

Chern classes enter prominently in the proof and applications of the Atiyah–Singer Index Theorem and in Donaldson theory initiated by Simon Donaldson and extended by Kronheimer–Mrowka to analyze four-manifold topology; they also occur in gauge theory contexts studied at Institute for Advanced Study and Princeton University. In algebraic geometry, Chern classes underpin the Riemann–Roch theorem framework advanced by Grothendieck, Hirzebruch, and Fulton and are used in enumerative geometry problems by Kontsevich and Gromov–Witten theory. Intersections of Chern classes yield characteristic numbers employed in classification problems by Hirzebruch and in rigidity theorems studied by Bott and Segal.

Generalizations and Related Theories

Generalizations include Chern classes in complex K-theory developed by Michael Atiyah and Friedrich Hirzebruch, equivariant Chern classes associated with group actions studied by Atiyah–Bott and Berline–Vergne, and Chern classes in algebraic cobordism and motivic cohomology explored by Voevodsky and Levine–Morel. Secondary invariants such as Chern–Simons invariants relate to three-dimensional topology investigated by Witten and Reshetikhin–Turaev; categorical and derived enhancements appear in the work of Jacob Lurie and Maxim Kontsevich on derived algebraic geometry and homological mirror symmetry involving Strominger–Yau–Zaslow conjectures.

Category:Characteristic classesCategory:Algebraic topologyCategory:Differential geometry