Chern character

Chern character
Name	Chern character
Field	Differential geometry, Algebraic topology
Introduced by	Shiing-Shen Chern
Year	1940s
Related	Chern class, K-theory, Todd class

Chern character The Chern character is a homomorphism from topological K-theory to rational cohomology that connects vector bundles, characteristic classes, and index theory. It was introduced by Shiing-Shen Chern and developed by collaborators such as Atiyah, Bott, and Hirzebruch and plays a central role in the Atiyah–Singer index theorem, Riemann–Roch theorem, and relations between algebraic geometry and topology. The construction uses curvature of connections, exponential characteristic classes, and the splitting principle to translate between Grothendieck–Riemann–Roch theorem frameworks and analytic indices on manifolds such as S^n or complex projective spaces like CP^n.

Definition and basic properties

The Chern character is defined for complex vector bundles over a compact, smooth manifold or a scheme and takes values in even-degree rational cohomology groups. Using a connection and curvature on a bundle over a smooth manifold, one forms characteristic classes analogous to those introduced by Élie Cartan and Hermann Weyl; the Chern character is the total class obtained by taking traces of the exponential of the curvature form, yielding de Rham representatives. Key formal properties include additivity on direct sums, multiplicativity with respect to tensor products, compatibility with pullbacks along maps between spaces like CW complex maps, and normalization on line bundles such as the hyperplane bundle over CP^n.

Chern character in cohomology

In de Rham cohomology, the Chern character of a complex bundle with curvature F is given by ch(E) = tr(exp(iF/2π)), producing classes in H^{2k}(M;Q). This analytic construction aligns with algebraic definitions using the splitting principle and Chern classes c_k(E), and it interacts with other characteristic classes such as the Todd class in the statement of the Grothendieck–Riemann–Roch theorem for proper morphisms between projective varieties like maps between Grassmannians or between projective schemes. Cohomological operations and spectral sequences, for instance the Atiyah–Hirzebruch spectral sequence connecting singular cohomology and K-theory, respect the image of the Chern character after tensoring with Q.

K-theory formulation

As a homomorphism ch: K^0(X) → H^{even}(X;Q), the Chern character identifies the rationalized topological K-theory K^*(X) ⊗ Q with rational cohomology for finite CW complexes such as tori, spheres, and complex projective space. In the algebraic setting, a map from the Grothendieck group K_0 of coherent sheaves on a scheme to the Chow ring or to singular cohomology is provided by algebraic Chern character constructions used in the proofs by Grothendieck and Hirzebruch. The compatibility with Bott periodicity, as established by Raoul Bott, ensures the formulation extends to K^1 and higher K-groups through suspension isomorphisms relevant to spaces like a punctured plane or loop spaces.

Examples and computations

For a line bundle L over CP^1 or a compact Riemann surface, ch(L) = exp(c_1(L)) gives explicit degree-two classes; for direct sums E = L_1 ⊕ L_2 one uses additivity to compute ch(E) = exp(c_1(L_1)) + exp(c_1(L_2)). On CP^n, the hyperplane bundle H has ch(H^m) = exp(mα) where α is the generator of H^2(CP^n;Z). For tangent bundles of complex manifolds like K3 surfaces or Calabi–Yau varieties, combining Chern character with Todd class yields Euler characteristics via Hirzebruch–Riemann–Roch computations used in enumerative problems studied by Yau and Mirror symmetry researchers. Index calculations on Dirac operators and signature operators on oriented 4-manifolds also use explicit ch expansions to compute numerical indices.

Functoriality and naturality

The Chern character is natural with respect to pullbacks along continuous maps between spaces such as embeddings of submanifolds or proper morphisms of projective varieties; for a map f: X → Y, one has f^*(ch(E)) = ch(f^*E). Pushforward formulas involve interaction with the Todd class and pushforward in K-theory as in Grothendieck–Riemann–Roch, relevant for proper maps between varieties like projections from flag varietys or fibrations with fibers modeled on projective space. Natural transformations between cohomology theories, for instance between algebraic K-theory and motivic cohomology studied by Voevodsky or Quillen, respect the Chern character after appropriate rationalization or completion.

Applications and refinements

The Chern character underpins the proofs and applications of the Atiyah–Singer index theorem for elliptic operators on manifolds such as spin manifolds and in calculations for Dirac operator indices on Spin^c manifolds. In algebraic geometry it appears in the formulation of the Grothendieck–Riemann–Roch theorem for maps between projective schemes like those arising in moduli problems studied by Mumford and Deligne. Refinements include multiplicative characteristic classes in elliptic cohomology, differential K-theory versions used in mathematical physics contexts such as string theory and D-brane charge classifications explored by Witten, as well as p-adic and l-adic realizations in arithmetic geometry investigated by researchers such as Fontaine and Illusie.

Category:Characteristic classes