Todd class

Todd class
Name	Todd class
Field	Algebraic topology; Algebraic geometry; Differential geometry
Introduced by	* Julius Todd * W. P. Thurston (contextual developments)
First defined	20th century
Related concepts	Chern class, Pontryagin class, Euler characteristic, Hirzebruch–Riemann–Roch theorem, Grothendieck–Riemann–Roch theorem

Todd class.

The Todd class is a characteristic class defined for complex vector bundles and complex manifolds that plays a central role in intersection theory, index theory, and enumerative geometry. It appears in the statement of the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem, connecting topological invariants of bundles to holomorphic and algebraic invariants on Riemann surfaces, complex manifolds, and algebraic varietys. Its construction uses formal power series in Chern classes and integrates naturally with operations considered by Atiyah–Singer index theorem, Grothendieck and Hirzebruch.

Definition and basic properties

The Todd class is a multiplicative characteristic class associated to complex vector bundles and to complex-oriented cohomology theories such as complex cobordism and K-theory. For a complex vector bundle E over a manifold or variety X, the Todd class td(E) lies in the even-degree part of the cohomology ring H^*(X; Q) or in the Chow ring A^*(X) after rationalization. It satisfies naturality with respect to bundle pullback along maps between manifolds or morphisms of algebraic varietys, and multiplicativity for direct sums: td(E ⊕ F) = td(E) ∪ td(F), with td(trivial line bundle) = 1. The Todd class is normalized so that for a complex line bundle L with first Chern class c1(L), td(L) = c1(L) / (1 − e^{−c1(L)}) when expressed via the exponential power series; this normalization ties the class to the exponential map used in the formulation of the Hirzebruch–Riemann–Roch theorem.

Todd class for complex vector bundles

Given a rank-r complex vector bundle E → X, one may define td(E) either axiomatically via multiplicativity and normalization on line bundles or concretely via splitting after passing to a suitable space where E splits as a sum of line bundles (splitting principle). Under the splitting principle, if E = L1 ⊕ ... ⊕ Lr with first Chern classes x_i = c1(L_i), then td(E) is the symmetric polynomial in the x_i obtained by applying the Todd power series to each x_i and taking the cup product. In algebraic geometry, for a smooth projective algebraic variety X, td(TX) for the tangent bundle TX is used to form intersection numbers with Chern character classes in the Grothendieck–Riemann–Roch formula relating pushforward of coherent sheafs and their Euler characteristics.

Computation via Chern roots and power series

Computation of the Todd class is most easily described using Chern roots and the formal power series t(z) = z / (1 − e^{−z}). Under the splitting principle, for Chern roots x_1,...,x_r one has: td(E) = ∏_{i=1}^r t(x_i) = ∏_{i=1}^r x_i / (1 − e^{−x_i}). Expanding t(z) as a power series in z yields polynomial expressions in elementary symmetric functions of the x_i, i.e. in the Chern classes c_k(E). Explicitly, t(z) = 1 + z/2 + z^2/12 − z^4/720 + ... so that td(E) = 1 + (1/2) c1(E) + (1/12)(c1(E)^2 + c2(E)) + ... These expansions are used in characteristic number computations on complex surfaces and higher-dimensional Kähler manifolds and match formal manipulations in Chern–Weil theory when working with curvature forms of holomorphic connections on hermitian bundles.

Todd genus and Riemann–Roch theorems

The Todd genus is the degree-zero characteristic number obtained by integrating the top-degree component td_n(X) of td(TX) over a compact complex manifold X of complex dimension n. The Hirzebruch–Riemann–Roch theorem expresses the holomorphic Euler characteristic χ(X, E) of a holomorphic vector bundle E as the pairing χ(X, E) = ∫_X ch(E) ∪ td(TX), where ch(E) is the Chern character. Grothendieck generalized this to the Grothendieck–Riemann–Roch theorem for proper morphisms between algebraic varietys, replacing integrals by pushforwards in Chow rings and relating K-theory classes and cohomological characteristic classes via td. The Todd genus differentiates among complex cobordism classes and is invariant under holomorphic deformation in families considered in moduli space theory.

Examples and calculations

For a complex projective line P^1 (the Riemann sphere), with tangent bundle T P^1 ≅ O(2), c1(T P^1) = 2ξ where ξ generates H^2(P^1; Z), and td(T P^1) = 1 + ξ. Integration gives the Todd genus ∫_{P^1} td_1 = 1, matching χ(O_{P^1}) = 1. For complex tori and abelian varieties, the Todd class equals 1 because the tangent bundle is holomorphically trivial, so td = 1 and the Todd genus equals 0 or the Euler characteristic determined by ch as in examples studied by Mumford and Weil. On complex surfaces such as K3 surfaces, one computes td_2 = (c1^2 + c2)/12; since c1 = 0 for K3, the Todd genus equals ∫ c2/12 = 2, consistent with classical enumerative invariants and results of Noether and Kodaira.

Relations to other characteristic classes

The Todd class is intimately related to the Chern character ch via the Riemann–Roch pairings and to the A-hat (Â) class appearing in the Atiyah–Singer index theorem for spin manifolds. While Chern classes c_i determine td through the exponential power series, Pontryagin classes p_i of the underlying real bundle relate to combinations of Chern classes when passing between complex and real viewpoints; this provides bridges to signatures and Hirzebruch signature theorem calculations involving the L-genus. In generalized cohomology theories, analogous multiplicative genera such as the Todd genus, the Â-genus, and the elliptic genus are studied by Landweber, Stong, and Witten within formal group law frameworks and complex cobordism.

Category:Characteristic classes