Todd ClassEdit

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The Todd class is a fundamental construction in differential geometry and algebraic topology that attaches to a complex vector bundle a characteristic class. It plays a central role in the Hirzebruch–Riemann–Roch framework and, in particular, links topology to analytic indices via the Todd genus. Conceptually, the Todd class captures information about how sections of a complex vector bundle behave and contribute to holomorphic invariants. In its most practical form, the Todd class is defined for complex vector bundles through a product over Chern roots, and for a complex manifold it becomes a class of its tangent bundle that feeds into index formulas.

Definition and construction

Let E be a complex vector bundle over a smooth manifold or, more generally, a complex manifold. If the Chern roots of E are denoted by x1, x2, ..., xr, the total Todd class td(E) is defined by td(E) = ∏ i (xi / (1 − e^(−xi))). Equivalently, td(E) is the multiplicative characteristic class associated with the power series t(z) = z/(1 − e^(−z)). The total Todd class td(E) lives in the rational cohomology of the base space and is determined by the splitting principle, which allows one to think of E formally as a direct sum of line bundles with Chern roots xi.

For a complex manifold M, td(TM) refers to the Todd class of the tangent bundle. The Todd class is a characteristic class in the sense that it depends only on the isomorphism class of E and behaves predictably under standard bundle operations.

The Todd genus is the numerical invariant obtained by integrating the top-degree part of td(TM) against the fundamental class of a compact complex manifold M: Td-genus(M) = ⟨td(TM), [M]⟩. In the algebraic setting, this interpretation is closely tied to holomorphic Euler characteristics and index theory.

Basic properties

  • Multiplicativity: td(E ⊕ F) = td(E) · td(F) for complex vector bundles E and F.
  • Natural with respect to pullback: if f: N → M is a smooth map and E is a complex vector bundle over M, then td(f^E) = f^(td(E)).
  • Expression in Chern classes: td(E) can be expanded in terms of the Chern classes c_i(E). For example, when E has Chern roots x_i, td(E) expands as: td(E) = 1 + (1/2)c1(E) + (1/12)(c1(E)^2 + c2(E)) + (1/24)(c1(E)^3 + 3c1(E)c2(E) + 2c3(E)) + … Here c1, c2, c3 are the Chern classes of E. The precise coefficients follow from the product formula with the generating function xi/(1 − e^(−xi)).
  • Relation to the Chern character: td(E) interacts with the Chern character ch(E) in the Hirzebruch–Riemann–Roch framework, which expresses holomorphic Euler characteristics in terms of characteristic classes.

Computation and examples

  • For the tangent bundle of a compact complex manifold M, td(TM) is used in index formulas that relate topology to holomorphic data. In particular, Hirzebruch–Riemann–Roch expresses the holomorphic Euler characteristic χ(O_M) as the integral of td(TM) over M: χ(O_M) = ⟨td(TM), [M]⟩.
  • A classical benchmark is projective space [CP^n]. The Todd genus of CP^n equals 1, reflecting the holomorphic simplicity of projective space and its structure sheaf O_CP^n. This yields χ(O_CP^n) = 1 and aligns with the broader Riemann–Roch predictions for simple homogeneous spaces.
  • For more complicated complex manifolds, the explicit computation of td(TM) requires knowledge of the Chern classes of the tangent bundle, which in turn depend on the geometry of the manifold (for example, line bundle constructions, fibrations, or homogeneous spaces).

Relation to other concepts

  • Chern classes: The Todd class is defined in terms of the Chern roots and can be expressed as a symmetric polynomial in the Chern classes c_i(E). Thus td(E) is a particular polynomial invariant constructed from the same basic ingredients as other characteristic classes.
  • Chern character and Hirzebruch–Riemann–Roch: The Todd class is a key ingredient in the Hirzebruch–Riemann–Roch theorem, which computes holomorphic Euler characteristics in terms of characteristic classes and the Chern character.
  • Complex and real vector bundles: While td is most naturally defined for complex vector bundles, one often applies it to the complexification of real tangent bundles in geometric settings, or to complexified bundles arising from additional geometric structure.
  • Applications in topology and algebraic geometry: td(TM) and the Todd genus appear in index theory, the study of elliptic operators, and enumerative geometry, serving as a bridge between topology, geometry, and analysis.

Historical notes

The Todd class is part of a broader family of genus and characteristic-class constructions developed in the 20th century to generalize classical index theorems. It is named in association with the Todd genus and features prominently in the development of the Hirzebruch–Riemann–Roch theorem, which generalizes the classical Riemann–Roch theorem to higher-dimensional complex manifolds and to vector bundles beyond line bundles.

See also