Todd class

In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle. The Todd class plays a fundamental role in generalising the classical Riemann–Roch theorem to higher dimensions, in the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Hirzebruch–Riemann–Roch theorem.

In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle. The Todd class plays a fundamental role in generalising the classical Riemann–Roch theorem to higher dimensions, in the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Hirzebruch–Riemann–Roch theorem. Q(x) = \frac{x}{1 - e^{-x}}=\sum_{i=0}^\infty \frac{B_i}{i!}x^i = 1 +\dfrac{x}{2}+\dfrac{x^2}{12}-\dfrac{x^4}{720}+\cdots be the formal power series with the property that the coefficient of x^n in Q(x)^{n+1} is 1, where B_i denotes the i -th Bernoulli number (with B_1 = +\frac{1}{2} ). Consider the coefficient of x^j in the product \prod_{i=1}^m Q(\beta_i x) \ for any m > j . This is symmetric in the \beta_i s and homogeneous of weight j : so can be expressed as a polynomial \operatorname{td}_j(p_1,\ldots, p_j) in the elementary symmetric functions p of the \beta_i s. Then \operatorname{td}_j defines the Todd polynomials: they form a multiplicative sequence with Q as characteristic power series. If E has the \alpha_i as its Chern roots, then the Todd class \operatorname{td}(E) = \prod Q(\alpha_i) which is to be computed in the cohomology ring of X (or in its completion if one wants to consider infinite-dimensional manifolds). The Todd class can be given explicitly as a formal power series in the Chern classes as follows: \operatorname{td}(E) = 1 + \frac{c_1}{2} + \frac{c_1^2 +c_2}{12} + \frac{c_1c_2}{24} + \frac{-c_1^4 + 4 c_1^2 c_2 + c_1c_3 + 3c_2^2 - c_4}{720} + \cdots where the cohomology classes c_i are the Chern classes of E , and lie in the cohomology group H^{2i}(X) . If X is finite-dimensional then most terms vanish and \operatorname{td}(E) is a polynomial in the Chern classes.

Properties of the Todd class

The Todd class is multiplicative: \operatorname{td}(E\oplus F) = \operatorname{td}(E)\cdot \operatorname{td}(F). Let \xi \in H^2({\mathbb C} P^n) be the fundamental class of the hyperplane section. From multiplicativity and the Euler exact sequence for the tangent bundle of {\mathbb C} P^n 0 \to {\mathcal O} \to {\mathcal O}(1)^{n+1} \to T {\mathbb C} P^n \to 0, one obtains \operatorname{td}(T {\mathbb C}P^n) = \left( \dfrac{\xi}{1-e^{-\xi}} \right)^{n+1}. For any algebraic curve C the Todd class is just \operatorname{td}(C) = 1 + \frac{1}{2} c_1(T_C) . Since C is projective, it can be embedded into some \mathbb{P}^n and we can find c_1(T_C) using the normal sequence 0 \to T_C \to T_\mathbb{P^n}|_C \to N_{C/\mathbb{P}^n} \to 0 and properties of chern classes. For example, if we have a degree d plane curve in \mathbb{P}^2 , we find the total chern class is \begin{align} c(T_C) &= \frac{c(T_{\mathbb{P}^2}|_C)}{c(N_{C/\mathbb{P}^2})} \\ &= \frac{1+3[H]}{1+d[H]} \\ &= (1+3[H])(1-d[H]) \\ &= 1 + (3-d)[H] \end{align} where [H] is the hyperplane class in \mathbb{P}^2 restricted to C .

Hirzebruch-Riemann-Roch formula

For any coherent sheaf F on a smooth compact complex manifold M, one has \chi(F)=\int_M \operatorname{ch}(F) \wedge \operatorname{td}(TM), where \chi(F) is its holomorphic Euler characteristic, \chi(F):= \sum_{i=0}^{\text{dim}_{\mathbb{C}} M} (-1)^i \text{dim}_{\mathbb{C}} H^i(M,F), and \operatorname{ch}(F) its Chern character.

See also

Genus of a multiplicative sequence

Notes

References

Friedrich Hirzebruch, Topological methods in algebraic geometry, Springer (1978)

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