Editing Todd class

In [[mathematics]], the '''Todd class''' is a certain construction now considered a part of the theory in [[algebraic topology]] of [[characteristic class]]es.  The Todd class of a [[vector bundle]] can be defined by means of the theory of [[Chern class]]es, and is encountered where Chern classes exist &mdash; most notably in [[differential topology]], the theory of [[complex manifold]]s 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–Riemann–Roch theorem|Grothendieck–Hirzebruch–Riemann–Roch theorem]].

== History ==
It is named for [[J. A. Todd]], who introduced a special case of the concept in algebraic geometry in 1937, before the Chern classes were defined. The geometric idea involved is sometimes called the '''Todd-Eger class'''. The general definition in higher dimensions is due to [[Friedrich Hirzebruch]].

== Definition ==
To define the Todd class <math>\operatorname{td}(E)</math> where <math>E</math> is a complex vector bundle on a [[topological space]] <math>X</math>, it is usually possible to limit the definition to the case of a [[Whitney sum]] of [[line bundle]]s, by means of a general device of characteristic class theory, the use of [[Chern roots]] (aka, the [[splitting principle]]). For the definition, let 

::<math> 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</math>
be the [[formal power series]] with the property that the coefficient of <math>x^n</math> in <math>Q(x)^{n+1}</math> is 1, where <math>B_i</math> denotes the <math>i</math>-th [[Bernoulli number]] (with <math>B_1 = +\frac{1}{2}</math>).  Consider the coefficient of <math>x^j</math> in the product 

:<math> \prod_{i=1}^m Q(\beta_i x) \ </math>

for any <math>m > j</math>.  This is symmetric in the <math>\beta_i</math>s and homogeneous of weight <math>j</math>: so can be expressed as a polynomial <math>\operatorname{td}_j(p_1,\ldots, p_j)</math> in the [[elementary symmetric function]]s <math>p</math> of the <math>\beta_i</math>s.  Then <math>\operatorname{td}_j</math> defines the '''Todd polynomials''': they form a [[multiplicative sequence]] with <math>Q</math> as characteristic [[power series]].
  
If <math>E</math> has the <math>\alpha_i</math> as its [[Chern roots]], then the '''Todd class'''

:<math>\operatorname{td}(E) = \prod Q(\alpha_i)</math>

which is to be computed in the [[cohomology ring]] of <math>X</math> (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:

:<math>\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 </math>

where the cohomology classes <math>c_i</math> are the Chern classes of <math>E</math>, and lie in the cohomology group <math>H^{2i}(X)</math>. If <math>X</math> is finite-dimensional then most terms vanish and <math>\operatorname{td}(E)</math> is a polynomial in the Chern classes.

==Properties of the Todd class==
The Todd class is multiplicative: 
::<math>\operatorname{td}(E\oplus F) = \operatorname{td}(E)\cdot \operatorname{td}(F).</math>

Let <math>\xi \in H^2({\mathbb C} P^n)</math> be the fundamental class of the hyperplane section.
From multiplicativity and the Euler exact sequence for the tangent bundle of <math> {\mathbb C} P^n</math>
:: <math> 0 \to  {\mathcal O} \to {\mathcal O}(1)^{n+1}   \to T {\mathbb C} P^n \to 0,</math>
one obtains
<ref>[http://math.stanford.edu/~vakil/245/245class18.pdf Intersection  Theory Class 18], by [[Ravi Vakil]]</ref> 
:: <math> \operatorname{td}(T {\mathbb C}P^n) =  \left( \dfrac{\xi}{1-e^{-\xi}} \right)^{n+1}.</math>

== Computations of the Todd class ==
For any algebraic curve <math>C</math> the Todd class is just <math>\operatorname{td}(C) = 1 + \frac{1}{2} c_1(T_C)</math>. Since <math>C</math> is projective, it can be embedded into some <math>\mathbb{P}^n</math> and we can find <math>c_1(T_C)</math> using the normal sequence<blockquote><math>0 \to T_C \to T_\mathbb{P^n}|_C \to N_{C/\mathbb{P}^n} \to 0</math></blockquote>and properties of chern classes. For example, if we have a degree <math>d</math> plane curve in <math>\mathbb{P}^2</math>, we find the total chern class is<blockquote><math>\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}</math></blockquote>where <math>[H]</math> is the hyperplane class in <math>\mathbb{P}^2</math> restricted to <math>C</math>.

==Hirzebruch-Riemann-Roch formula==
{{Main|Hirzebruch–Riemann–Roch theorem}}
For any [[coherent sheaf]] ''F'' on a smooth 
compact [[complex manifold]] ''M'', one has
::<math>\chi(F)=\int_M \operatorname{ch}(F) \wedge \operatorname{td}(TM),</math>
where <math>\chi(F)</math> is its [[holomorphic Euler characteristic]],
::<math>\chi(F):= \sum_{i=0}^{\text{dim}_{\mathbb{C}}  M} (-1)^i \text{dim}_{\mathbb{C}} H^i(M,F),</math>
and <math>\operatorname{ch}(F)</math> its [[Chern character]].

==See also==
* [[Genus of a multiplicative sequence]]

==Notes==

<references/>

==References==

*{{Citation | last1=Todd | first1=J. A. |authorlink=J. A. Todd | title=The Arithmetical Invariants of Algebraic Loci  | doi=10.1112/plms/s2-43.3.190  | zbl=0017.18504 | year=1937 | journal=[[Proceedings of the London Mathematical Society]] | volume=43 | issue=1 | pages=190–225}}
* [[Friedrich Hirzebruch]], ''Topological methods in algebraic geometry'', Springer  (1978)
*{{springer|id=T/t092930|title=Todd class|author=M.I. Voitsekhovskii}}

[[Category:Characteristic classes]]