---
tag: 2YDB5089FR
kind: concept
name: Todd class
msc: [14C17, 14C40, 57R20]
sources:
  - url: https://en.wikipedia.org/wiki/Todd_class
same_as:
  wikidata: Q2438393
  wikipedia: Todd class
---

## Summary

The Todd class is a characteristic class of vector bundles, built from the [[Chern classes|PAD37GKDPD]] by a fixed power series.
The Todd class of a smooth variety is the Todd class of its tangent bundle.
It is the correction term in the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]] and the [[Grothendieck–Riemann–Roch theorem|3HKREN3Y3F]], which compute Euler characteristics of sheaves from Chern classes.

## Explanations

### L1 human

The Riemann–Roch theorem expresses the dimension of a space of sections, an analytic quantity, through numbers such as degrees and the genus.
In higher dimensions the right expression needs a universal polynomial in the Chern classes of the tangent bundle, and this polynomial is the Todd class.
For a curve of genus $g$, the Todd class is $1 + \frac{1}{2} c_1$, and its degree part gives the term $1 - g$ in the Riemann–Roch formula $\chi(L) = \deg L + 1 - g$.