Next: The Atiyah-Bott fixed point Up: The life and works Previous: Nevanlinna theory and the

# Characteristic numbers and the Bott residue

According to the celebrated Hopf index theorem, the Euler characteristic of a smooth manifold is equal to the number of zeros of a vector field on the manifold, each counted with its index. In [41] and [43], Bott generalized the Hopf index theorem to other characteristic numbers such as the Pontryagin numbers of a real manifold and the Chern numbers of a complex manifold.

We will describe Bott's formula only for Chern numbers. Let be a compact complex manifold of dimension , and the Chern classes of the tangent bundle of . The Chern numbers of are the integrals , as ranges over all weighted homogeneous polynomials of degree . Like the Hopf index theorem, Bott's formula computes a Chern number in terms of the zeros of a vector field on , but the vector field must be holomorphic and the counting of the zeros is a little more subtle.

For any vector field and any function on , the Lie derivative satisfies:

It follows that at a zero of ,

Thus, at , the Lie derivative induces an endomorphism

of the tangent space of at . The zero is said to be nondegenerate if is nonsingular.

For any endomorphism of a vector space , we define the numbers to be the coefficients of its characteristic polynomial:

Bott's Chern number formula is as follows. Let be a compact complex manifold of complex dimension and a holomorphic vector field having only isolated nondegenerate zeros on . For any weighted homogeneous polynomial , ,

 (2)

summed over all the zeros of the vector field. Note that by the definition of a nondegenerate zero, , which is , is nonzero.

In Bott's formula, if the polynomial does not have degree , then the left-hand side of (2) is zero, and the formula gives an identity among the numbers . For the polynomial , Bott's formula recovers the Hopf index theorem:

zeros of

Bott's formula (2) is reminiscent of Cauchy's residue formula and so the right-hand side of (2) may be viewed as a residue of at .

In [43] Bott generalized his Chern number formula (2), which assumes isolated zeros, to holomorphic vector fields with higher-dimensional zero sets and to bundles other than the tangent bundle (a vector field is a section of the tangent bundle).

Next: The Atiyah-Bott fixed point Up: The life and works Previous: Nevanlinna theory and the
HTML generated on 2005-12-21