next up previous
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 $ M$ be a compact complex manifold of dimension $ n$ , and $ c_1(M), \dots, c_n(M)$ the Chern classes of the tangent bundle of $ M$ . The Chern numbers of $ M$ are the integrals $ \int_M \phi (c_1(M), \dots , c_n(M))$ , as $ \phi$ ranges over all weighted homogeneous polynomials of degree $ n$ . Like the Hopf index theorem, Bott's formula computes a Chern number in terms of the zeros of a vector field $ X$ on $ M$ , but the vector field must be holomorphic and the counting of the zeros is a little more subtle.

For any vector field $ Y$ and any $ C^{\infty}$ function $ f$ on $ M$ , the Lie derivative $ \mathcal{L}_{X}$ satisfies:

$\displaystyle \mathcal{L}_{X}(fY)= (Xf)Y+f\mathcal{L}_{X}Y.

It follows that at a zero $ p$ of $ X$ ,

$\displaystyle (\mathcal{L}_{X}fY)_p = f(p) (\mathcal{L}_{X}Y)p.

Thus, at $ p$ , the Lie derivative $ \mathcal{L}_{X}$ induces an endomorphism

$\displaystyle L_p : T_pM \to T_p M

of the tangent space of $ M$ at $ p$ . The zero $ p$ is said to be nondegenerate if $ L_p$ is nonsingular.

For any endomorphism $ A$ of a vector space $ V$ , we define the numbers $ c_i (A)$ to be the coefficients of its characteristic polynomial:

$\displaystyle \det (I+tA) = \sum c_i (A) t^i.

Bott's Chern number formula is as follows. Let $ M$ be a compact complex manifold of complex dimension $ n$ and $ X$ a holomorphic vector field having only isolated nondegenerate zeros on $ M$ . For any weighted homogeneous polynomial $ \phi (x_1, \dots, x_n)$ , $ \deg x_i = 2i$ ,

$\displaystyle \int_M \phi (c_1(M), \dots, c_n(M)) = \sum_p \dfrac{\phi (c_1 (L_p), \dots, c_n(L_p))}{c_n(L_p)},$ (2)

summed over all the zeros of the vector field. Note that by the definition of a nondegenerate zero, $ c_n(L_p)$ , which is $ \det L_p$ , is nonzero.

In Bott's formula, if the polynomial $ \phi$ does not have degree $ 2n$ , then the left-hand side of (2) is zero, and the formula gives an identity among the numbers $ c_i(L_p)$ . For the polynomial $ \phi
(x_1, \dots, x_n) = x_n$ , Bott's formula recovers the Hopf index theorem:

$\displaystyle \int_M c_n (M) = \sum_p \dfrac{c_n(L_p)}{c_n(L_p)} = \char93 $    zeros of $\displaystyle X.

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 $ \phi$ at $ p$ .

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