Bott residue formula

In mathematics, the Bott residue formula, introduced by Template:Harvs, describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold.

Statement

If v is a holomorphic vector field on a compact complex manifold M, then

$\sum _{v(p)=0}\frac{P(A_p)}{\det A_p}=\underset{M}{\int }P(i\mathrm{\Theta }/2\pi )$

where

• The sum is over the fixed points p of the vector field v
• The linear transformation A_p is the action induced by v on the holomorphic tangent space at p
• P is an invariant polynomial function of matrices of degree dim(M)
• Θ is a curvature matrix of the holomorphic tangent bundle

See also

• Atiyah–Bott fixed-point theorem
• Holomorphic Lefschetz fixed-point formula

References

• Template:Citation
• Template:Citation