Bott residue formula explained

In mathematics, the Bott residue formula, introduced by, 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} (P(A _{p)})/(\det A _{p}) = \int _{M} P(i\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