In mathematics, the Bott residue formula, introduced by Bott (1967), 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

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

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