Bott residue formula

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

See also

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

References

• Bott, Raoul (1967), "Vector fields and characteristic numbers", The Michigan Mathematical Journal, 14: 231–244, doi:10.1307/mmj/1028999721, ISSN 0026-2285, MR 0211416
• Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523