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# 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

\( \sum_{v(p)=0}\frac{P(A_p)}{\det A_p} = \int_M P(i\Theta/2\pi) \)

where

The sum is over the fixed points p of the vector field v

The linear transformation Ap 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

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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