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
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
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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
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