Puneet Varma (Editor)

Bott residue formula

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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

∑ v ( p ) = 0 P ( A p ) det A p = ∫ M P ( i Θ / 2 π )

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

    • References

    Bott residue formula Wikipedia
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