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Holomorphic Lefschetz fixed point formula

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In mathematics, the Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the fixed points of a holomorphic vector field of a compact complex manifold to a sum over its Dolbeault cohomology groups.

Statement

If f is an automorphism of a compact complex manifold M with isolated fixed points, then

∑ f ( p ) = p 1 det ( 1 − A p ) = ∑ q ( − 1 ) q trace ⁡ ( f ∗ | H ∂ ¯ 0 , q ( M ) )

where

  • The sum is over the fixed points p of f
  • The linear transformation Ap is the action induced by f on the holomorphic tangent space at p

    • References

    Holomorphic Lefschetz fixed-point formula Wikipedia
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