Holomorphic Lefschetz fixed-point formula

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

\sum _{f(p)=p}{\frac {1}{\det(1-A_{p})}}=\sum _{q}(-1)^{q}\operatorname {trace} (f^{*}|H_{\overline {\partial }}^{0,q}(M))

where

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

See also

Bott residue formula

References

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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