Holomorphic Lefschetz fixed-point formula

In mathematics, the Holomorphic Lefschetz formula, introduced by Lefschetz (????), describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold.

Statement

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

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

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

References

• Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR1288523