Holomorphic Lefschetz fixed-point formula
# 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
