title: Holomorphic Lefschetz fixed-point formula
text: # 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
  