Holomorphic Lefschetz fixed-point formula explained

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} (1)/(\det ( 1- A _{p)}) = \sum ^{q\operatorname{trace}(f} _{q( -1 )} ^{*|H} ^{0 ,q} ^{\overline\partial} (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