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

$\sum_{f(p)=p}\frac{1}{\det (1-A_p)} = \sum_q(-1)^q\operatorname{trace}(f^*|H^{0,q}_{\overline\partial}(M))$

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

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