- Source: 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
∑
f
(
p
)
=
p
1
det
(
1
−
A
p
)
=
∑
q
(
−
1
)
q
trace
(
f
∗
|
H
∂
¯
0
,
q
(
M
)
)
{\displaystyle \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 Ap 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
Kata Kunci Pencarian:
- Lefschetz fixed-point theorem
- Holomorphic Lefschetz fixed-point formula
- Atiyah–Bott fixed-point theorem
- Bott residue formula
- Kähler manifold
- Hodge theory
- Möbius transformation
- Raoul Bott
- Michael Atiyah
- Triangulation (topology)
