From Wikipedia, the free encyclopedia
In mathematics, the Bott residue formula, introduced by
Bott (
1967), describes a sum over the
fixed points of a
holomorphic
vector field of a compact
complex manifold.
Statement
If v is a holomorphic vector field on a compact complex manifold M, then
![{\displaystyle \sum _{v(p)=0}{\frac {P(A_{p})}{\det A_{p}}}=\int _{M}P(i\Theta /2\pi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b737ad539c1b22ecad7efc586346378eee395668)
where
- The sum is over the fixed points p of the vector field v
- The linear transformation Ap is the action induced by v on the holomorphic tangent space at p
- P is an invariant polynomial function of matrices of degree dim(M)
- Θ is a curvature matrix of the holomorphic tangent bundle
See also
References
-
Bott, Raoul (1967),
"Vector fields and characteristic numbers", The Michigan Mathematical Journal, 14: 231–244,
doi:
10.1307/mmj/1028999721,
ISSN
0026-2285,
MR
0211416
-
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