In mathematics, the Bott residue formula, introduced by , describes a sum over the
fixed points of a
holomorphic vector field of a compact
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
.
Statement
If ''v'' is a holomorphic vector field on a compact complex manifold ''M'', then
:
where
*The sum is over the fixed points ''p'' of the vector field ''v''
*The linear transformation ''A''
''p'' 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
*
Atiyah–Bott fixed-point theorem
In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds ''M'', which uses an elliptic complex on ''M''. This is a sys ...
*
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 point (mathematics), fixed points of a holomorphic vector field of a compact complex manifo ...
References
*
*{{Citation , last1=Griffiths , first1=Phillip , author1-link=Phillip Griffiths , last2=Harris , first2=Joseph , author2-link=Joe Harris (mathematician) , title=Principles of algebraic geometry , publisher=
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in p ...
, location=New York , series=Wiley Classics Library , isbn=978-0-471-05059-9 , mr=1288523 , year=1994
Complex manifolds