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In mathematics, the '''Bott residue formula''', introduced by {{harvs|txt|last=Bott|year=1967|authorlink=Raoul Bott}}, 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
:<math> \sum_{v(p)=0}\frac{P(A_p)}{\det A_p} = \int_M P(i\Theta/2\pi)</math>
where
*The sum is over the fixed points ''p'' of the vector field ''v''
*The linear transformation ''A''<sub>''p''</sub> 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]]
*[[Holomorphic Lefschetz fixed-point formula]]
 
==References==
 
*{{Citation | last1=Bott | first1=Raoul | author1-link=Raoul Bott | title=Vector fields and characteristic numbers | url=http://projecteuclid.org/euclid.mmj/1028999721 | doi=10.1307/mmj/1028999721 | mr=0211416  | year=1967 | journal=The Michigan Mathematical Journal | issn=0026-2285 | volume=14 | pages=231–244}}
*{{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]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523  | year=1994}}
 
[[Category:Complex manifolds]]

Revision as of 05:18, 21 September 2013

In mathematics, the Bott residue formula, introduced by Template:Harvs, 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

v(p)=0P(Ap)detAp=MP(iΘ/2π)

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

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