mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Holomorphic Lefschetz formula is an analogue for

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 ...

s of the

Lefschetz fixed-point formula In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...

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 In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohom ...

groups.

Statement

If ''f'' is an

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...

of a compact complex manifold ''M'' with isolated fixed points, then :

\sum_\frac = \sum_q(-1)^q\operatorname(f^*, H^_(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''

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 Theorems in algebraic geometry

Statement

See also

References