“Arithmetic genus”的意思、由来-开放百科全书

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Arithmetic genus

释义

Complex projective manifolds
Kähler manifolds
See also
References
Further reading

In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.

Complex projective manifolds

The arithmetic genus of a complex projective manifold

of dimension n can be defined as a combination of Hodge numbers, namely

p_a = h^n,0 − h^{n − 1, 0} + ... + (−1)^{n − 1}h^{1, 0}.

When n = 1 we have{{Clarify|relation between χ and p_a?|date=August 2016}} χ = 1 − g where g is the usual (topological) meaning of genus of a surface, so the definitions are compatible.

Kähler manifolds

By using h^p,q = h^q,p for compact Kähler manifolds this can be

reformulated as the Euler characteristic in coherent cohomology for the structure sheaf :

This definition therefore can be applied to some other

locally ringed spaces.

References

{{cite book | author=P. Griffiths | authorlink=Phillip Griffiths |author2=J. Harris |authorlink2=Joe Harris (mathematician) | title=Principles of Algebraic Geometry | edition=2nd | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | zbl=0836.14001 | page=494 }}
{{Citation | last1=Rubei | first1=Elena | title=Algebraic Geometry, a concise dictionary | publisher=Walter De Gruyter | location=Berlin/Boston | isbn=978-3-11-031622-3 | year=2014}}

Complex projective manifolds

Kähler manifolds

See also

References

Further reading