Arithmetic genus

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 manifoldsEdit

The arithmetic genus of a complex projective manifold of dimension n can be defined as a combination of Hodge numbers, namely

pa = hn,0hn − 1, 0 + ... + (−1)n − 1h1, 0.

When n = 1 we have[clarification needed] χ = 1 − g where g is the usual (topological) meaning of genus of a surface, so the definitions are compatible.

Kähler manifoldsEdit

By using hp,q = hq,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.

See alsoEdit

ReferencesEdit

  • P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library (2nd ed.). Wiley Interscience. p. 494. ISBN 0-471-05059-8. Zbl 0836.14001.
  • Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Berlin/Boston: Walter De Gruyter, ISBN 978-3-11-031622-3

Further readingEdit