Arithmetic genus explained

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.

Projective varieties

Let X be a projective scheme of dimension r over a field k, the arithmetic genus

p _{a} of X is defined as$$p\_a(X)=(-1)^r\; (\backslash chi(\backslash mathcal\_X)-1).$$Here \chi( l{O} _{X)} is the Euler characteristic of the structure sheaf l{O} _{X} .^[1]

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=\sum} ^{n -1} ^{j =0} ( -1 ) ^{j} h ^{n-j,0} .

When n=1, the formula becomes

p ^{1 , 0} _{a = h} . According to the Hodge theorem, h ^{0,1} = h ^{1,0} . Consequently h ^{0,1} = h ^{1(X)/2=g} , where g is the usual (topological) meaning of genus of a surface, so the definitions are compatible.

When X is a compact Kähler manifold, applying h^p,q = h^q,p recovers the earlier definition for projective varieties.

Kähler manifolds

l{O} _{M} :p ^{n(\chi( l{O}} _{M) -1 ).}

This definition therefore can be applied to some other locally ringed spaces.

See also

• Genus (mathematics)
• Geometric genus

References

• Book: Phillip Griffiths

. P. Griffiths . Phillip Griffiths . J. Harris . Joe Harris (mathematician) . Principles of Algebraic Geometry . 2nd . Wiley Classics Library . Wiley Interscience . 1994 . 0-471-05059-8 . 0836.14001 . 494 .

1. Book: Hartshorne, Robin . Robin Hartshorne . Algebraic Geometry . 1977 . Springer New York . 978-1-4419-2807-8 . Graduate Texts in Mathematics . 52 . New York, NY . 230 . 10.1007/978-1-4757-3849-0. 197660097 .

Further reading

• Book: Hirzebruch, Friedrich . Friedrich Hirzebruch

. Friedrich Hirzebruch . Topological methods in algebraic geometry . Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel . Reprint of the 2nd, corr. print. of the 3rd . 1978 . Classics in Mathematics . Berlin . . 1995 . 3-540-58663-6 . 0843.14009 .