In mathematics, the arithmetic genus of an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...

is one of a few possible generalizations of the

genus of an algebraic curve In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a Surface (topology), surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surf ...

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...

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 (\chi(\mathcal_X)-1).

Here

\chi(\mathcal_X)

is the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...

of the structure sheaf

\mathcal_X

Complex projective manifolds

The arithmetic genus of a

complex projective manifold __notoc__ In mathematics, an algebraic manifold is an algebraic variety which is also a manifold. As such, algebraic manifolds are a generalisation of the concept of smooth curves and surfaces defined by polynomials. An example is the sphere, which ...

of dimension ''n'' can be defined as a combination of

Hodge number In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohomo ...

s, namely :

p_a=\sum_^ (-1)^j h^.

When ''n=1'', the formula becomes

p_a=h^

. According to the Hodge theorem,

h^=h^

. Consequently

h^=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

By using ''h''^''p'',''q'' = ''h''^''q'',''p'' for compact Kähler manifolds this can be reformulated as the

coherent cohomology In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the exist ...

for the structure sheaf

\mathcal_M

: :

p_a=(-1)^n(\chi(\mathcal_M)-1).\,

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

References

* *

Projective varieties

Complex projective manifolds

Kähler manifolds

See also

References

Further reading