mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 solution set, set of solutions of a system of polynomial equations over the real number, ...

is one of a few possible generalizations of the genus of an algebraic curve or

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 vers ...

Projective varieties

Let ''X'' be a

projective scheme In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the de ...

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 space's ...

of the structure sheaf

\mathcal_X

Complex projective manifolds

The arithmetic genus of a complex projective manifold 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 coho ...

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 In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...

, 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

in coherent cohomology for the

structure sheaf In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...

\mathcal_M

: :

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

This definition therefore can be applied to some other

locally ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...

References

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Projective varieties

Complex projective manifolds

Kähler manifolds

See also

References

Further reading