In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the geometric genus is a basic
birational invariant In algebraic geometry, a birational invariant is a property that is preserved under birational equivalence.
Formal definition
A birational invariant is a quantity or object that is well-defined on a birational equivalence class of algebraic varieti ...
of
algebraic varieties
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. ...
and
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
s.
Definition
The geometric genus can be defined for
non-singular complex projective varieties and more generally for
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
s as the
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 co ...
(equal to by
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Al ...
), that is, the dimension of the
canonical linear system plus one.
In other words for a variety of
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a ...
it is the number of linearly independent holomorphic -
forms
Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens.
Form also refers to:
*Form (document), a document (printed or electronic) with spaces in which to write or enter data
* ...
to be found on .
[Danilov & Shokurov (1998), p. 53/ref> This definition, as the dimension of
:
then carries over to any base field, when is taken to be the sheaf of ]Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic ...
s and the power is the (top) exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
, the canonical line bundle.
The geometric genus is the first invariant of a sequence of invariants called the plurigenera In mathematics, the pluricanonical ring of an algebraic variety ''V'' (which is non-singular), or of a complex manifold, is the graded ring
:R(V,K)=R(V,K_V) \,
of sections of powers of the canonical bundle ''K''. Its ''n''th graded component (f ...
.
Case of curves
In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces
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 ...
. The algebraic definition of genus agrees with the topological notion. On a nonsingular curve, the canonical line bundle has degree .
The notion of genus features prominently in the statement of the Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
(see also Riemann–Roch theorem for algebraic curves) and of the Riemann–Hurwitz formula In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramifi ...
. By the Riemann-Roch theorem, an irreducible plane curve of degree ''d'' has geometric genus
:
where ''s'' is the number of singularities when properly counted
If is an irreducible (and smooth) hypersurface in the projective plane cut out by a polynomial equation of degree , then its normal line bundle is the Serre twisting sheaf
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functo ...
, so by the adjunction formula In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedde ...
, the canonical line bundle of is given by
:
Genus of singular varieties
The definition of geometric genus is carried over classically to singular curves , by decreeing that
:
is the geometric genus of the normalization . That is, since the mapping
:
is birational
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational f ...
, the definition is extended by birational invariance.
See also
*Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
Topology
Orientable surfaces
The genus of ...
*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.
Projective varieties
Let ''X'' be a projective scheme of dimension ''r'' over a field '' ...
* Invariants of surfaces
Notes
References
*
* {{cite book , author1=V. I. Danilov , author2=Vyacheslav V. Shokurov , title=Algebraic curves, algebraic manifolds, and schemes , publisher=Springer , year=1998 , isbn=978-3-540-63705-9
Algebraic varieties