In
algebraic geometry, the Néron–Severi group of a
variety
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
is
the group of divisors modulo
algebraic equivalence In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined inte ...
; in other words it is the group of
components of the
Picard scheme
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global v ...
of a variety. Its rank is called the
Picard number. It is named after
Francesco Severi and
André Néron
André Néron (November 30, 1922, La Clayette, France – April 6, 1985, Paris, France) was a French mathematician at the Université de Poitiers who worked on elliptic curves and abelian varieties. He discovered the Néron minimal model of an ...
.
Definition
In the cases of most importance to classical algebraic geometry, for a
complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
:X \times Y \to Y
is a closed map (i.e. maps closed sets onto closed sets). Thi ...
''V'' that is
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
, the
connected component of the Picard scheme is an
abelian variety written
:Pic
0(''V'').
The quotient
:Pic(''V'')/Pic
0(''V'')
is an abelian group NS(''V''), called the Néron–Severi group of ''V''. This is a
finitely-generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
by the Néron–Severi theorem, which was proved by Severi over the complex numbers and by Néron over more general fields.
In other words, the
Picard group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
fits into an
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
:
The fact that the rank is finite is
Francesco Severi's theorem of the base; the rank is the Picard number of ''V'', often denoted ρ(''V''). The elements of finite order are called Severi divisors, and form a finite group which is a birational invariant and whose order is called the Severi number. Geometrically NS(''V'') describes the
algebraic equivalence In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined inte ...
classes of
divisors
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
on ''V''; that is, using a stronger, non-linear equivalence relation in place of
linear equivalence of divisors, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to
numerical equivalence, an essentially topological classification by
intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...
s.
First Chern class and integral valued 2-cocycles
The
exponential sheaf sequence In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.
Let ''M'' be a complex manifold, and write ''O'M'' for the sheaf of holomorphic functions on ''M''. Let ''O'M''* be ...
:
gives rise to a long exact sequence featuring
:
The first arrow is the
first Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
on the
Picard group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
:
and the Neron-Severi group can be identified with its image.
Equivalently, by exactness, the Neron-Severi group is the kernel of the second arrow
:
In the complex case, the Neron-Severi group is therefore the group of 2-cocycles whose
Poincaré dual
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* L ...
is represented by a complex hypersurface, that is, a
Weil divisor.
For complex tori
Complex tori are special because they have multiple equivalent definitions of the Neron-Severi group. One definition uses its complex structure for the definition
pg 30. For a complex torus
, where
is a complex vector space of dimension
and
is a lattice of rank
embedding in
, the first Chern class
makes it possible to identify the Neron-Severi group with the group of Hermitian forms
on
such that
Note that
is an alternating integral form on the lattice
.
See also
*
Complex torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
References
*
*A. Néron, ''Problèmes arithmétiques et géometriques attachée à la notion de rang d'une courbe algébrique dans un corps'' Bull. Soc. Math. France, 80 (1952) pp. 101–166
*A. Néron, ''La théorie de la base pour les diviseurs sur les variétés algébriques'', Coll. Géom. Alg. Liège, G. Thone (1952) pp. 119–126
* F. Severi, ''La base per le varietà algebriche di dimensione qualunque contenute in una data e la teoria generale delle corrispondénze fra i punti di due superficie algebriche'' Mem. Accad. Ital., 5 (1934) pp. 239–283
{{DEFAULTSORT:Neron-Severi Group
Algebraic geometry