Abelian surface
An Abelian variety of dimension two, i.e. a complete connected group variety of dimension two over a field . The group law is commutative. In the sequel,
is assumed to be algebraically closed (cf. Algebraically closed field).
In the classification of algebraic surfaces, Abelian surfaces are exactly the smooth complete surfaces with Kodaira dimension
, geometric genus
and irregularity
.
For an Abelian surface , the dual Abelian variety
is again an Abelian surface. An invertible sheaf
on
defines the homomorphism
,
. The homomorphism
depends only on the algebraic equivalence class of
. The invertible sheaf
is ample (cf. Ample sheaf) if and only if
is an isogeny (i.e.,
is surjective and has finite kernel) and
. In this case,
with a positive integer
and the Riemann–Roch theorem says that
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where denotes the self-intersection number. Every Abelian surface admits an ample invertible sheaf and hence is projective (cf. Projective scheme).
A polarization on
is the algebraic equivalence class
of an ample invertible sheaf
. The degree
of
is by definition
. An Abelian surface
together with a polarization
is a polarized Abelian surface. A principal polarization is a polarization of degree
. A principally polarized Abelian surface
is either the Jacobi variety
of a smooth projective curve
of genus
, and
is the class of the theta divisor, or
is the product of two elliptic curves (Abelian varieties of dimension one, cf. also Elliptic curve) with
the product polarization.
If the degree of is prime to
, then
is said to be a separable polarization and the kernel of
is isomorphic to the group
, where
and
are positive integers such that
divides
and
. The pair
is the type of the polarized Abelian surface
.
A polarization of type
on
defines a polarization
on the dual Abelian surface
. The polarization
is again of type
and it is characterized by each of the following two equivalent properties:
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For a polarized Abelian surface of type
, the assignment
defines a rational mapping from
into the projective space of hyperplanes in
:
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If , then
is everywhere defined. The Lefschetz theorem says that for
the morphism
is an embedding. Suppose
; then
with a polarization
of type
. If the linear system
has no fixed components, then
is an embedding.
Complex Abelian surfaces.
An Abelian surface over the field of complex numbers is a complex torus
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(with a lattice in
) admitting a polarization. A polarization
on
can be considered as a non-degenerate alternating form
, the elementary divisors of which are given by the type
of
.
In the sequel, the field is assumed to be
, although some of the following results are also valid for arbitrary algebraically closed fields.
Suppose is of type
and the linear system
has no fixed components. The Reider theorem states that for
the invertible sheaf
is very ample if and only if there is no elliptic curve
on
with
(see [a14] and [a10]). For arbitrary
there exist finitely many isogenies
of degree
onto principally polarized Abelian surfaces (cf. also Isogeny). Suppose
with a symmetric invertible sheaf
(i.e.,
) and let
be the unique divisor in the linear system
. The divisor
on
defines a symmetric invertible sheaf
with class
and the covering
is étale of degree
. One distinguishes two cases: i)
is smooth of genus
,
and
is smooth of genus
; and ii)
is the sum
of two elliptic curves with intersection number
,
and
is the sum
of two elliptic curves with
.
In the following list, is a polarized Abelian surface of type
such that
admits no fixed components
Type — The linear system
has exactly
base points. The blow-up
of
in these points admits a morphism
induced by
. The general fibre of
is a smooth curve of genus
. The curve
on
defining
as above is double elliptic:
with an elliptic curve
, and
is isomorphic to
(see [a3]).
Type —
defines a
-fold covering
ramified along a curve
of degree
. The general divisor in the linear system
is a smooth curve of genus
. There are
isogenies
of degree
onto principally polarized Abelian surfaces. In case i) the smooth genus-
curve
is double elliptic:
, and the embedding of
into the Jacobian
induces an exact sequence
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The étale -fold covering
induces a morphism
with image
, the dual Abelian surface of
(see [a7]).
Type — There are
isogenies
of degree
onto principally polarized Abelian surfaces. If the curves
and
do not admit elliptic involutions compatible with
, then
is a birational morphism onto a singular octic
. In the exceptional case,
is a double covering of a singular quartic
, which is birational to an elliptic scroll. In the first case the octic
is smooth outside the four coordinate planes of
and touches the coordinate planes in curves
,
, of degree
. Each of the curves
has
double points and passes through
pinch points of
. The octic is a
covering of a Kummer surface:
(see also Type
below). The restrictions
are
-fold coverings of four double conics of
lying on a coordinate tetrahedron. The three double points of
map to three double points of
on the conic
and the
pinch points on
map to the other three double points on the double conic
(see [a6]).
Type — The invertible sheaf
is very ample, i.e.
is an embedding if and only if the curves
and
do not admit elliptic involutions compatible with
. In the exceptional case
is a double covering of an elliptic scroll (see [a13] and [a9]). If
is very ample,
is a smooth surface of degree
in
. It is the zero locus of a section of the Horrocks–Mumford bundle
(see [a8]). Conversely, the zero set
of a general section
is an Abelian surface of degree
, i.e. of type
.
Type —
is twice a principal polarization on
. The morphism
is a double covering of the Kummer surface
associated with
. It is isomorphic to
.
Type — The ideal sheaf
of the image of the embedding
is generated by
quadrics (see [a3]).
Type — Suppose
is very ample and let
be the associated Kummer surface. The subvector space
of odd sections induces an embedding of
, the blow-up of
in the
double points, as a smooth quartic surface into
.
is invariant under the action of the level-
Heisenberg group (cf. also Heisenberg representation)
. The
blown-up double points become skew lines on the quartic surface. Any
-invariant quartic surface in
with
skew lines comes from a polarized Abelian surface
of type
in this way (see [a5], [a11] and [a12]).
Type —
is three times a principal polarization and
is an embedding. If
is not a product, then the quadrics
vanishing on
generate the ideal sheaf
. In the product case,
is generated by quadrics and cubics (see [a4]).
Algebraic completely integrable systems.
An algebraic completely integrable system in the sense of M. Adler and P. van Moerbeke is a completely integrable polynomial Hamiltonian system on (with Casimir functions
and
independent constants of motion
in involution) such that:
a) for a general point the invariant manifold
is an open affine part of an Abelian variety
;
b) the flows of the integrable vector fields linearize on the Abelian varieties
[a2].
The divisor at infinity defines a polarization on
. In this way the mapping
defines a family of polarized Abelian varieties (cf. Moduli problem). Some examples of
-dimensional algebraic completely integrable systems and their associated Abelian surfaces are:
the three-body Toda lattice and the even, respectively odd, master systems (cf. also Master equations in cooperative and social phenomena) linearize on principally polarized Abelian surfaces;
the Kowalewski top, the Hénon–Heiles system and the Manakov geodesic flow on linearize on Abelian surfaces of type
[a1];
the Garnier system linearizes on Abelian surfaces of type [a15].
References
[a1] | M. Adler, P. van Moerbeke, "The Kowalewski and Hénon–Heiles motions as Manakov geodesic flows on ![]() |
[a2] | M. Adler, P. van Moerbeke, "The complex geometry of the Kowalewski–Painlevé analysis" Invent. Math. , 97 (1989) pp. 3–51 |
[a3] | W. Barth, "Abelian surfaces with ![]() |
[a4] | W. Barth, "Quadratic equations for level-![]() |
[a5] | W. Barth, I. Nieto, "Abelian surfaces of type ![]() ![]() |
[a6] | Ch. Birkenhake, H. Lange, D. van Straten, "Abelian surfaces of type ![]() |
[a7] | Ch. Birkenhake, H. Lange, "Moduli spaces of Abelian surfaces wih isogeny" , Geometry and Analysis, Bombay Colloquium 1992 , Tata Inst. Fundam. Res. (1995) pp. 225–243 |
[a8] | G. Horrocks, D. Mumford, "A rank ![]() ![]() ![]() |
[a9] | K. Hulek, H. Lange, "Examples of abelian surfaces in ![]() |
[a10] | H. Lange, Ch. Birkenhake, "Complex Abelian varieties" , Grundlehren math. Wiss. , 302 , Springer (1992) |
[a11] | I. Naruki, "On smooth quartic embeddings of Kummer surfaces" Proc. Japan Acad. , 67 A (1991) pp. 223–224 |
[a12] | V. V. Nikulin, "On Kummer surfaces" Math USSR Izv. , 9 (1975) pp. 261–275 (In Russian) |
[a13] | S. Ramanan, "Ample divisors on abelian surfaces" Proc. London Math. Soc. , 51 (1985) pp. 231–245 |
[a14] | I. Reider, "Vector bundles of rank ![]() |
[a15] | P. Vanhaecke, "A special case of the Garnier system, ![]() |
Abelian surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abelian_surface&oldid=16005