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An [[Abelian variety|Abelian variety]] of dimension two, i.e. a complete connected group variety of dimension two over a field  $  k $. 
The group law is commutative. In the sequel,  $  k $
is assumed to be algebraically closed (cf. [[Algebraically closed field|Algebraically closed field]]).

In the classification of algebraic surfaces, Abelian surfaces are exactly the smooth complete surfaces  $  A $
with [[Kodaira dimension|Kodaira dimension]]  $  \kappa = 0 $, 
[[Geometric genus|geometric genus]]  $  p _ {g} = h  ^ {2} ( A, {\mathcal O} _ {A} ) =1 $
and [[Irregularity|irregularity]]  $  q = h  ^ {1} ( A, {\mathcal O} _ {A} ) = 2 $.

For an Abelian surface  $  A $, 
the dual Abelian variety  $  {\widehat{A}  } = { \mathop{\rm Pic} }  ^ {0} ( A ) $
is again an Abelian surface. An invertible [[Sheaf|sheaf]]  $  L $
on  $  A $
defines the homomorphism  $  {\phi _ {L} } : A \rightarrow { {\widehat{A}  } } $, 
$  a \mapsto t _ {a}  ^ {*} L \otimes L ^ {- 1 } $. 
The homomorphism  $  \phi _ {L} $
depends only on the algebraic equivalence class of  $  L $. 
The invertible sheaf  $  L $
is ample (cf. [[Ample sheaf|Ample sheaf]]) if and only if  $  \phi _ {L} $
is an [[Isogeny|isogeny]] (i.e.,  $  \phi _ {L} $
is surjective and has finite kernel) and  $  h  ^ {0} ( A,L ) \neq0 $. 
In this case,  $  { \mathop{\rm deg} } \phi _ {L} = d  ^ {2} $
with a positive integer  $  d $
and the [[Riemann–Roch theorem|Riemann–Roch theorem]] says that

$$ 
h  ^ {0} ( A,L ) = {
\frac{1}{2}
 } ( L  ^ {2} ) = d,
$$

where  $  ( L  ^ {2} ) $
denotes the self-intersection number. Every Abelian surface admits an ample invertible sheaf and hence is projective (cf. [[Projective scheme|Projective scheme]]).

A polarization  $  \lambda $
on  $  A $
is the algebraic equivalence class  $  [ L ] $
of an ample invertible sheaf  $  L $. 
The degree  $  { \mathop{\rm deg} } \lambda $
of  $  \lambda $
is by definition  $  d = \sqrt { { \mathop{\rm deg} } \phi _ {L} } $. 
An Abelian surface  $  A $
together with a polarization  $  \lambda $
is a polarized Abelian surface. A principal polarization is a polarization of degree  $  1 $. 
A principally polarized Abelian surface  $  ( A, \lambda ) $
is either the [[Jacobi variety|Jacobi variety]]  $  J ( H ) $
of a smooth projective curve  $  H $
of genus  $  2 $, 
and  $  \lambda = \theta $
is the class of the theta divisor, or  $  A $
is the product of two elliptic curves (Abelian varieties of dimension one, cf. also [[Elliptic curve|Elliptic curve]]) with  $  \lambda $
the product polarization.

If the degree of  $  \lambda = [ L ] $
is prime to  $  { \mathop{\rm char} } ( k ) $, 
then  $  \lambda $
is said to be a separable polarization and the kernel of  $  \phi _ {L} $
is isomorphic to the group  $  ( \mathbf Z/d _ {1} \mathbf Z )  ^ {2} \times ( \mathbf Z/d _ {2} \mathbf Z )  ^ {2} $, 
where  $  d _ {1} $
and  $  d _ {2} $
are positive integers such that  $  d _ {1} $
divides  $  d _ {2} $
and  $  d _ {1} d _ {2} = { \mathop{\rm deg} } \lambda $. 
The pair  $  ( d _ {1} ,d _ {2} ) $
is the type of the polarized Abelian surface  $  ( A, \lambda ) $.

A polarization  $  \lambda = [ L ] $
of type  $  ( d _ {1} ,d _ {2} ) $
on  $  A $
defines a polarization  $  {\widehat \lambda   } = [ {\widehat{L}  } ] $
on the dual Abelian surface  $  {\widehat{A}  } $. 
The polarization  $  {\widehat \lambda   } $
is again of type  $  ( d _ {1} ,d _ {2} ) $
and it is characterized by each of the following two equivalent properties:

$$ 
\phi _ {L}  ^ {*} {\widehat \lambda   } = d _ {1} d _ {2} \lambda  \iff  \phi _ { {\widehat{L}  }  } \phi _ {L} = d _ {1} d _ {2} { \mathop{\rm id} } _ {A} .
$$

For a polarized Abelian surface  $  ( A, \lambda = [ L ] ) $
of type  $  ( d _ {1} ,d _ {2} ) $, 
the assignment  $  A \ni a \mapsto \{ {\sigma \in H  ^ {0} ( A,L ) } : {\sigma ( a ) = 0 } \} \subset  H  ^ {0} ( A,L ) $
defines a rational mapping from  $  A $
into the projective space of hyperplanes in  $  H  ^ {0} ( A,L ) $:

$$ 
{\varphi _ {L} } : A \rightarrow {\mathbf P ( H  ^ {0} ( A,L )  ^ {*} ) } \simeq \mathbf P _ {k} ^ {d _ {1} d _ {2} - 1 } .
$$

If  $  d _ {1} \geq  2 $, 
then  $  \varphi _ {L} $
is everywhere defined. The Lefschetz theorem says that for  $  d _ {1} \geq  3 $
the morphism  $  \varphi _ {L} $
is an embedding. Suppose  $  d _ {1} = 2 $; 
then  $  \lambda = 2 \mu $
with a polarization  $  \mu = [ M ] $
of type  $  ( 1, { {d _ {2} } / 2 } ) $. 
If the linear system  $  | M | $
has no fixed components, then  $  \varphi _ {L} $
is an embedding.

==Complex Abelian surfaces.==
An Abelian surface over the field  $  \mathbf C $
of complex numbers is a [[Complex torus|complex torus]]

$$ 
A = { {\mathbf C  ^ {2} } / \Lambda }
$$

(with a lattice  $  \Lambda \simeq \mathbf Z  ^ {4} $
in  $  \mathbf C  ^ {2} $) 
admitting a polarization. A polarization  $  \lambda $
on  $  A $
can be considered as a non-degenerate alternating form  $  \Lambda \times \Lambda \rightarrow \mathbf Z $, 
the elementary divisors of which are given by the type  $  ( d _ {1} ,d _ {2} ) $
of  $  \lambda $.

In the sequel, the field  $  k $
is assumed to be  $  \mathbf C $, 
although some of the following results are also valid for arbitrary algebraically closed fields.

Suppose  $  ( A, \lambda = [ L ] ) $
is of type  $  ( 1,d ) $
and the linear system  $  | L | $
has no fixed components. The Reider theorem states that for  $  d \geq  5 $
the invertible sheaf  $  L $
is very ample if and only if there is no elliptic curve  $  E $
on  $  A $
with  $  ( E \cdot L ) = 2 $(
see [[#References|[a14]]] and [[#References|[a10]]]). For arbitrary  $  d \geq  1 $
there exist finitely many isogenies  $  f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $
of degree  $  d $
onto principally polarized Abelian surfaces (cf. also [[Isogeny|Isogeny]]). Suppose  $  \theta = [ \Theta ] $
with a symmetric invertible sheaf  $  \Theta $(
i.e.,  $  ( -1 ) _ {A}  ^ {*} \Theta \simeq \Theta $) 
and let  $  H $
be the unique divisor in the linear system  $  | \Theta | $. 
The divisor  $  C = f ^ {- 1 } ( H ) $
on  $  A $
defines a symmetric invertible sheaf  $  L = {\mathcal O} _ {A} ( C ) $
with class  $  [ L ] = \lambda $
and the covering  $  {f \mid  _ {C} } : C \rightarrow H $
is étale of degree  $  d $. 
One distinguishes two cases: i)  $  H $
is smooth of genus  $  2 $, 
$  B = J ( H ) $
and  $  C $
is smooth of genus  $  d + 1 $; 
and ii)  $  H $
is the sum  $  E _ {1} + E _ {2} $
of two elliptic curves with intersection number  $  ( E _ {1} \cdot E _ {2} ) = 1 $, 
$  B = E _ {1} \times E _ {2} $
and  $  C $
is the sum  $  F _ {1} + F _ {2} $
of two elliptic curves with  $  ( F _ {1} \cdot F _ {2} ) = d $.

In the following list,  $  ( A, \lambda = [ L ] ) $
is a polarized Abelian surface of type  $  ( d _ {1} ,d _ {2} ) $
such that  $  | L | $
admits no fixed components

Type  $  ( 1,2 ) $— 
The linear system  $  | L | $
has exactly  $  4 $
base points. The blow-up  $  {\widetilde{A}  } $
of  $  A $
in these points admits a morphism  $  { {\widetilde \varphi   } _ {L} } : { {\widetilde{A}  } } \rightarrow {\mathbf P  ^ {1} } $
induced by  $  \varphi _ {L} $. 
The general fibre of  $  {\widetilde \varphi   } _ {L} $
is a smooth curve of genus  $  3 $. 
The curve  $  C $
on  $  A $
defining  $  L $
as above is double elliptic:  $  C { \mathop \rightarrow \limits ^ { {2:1 }}  } E $
with an elliptic curve  $  E $, 
and  $  A $
is isomorphic to  $  { {J ( C ) } / E } $(
see [[#References|[a3]]]).

Type  $  ( 1,3 ) $— 
$  L $
defines a  $  6 $-
fold covering  $  {\varphi _ {L} } : A \rightarrow {\mathbf P  ^ {2} } $
ramified along a curve  $  R \subset  \mathbf P  ^ {2} $
of degree  $  18 $. 
The general divisor in the linear system  $  | L | $
is a smooth curve of genus  $  4 $. 
There are  $  4 $
isogenies  $  f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $
of degree  $  3 $
onto principally polarized Abelian surfaces. In case i) the smooth genus- $  4 $
curve  $  C \in | L | $
is double elliptic:  $  C { \mathop \rightarrow \limits ^ { {2:1 }}  } E $, 
and the embedding of  $  E $
into the Jacobian  $  J ( C ) $
induces an exact sequence

$$ 
0 \rightarrow E \times E \rightarrow J ( C ) \rightarrow A \rightarrow 0.
$$

The étale  $  3 $-
fold covering  $  {f \mid  _ {C} } : C \rightarrow H $
induces a morphism  $  {f  ^ {*} } : {J ( H ) } \rightarrow {J ( C ) } $
with image  $  {\widehat{A}  } $, 
the dual Abelian surface of  $  A $(
see [[#References|[a7]]]).

Type  $  ( 1,4 ) $— 
There are  $  24 $
isogenies  $  f : {( A, \lambda ) } \rightarrow {( B, \theta ) } $
of degree  $  4 $
onto principally polarized Abelian surfaces. If the curves  $  C $
and  $  H $
do not admit elliptic involutions compatible with  $  f $, 
then  $  \varphi _ {L} :A \twoheadrightarrow {\overline{A}\; } \subset  \mathbf P  ^ {3} $
is a birational morphism onto a singular octic  $  {\overline{A}\; } $. 
In the exceptional case,  $  \varphi _ {L} : A \twoheadrightarrow {\overline{A}\; } \subset  \mathbf P  ^ {3} $
is a double covering of a singular quartic  $  {\overline{A}\; } $, 
which is birational to an elliptic scroll. In the first case the octic  $  {\overline{A}\; } $
is smooth outside the four coordinate planes of  $  \mathbf P  ^ {3} $
and touches the coordinate planes in curves  $  D _ {i} $, 
$  i = 1 \dots 4 $, 
of degree  $  4 $. 
Each of the curves  $  D _ {i} $
has  $  3 $
double points and passes through  $  12 $
pinch points of  $  {\overline{A}\; } $. 
The octic is a  $  8:1 $
covering of a [[Kummer surface|Kummer surface]]:  $  p: {\overline{A}\; } \twoheadrightarrow K \subset  \mathbf P  ^ {3} $(
see also Type  $  ( 2,2 ) $
below). The restrictions  $  p \mid  _ {D _ {i}  } $
are  $  4 $-
fold coverings of four double conics of  $  K $
lying on a coordinate tetrahedron. The three double points of  $  D _ {i} $
map to three double points of  $  K $
on the conic  $  p ( D _ {i} ) $
and the  $  12 $
pinch points on  $  D _ {i} $
map to the other three double points on the double conic  $  p ( D _ {i} ) $(
see [[#References|[a6]]]).

Type  $  ( 1,5 ) $— 
The invertible sheaf  $  L $
is very ample, i.e.  $  {\varphi _ {L} } : A \rightarrow {\mathbf P  ^ {4} } $
is an embedding if and only if the curves  $  C $
and  $  H $
do not admit elliptic involutions compatible with  $  f $. 
In the exceptional case  $  \varphi _ {L} $
is a double covering of an elliptic scroll (see [[#References|[a13]]] and [[#References|[a9]]]). If  $  L $
is very ample,  $  \varphi _ {L} ( A ) $
is a smooth surface of degree  $  10 $
in  $  \mathbf P  ^ {4} $. 
It is the zero locus of a section of the Horrocks–Mumford bundle  $  F $(
see [[#References|[a8]]]). Conversely, the zero set  $  \{ \sigma = 0 \} \subset  \mathbf P  ^ {4} $
of a general section  $  \sigma \in H  ^ {0} ( \mathbf P  ^ {4} ,F ) $
is an Abelian surface of degree  $  10 $, 
i.e. of type  $  ( 1,5 ) $.

Type  $  ( 2,2 ) $— 
$  \lambda $
is twice a principal polarization on  $  A $. 
The morphism  $  \varphi _ {L} : A \twoheadrightarrow K _ {A} \subset  \mathbf P  ^ {3} $
is a double covering of the [[Kummer surface|Kummer surface]]  $  K _ {A} $
associated with  $  A $. 
It is isomorphic to  $  {A / {( - 1 ) _ {A} } } $.

Type  $  ( 2,4 ) $— 
The ideal sheaf  $  {\mathcal I} _ { {A / {\mathbf P  ^ {7}  } } } $
of the image of the embedding  $  \varphi _ {L} : A \hookrightarrow \mathbf P  ^ {7} $
is generated by  $  6 $
quadrics (see [[#References|[a3]]]).

Type  $  ( 2,6 ) $— 
Suppose  $  L $
is very ample and let  $  K _ {A} = {A / {( - 1 ) _ {A} } } $
be the associated Kummer surface. The subvector space  $  H  ^ {0} ( A,L )  ^ {-} \subset  H  ^ {0} ( A,L ) $
of odd sections induces an embedding of  $  {\widetilde{K}  } _ {A} $, 
the blow-up of  $  K _ {A} $
in the  $  16 $
double points, as a smooth quartic surface into  $  \mathbf P  ^ {3} $. 
$  {\widetilde{K}  } _ {A} \subset  \mathbf P  ^ {3} $
is invariant under the action of the level- $  2 $
Heisenberg group (cf. also [[Heisenberg representation|Heisenberg representation]])  $  H ( 2,2 ) $. 
The  $  16 $
blown-up double points become skew lines on the quartic surface. Any  $  H ( 2,2 ) $-
invariant quartic surface in  $  \mathbf P  ^ {3} $
with  $  16 $
skew lines comes from a polarized Abelian surface  $  ( A, \lambda ) $
of type  $  ( 2,6 ) $
in this way (see [[#References|[a5]]], [[#References|[a11]]] and [[#References|[a12]]]).

Type  $  ( 3,3 ) $— 
$  \lambda $
is three times a principal polarization and  $  \varphi _ {L} : A \hookrightarrow \mathbf P  ^ {8} $
is an embedding. If  $  ( A, \lambda ) $
is not a product, then the quadrics  $  Q \in H  ^ {0} ( \mathbf P  ^ {8} , {\mathcal I} _ { {A / {\mathbf P  ^ {8}  } } } ( 2 ) ) $
vanishing on  $  A $
generate the ideal sheaf  $  {\mathcal I} _ { {A / {\mathbf P  ^ {8}  } } } $. 
In the product case,  $  {\mathcal I} _ { {A / {\mathbf P  ^ {8}  } } } $
is generated by quadrics and cubics (see [[#References|[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|Hamiltonian system]] on  $  \mathbf C  ^ {N} $(
with Casimir functions  $  {H _ {1} \dots H _ {k} } : {\mathbf C  ^ {N} } \rightarrow \mathbf C $
and  $  m = { {( N - k ) } / 2 } $
independent constants of motion  $  H _ {k + 1 }  \dots H _ {k + m }  $
in involution) such that:

a) for a general point  $  c = {}  ^ {t} ( c _ {1} \dots c _ {k + m }  ) \in \mathbf C ^ {k + m } $
the invariant manifold  $  A _ {c}  ^ {o} = \cap _ {i = 1 }  ^ {m + k } \{ H _ {i} = c _ {i} \} \subset  \mathbf C  ^ {N} $
is an open affine part of an Abelian variety  $  A _ {c} $;

b) the flows of the integrable vector fields  $  X _ {u _ {i}  } $
linearize on the Abelian varieties  $  A _ {c} $[[#References|[a2]]].

The divisor at infinity  $  D _ {c} = A _ {c} - A _ {c}  ^ {o} $
defines a polarization on  $  A _ {c} $. 
In this way the mapping  $  {( H _ {1} \dots H _ {k + m }  ) } : {\mathbf C  ^ {N} } \rightarrow {\mathbf C ^ {k + m } } $
defines a family of polarized Abelian varieties (cf. [[Moduli problem|Moduli problem]]). Some examples of  $  2 $-
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|Master equations in cooperative and social phenomena]]) linearize on principally polarized Abelian surfaces;

the [[Kowalewski top|Kowalewski top]], the Hénon–Heiles system and the Manakov geodesic flow on  $  { \mathop{\rm SO} } ( 4 ) $
linearize on Abelian surfaces of type  $  ( 1,2 ) $[[#References|[a1]]];

the Garnier system linearizes on Abelian surfaces of type  $  ( 1,4 ) $[[#References|[a15]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Adler, P. van Moerbeke, "The Kowalewski and Hénon–Heiles motions as Manakov geodesic flows on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040262.png" />: a two-dimensional family of Lax pairs" ''Comm. Math. Phys.'' , '''113''' (1988) pp. 659–700</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Adler, P. van Moerbeke, "The complex geometry of the Kowalewski–Painlevé analysis" ''Invent. Math.'' , '''97''' (1989) pp. 3–51 {{MR|}} {{ZBL|0678.58020}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Barth, "Abelian surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040263.png" />-polarization" , ''Algebraic Geometry, Sendai, 1985'' , ''Advanced Studies in Pure Math.'' , '''10''' (1987) pp. 41–84 {{MR|946234}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> W. Barth, "Quadratic equations for level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040264.png" /> abelian surfaces" , ''Abelian Varieties, Proc. Workshop Egloffstein 1993'' , de Gruyter (1995) pp. 1–18 {{MR|1336597}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Barth, I. Nieto, "Abelian surfaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040265.png" /> and quartic surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040266.png" /> skew lines" ''J. Algebraic Geom.'' , '''3''' (1994) pp. 173–222 {{MR|1257320}} {{ZBL|0809.14027}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Ch. Birkenhake, H. Lange, D. van Straten, "Abelian surfaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040267.png" />" ''Math. Ann.'' , '''285''' (1989) pp. 625–646 {{MR|1027763}} {{ZBL|0714.14028}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> 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 {{MR|1351509}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> G. Horrocks, D. Mumford, "A rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040268.png" /> vector bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040269.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040270.png" /> symmetries" ''Topology'' , '''12''' (1973) pp. 63–81 {{MR|382279}} {{ZBL|0255.14017}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> K. Hulek, H. Lange, "Examples of abelian surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040271.png" />" ''J. Reine Angew. Math.'' , '''363''' (1985) pp. 200–216 {{MR|0814021}} {{ZBL|0593.14027}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> H. Lange, Ch. Birkenhake, "Complex Abelian varieties" , ''Grundlehren math. Wiss.'' , '''302''' , Springer (1992) {{MR|1217487}} {{ZBL|0779.14012}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> I. Naruki, "On smooth quartic embeddings of Kummer surfaces" ''Proc. Japan Acad.'' , '''67 A''' (1991) pp. 223–224 {{MR|1137912}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> V. V. Nikulin, "On Kummer surfaces" ''Math USSR Izv.'' , '''9''' (1975) pp. 261–275 (In Russian) {{MR|429917}} {{ZBL|0325.14015}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> S. Ramanan, "Ample divisors on abelian surfaces" ''Proc. London Math. Soc.'' , '''51''' (1985) pp. 231–245 {{MR|0794112}} {{ZBL|0603.14013}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> I. Reider, "Vector bundles of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040272.png" /> and linear systems on algebraic surfaces" ''Ann. of Math.'' , '''127''' (1988) pp. 309–316 {{MR|0932299}} {{ZBL|0663.14010}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P. Vanhaecke, "A special case of the Garnier system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110040/a110040273.png" />-polarized Abelian surfaces and their moduli" ''Compositio Math.'' , '''92''' (1994) pp. 157–203 {{MR|1283227}} {{ZBL|}} </TD></TR></table>