# Abelian surface

An 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).

In the classification of algebraic surfaces, Abelian surfaces are exactly the smooth complete surfaces $A$ with Kodaira dimension $\kappa = 0$, geometric genus $p _ {g} = h ^ {2} ( A, {\mathcal O} _ {A} ) =1$ and 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 $L$ 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) if and only if $\phi _ {L}$ is an 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 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).

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 $J ( H )$ 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) 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

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

(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 [a14] and [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). 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 [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 [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: $p: {\overline{A}\; } \twoheadrightarrow K \subset \mathbf P ^ {3}$( 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 [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 [a13] and [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 [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 $K _ {A}$ 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 [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) $H ( 2,2 )$. 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 [a5], [a11] and [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 [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 $\mathbf C ^ {N}$( 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 $A _ {c}$[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). 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) linearize on principally polarized Abelian surfaces;

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

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

How to Cite This Entry:
Abelian surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abelian_surface&oldid=53242
This article was adapted from an original article by Ch. Birkenhake (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article