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].

References

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