# 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

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:

For a polarized Abelian surface of type , the assignment defines a rational mapping from into the projective space of hyperplanes in :

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

(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

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 : a two-dimensional family of Lax pairs" Comm. Math. Phys. , 113 (1988) pp. 659–700 |

[a2] | M. Adler, P. van Moerbeke, "The complex geometry of the Kowalewski–Painlevé analysis" Invent. Math. , 97 (1989) pp. 3–51 Zbl 0678.58020 |

[a3] | W. Barth, "Abelian surfaces with -polarization" , Algebraic Geometry, Sendai, 1985 , Advanced Studies in Pure Math. , 10 (1987) pp. 41–84 MR946234 |

[a4] | W. Barth, "Quadratic equations for level- abelian surfaces" , Abelian Varieties, Proc. Workshop Egloffstein 1993 , de Gruyter (1995) pp. 1–18 MR1336597 |

[a5] | W. Barth, I. Nieto, "Abelian surfaces of type and quartic surfaces with skew lines" J. Algebraic Geom. , 3 (1994) pp. 173–222 MR1257320 Zbl 0809.14027 |

[a6] | Ch. Birkenhake, H. Lange, D. van Straten, "Abelian surfaces of type " Math. Ann. , 285 (1989) pp. 625–646 MR1027763 Zbl 0714.14028 |

[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 MR1351509 |

[a8] | G. Horrocks, D. Mumford, "A rank vector bundle on with symmetries" Topology , 12 (1973) pp. 63–81 MR382279 Zbl 0255.14017 |

[a9] | K. Hulek, H. Lange, "Examples of abelian surfaces in " J. Reine Angew. Math. , 363 (1985) pp. 200–216 MR0814021 Zbl 0593.14027 |

[a10] | H. Lange, Ch. Birkenhake, "Complex Abelian varieties" , Grundlehren math. Wiss. , 302 , Springer (1992) MR1217487 Zbl 0779.14012 |

[a11] | I. Naruki, "On smooth quartic embeddings of Kummer surfaces" Proc. Japan Acad. , 67 A (1991) pp. 223–224 MR1137912 |

[a12] | V. V. Nikulin, "On Kummer surfaces" Math USSR Izv. , 9 (1975) pp. 261–275 (In Russian) MR429917 Zbl 0325.14015 |

[a13] | S. Ramanan, "Ample divisors on abelian surfaces" Proc. London Math. Soc. , 51 (1985) pp. 231–245 MR0794112 Zbl 0603.14013 |

[a14] | I. Reider, "Vector bundles of rank and linear systems on algebraic surfaces" Ann. of Math. , 127 (1988) pp. 309–316 MR0932299 Zbl 0663.14010 |

[a15] | P. Vanhaecke, "A special case of the Garnier system, -polarized Abelian surfaces and their moduli" Compositio Math. , 92 (1994) pp. 157–203 MR1283227 |

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Abelian surface.

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