K3-surface

2020 Mathematics Subject Classification: Primary: 14J28 [MSN][ZBL]

A smooth projective algebraic surface $ X $ for which the canonical class is trivial and for which the dimension, $ \mathop{\rm dim} H ^ {1} ( X, \Omega ^ {1} ) $, of the space of one-dimensional differential forms on $ X $ is 0. The values of the following invariants for a $ K3 $-surface are known: the geometric genus, $ p _ {g} = \mathop{\rm dim} H ^ {2} ( X, \Omega ^ {2} ) = 1 $, the Euler characteristic of the structure sheaf, $ \chi ( {\mathcal O} ) = 2 $, the étale or (over the field of complex numbers) the topological Betti numbers (cf. Betti number) $ b _ {0} = b _ {4} = 1 $, $ b _ {1} = b _ {3} = 0 $ and $ b _ {2} = 22 $, and the Euler–Poincaré characteristic, $ e( X)= 24 $. The Riemann–Roch formula for a one-dimensional invertible sheaf $ D $ on a $ K3 $-surface takes the form

$$ \mathop{\rm dim} H ^ {0} ( X, D) + \mathop{\rm dim} H ^ {0} ( X, D ^ {-1} ) = \frac{( D) ^ {2} }{2} + 2 + \mathop{\rm dim} H ^ {1} ( X, D), $$

where $ ( D) ^ {2} $ is the index of self-intersection for the class of divisors corresponding to $ D $ (see Riemann–Roch theorem). If $ D $ corresponds to an effective irreducible divisor, then $ H ^ {1} ( X, D) = 0 $.

The formula for the arithmetic genus of an irreducible curve $ C $ on $ X $ also has a simple form:

$$ p _ {a} ( C) = \frac{( C) ^ {2} }{2} + 1 . $$

As a result one obtains $ ( C) ^ {2} \geq - 2 $, with $ ( C) ^ {2} = - 2 $ only for smooth rational curves. It also follows that $ ( D) ^ {2} $ is an even number for any divisor $ D $. Let $ N( X) $ be the Néron–Severi group of the surface $ X $, i.e. the group of divisor classes on $ X $ with respect to algebraic equivalence. Then $ N( X) $ is a free Abelian group of rank $ \rho $, where $ 1 \leq \rho \leq 20 $ if the characteristic of the ground field $ k $ is 0, and $ 1 \leq \rho \leq 20 $ or $ \rho = 22 $ if $ \mathop{\rm char} k > 0 $. The intersection index defines an integer-valued bilinear form on $ N( X) $ for which the square of any element is even. Surfaces with $ \rho = 20 $ (for $ \mathop{\rm char} k = 0 $) are called singular, while those with $ \rho = 22 $ (for $ \mathop{\rm char} k > 0 $) are called supersingular.

Another numerical invariant for a surface $ X $ is the minimal possible index $ \pi $ of self-intersection of an effective very ample divisor on $ X $, i.e. the minimum possible degree of a polarization on $ X $. If $ \pi = 2n- 2 $, the surface $ X $ can be imbedded in $ n $-dimensional projective space and cannot be imbedded in a projective space of lower dimension.

An important tool in studying $ K3 $-surfaces is to represent them as families (pencils) of elliptic curves. A surface $ X $ is represented as a family of elliptic curves if one is given a regular mapping $ \tau : X \rightarrow P ^ {1} $ all fibres of which, apart from a finite number, are non-singular elliptic curves. A surface $ X $ can be represented in that form if and only if the group $ N( X) $ contains a non-zero element with self-intersection index 0, where all such representations correspond to the classes of effective divisors with self-intersection index 0. If a surface represented as a family of elliptic curves is a $ K3 $-surface, it does not have multiple fibres. The Jacobi elliptic family constructed from such a family is again a $ K3 $-surface.

An important class of $ K3 $-surfaces is provided by the Kummer surfaces. A Kummer surface is a non-singular model for the quotient of a two-dimensional Abelian variety $ A $ with respect to the automorphism subgroup generated by the sign-change mapping. In particular, the surface given by the equation $ x _ {0} ^ {4} + x _ {1} ^ {4} + x _ {2} ^ {4} + x _ {3} ^ {4} = 0 $ in $ P ^ {3} $ is a Kummer surface. Any smooth surface of order four in $ P ^ {3} $ is a $ K3 $-surface. A smooth surface obtained as the intersection of three hypersurfaces of order two (quadrics) in $ P ^ {5} $ and a double covering of $ P ^ {2} $ ramified in a smooth curve of order six are $K3$-surfaces.

All $ K3 $-surfaces over the field of complex numbers are diffeomorphic; their moduli variety is connected and has dimension 19. One examines the structure of this variety and of $ K3 $-surface automorphisms by means of the period mapping. The period mapping for a $ K3 $-surface over the field of complex numbers is bijective (a theorem of Torelli type) [2].

If one is given a one-dimensional family of $ K3 $-surfaces (over $ \mathbf C $) with one singular fibre, then the surface can be rearranged after covering the base, without alterations outside the singular fibre, so that that fibre either becomes non-singular or else is one of two types: a) the components of the singular fibre and the intersection curves are rational, and the dual polyhedron for the singular fibre has the topological type of a two-dimensional sphere; or b) the components of the singular fibre constitute a chain, and non-empty intersections occur only for adjacent surfaces, while the two extreme surfaces are rational surfaces, the middle ones are elliptic ruled surfaces and the intersection curves are elliptic. Types a) or b) arise when the monodromy of the family is non-trivial [2].

$ K3 $-surfaces over an algebraically closed field of positive characteristic allow of a lifting into characteristic zero, their crystalline cohomology spaces do not have torsion, and their ranks coincide with the corresponding Betti numbers. For supersingular surfaces, an analogue of the period mapping has been constructed and a theorem of Torelli type has been proved. In this case the period variety is irreducible, complete, has dimension 9, and is unirational. All possible intersection forms on $ N( X) $ have been described for supersingular surfaces, and there are nine of them for each value of the characteristic of the ground field [4].

References

[1]	I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001
[2]	V.S. Kulikov, "Degenerations of $K3$-surfaces and Enriques surfaces" Math. USSR Izv. , 11 (1977) pp. 957–989 Izv. Akad. Nauk SSSR Ser. Mat. , 41 : 5 (1977) pp. 1008–1042
[3]	A.N. Rudakov, I.R. Shafarevich, "On the degeneration of $K3$ surfaces over fields of finite characteristic" Math. USSR Izv. , 18 (1982) pp. 561–574 Izv. Akad. Nauk SSSR Ser. Mat. , 45 : 3 (1981) pp. 646–661 Zbl 0489.14015
[4]	A.N. Rudakov, I.R. Shafarevich, "Surfaces of type $K3$ over fields of finite characteristic" J. Soviet Math. , 22 : 4 (1983) pp. 1476–1533 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 18 (1981) Zbl 0518.14015
[5]	D. Husemoller, "Classification and embeddings of surfaces" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 329–420 MR0506292 Zbl 0326.14009

Comments

Proofs of some of the results mentioned about $ K 3 $-surfaces over fields of positive characteristic can be found in [a3] and [a5]. The notion of a $ K 3 $-surface applies also to (not necessarily algebraic) complex surfaces. Each such a surface has a structure of a Kählerian variety (cf. Kähler manifold) [a4]. There is an analogue of Torelli's theorem (cf. Torelli theorems) for complex $ K 3 $-surfaces [a1].

References

[a1]	D. Burns, M. Rappoport, "On the Torelli theorem for Kählerian $K3$ surfaces" Ann. Sci. Ec. Norm. Sup. , 8 (1975) pp. 235–274
[a2]	A., et al. (ed.) Beauville, "Géométrie des surfaces $K3$: modules et périodes. Sem. Palaiseau" Astérisque , 126 (1985)
[a3]	A. Ogus, "Periods of integrals in characteristic $p$" , Internat. Congress Mathematicians (Warszawa, 1982) , PWN & North-Holland (1983) pp. 753–762 MR0804731 Zbl 0585.14015
[a4]	Y. Siu, "Every $K3$ surface is Kähler" Invent. Math. , 73 (1983) pp. 139–150 Zbl 0557.32004
[a5]	J. Giraud (ed.) et al. (ed.) , Surfaces algébrique , Lect. notes in math. , 868 , Springer (1981) MR0638596 Zbl 0452.00012
[a6]	A. Beauville, "Surfaces algébriques complexes" Astérisque , 54 (1978) MR0485887 Zbl 0394.14014
[a7]	A. van de Ven, "Compact complex surfaces" , Springer (1984) Zbl 0718.14023