Contractible affine surfaces with quotient singularities

Abstract

We consider contractible affine surfaces of negative Kodaira dimension with only quotient singularities. We prove that the smooth locus of such a surface has negative Kodaira dimension. It follows that if such a surface has only one singular point, then it is isomorphic to a quotient C 2 /G, where G is a finite group acting linearly on C 2 . The main resultIn [KR] we studied surfaces of the form C 3 / /C * and showed that they are isomorphic to C 2 /G, G a finite cyclic group acting linearly on C 2 . (If H is a reductive algebraic group acting algebraically on the affine variety X = Spec A, we denote by X/ /H the categorical quotient Y = Spec A H with A H the ring of invariants in A. If H is finite, we can identify X/ /H with X/H, the space of H-orbits.) Our results in [KR] came close to a characterization of C 2 /G via Kodaira dimension in case G is cyclic. Here we take up this problem for general finite subgroups of GL 2 (C). Note that we can assume that G is "small", i.e., does not contain any pseudoreflections. We prove the following main result.1.1. Theorem. Let S be a normal, topologically contractible affine surface over C of negative Kodaira dimension and smooth except for quotient singular points. Then:(i) the smooth locus S 0 of S has negative Kodaira dimension; and (ii) if S has at most one singular point, then S C 2 /G where G is a finite group acting linearly on C 2 .A complete description of S in the case of several singular points is given in Section 3. We note that by the Kodaira dimension of a surface X we mean, by definition, the Kodaira dimension of a desingularization X of X (see Paragraph 2.4.2 for more details).If S is smooth, then G = {1} and the fact that S is isomorphic to C 2 is a fundamental result of Miyanishi and Sugie [MS1] and Fujita [Fu1]. Other characterizations of C 2 /G, not involving Kodaira dimension, have been given. Miyanishi [M2] has shown that an 294 MARIUSZ KORAS AND PETER RUSSELL affine surface S dominated by C 2 via a finite morphism is isomorphic to C 2 /G, and Gurjar and Shastri [GS] proved the same if S is contractible and its fundamental group at infinity is finite. (In case S is smooth, this is a celebrated result of Ramanujam [R].) It appears, however, that these conditions are very difficult to verify in applications, even in situations where our condition on the Kodaira dimension is easily checked. As an example we discuss some consequences of our result for surfaces of the form C n / /H in Section 8 (see Theorem 8.1).The proof of Theorem 1.1 occupies Sections 2 to 7. Let κ be the Kodaira dimension of S 0 . We rule out κ = 0, 1 in Section 4, and κ = 2, as usual the most difficult case, in Sections 5, 6, and 7. In basic outline, our arguments follow those of [K] for κ = 0, 1 and of [KR] for κ = 2, and we do not hesitate to use results from those papers where they are directly applicable to our present more general situation. In case κ < 0, the proof of Theorem 1.1(ii), given in Section 3, is a quite direct consequence of the important result of Miyanishi and Tsu...

Two-dimensional quotients of Cn are isomorphic to C2/Γ

“…This proof made essential use of Fujita's result from [4] where Fujita has classified NC-minimal compactifications of a smooth affine surface S with κ(S) = 0. The proof of the result in [11] mentioned in the Introduction also uses Fujita's classification. Since we have to use the result in [11] to show that in case V has at most one singular point, then V ∼ = C 2 /Γ, we will not give the proof in [6].…”

“…The proof of the result in [11] mentioned in the Introduction also uses Fujita's classification. Since we have to use the result in [11] to show that in case V has at most one singular point, then V ∼ = C 2 /Γ, we will not give the proof in [6]. In [9] a proof is given when V has at most one (cyclic quotient) singular point.…”

“…The result of Koras-Russell is used in an important way. It is possible to avoid the use of part (2) of their result by using arguments from [6] but we will not do so since the arguments in [6] and [11] both use Fujita's work [4] in an essential way.…”

Affine normal surfaces with simply-connected smooth locus

“…Theorem 1 [12] V is isomorphic to C 2 /G if and only if V is topologically contractible, κ(V ) = −∞ and V has at most one quotient singularity.…”

Exceptional singular \mathbb{Q}-homology planes

“…Singular Q-homology planes appear for example as quotients of smooth ones by the actions of finite groups or as twodimensional quotients of C n by the actions of reductive groups (cf. [13], [5]). Let S be a Q-homology plane and let S 0 be its smooth locus (S = S 0 if S is smooth).…”