Non-complete algebraic surfaces with logarithmic Kodaira dimension - ∞ and with non-connected boundaries at infinity

Miyanishi, Masayoshi; Tsunoda, Shuichiro

doi:10.4099/math1924.10.195

Cited by 46 publications

(37 citation statements)

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“…Let h : Σ → Σ be the minimal resolution of the singularity O. Then the exceptional locus h −1 (O) is an admissible rational fork F = B 0 + T 1 + T 2 + T 3 (see [6] for the definitions), where B 0 is the central component and T 1 , T 2 , T 3 are three maximal rational twigs meeting B 0 . The A 1 * -fibration π : X → B extends to an A 1 -fibration π : Σ → B for which B 0 is a cross-section.…”

Section: A 1 * -Fibrations Given By G M -Actionsmentioning

confidence: 99%

Étale endomorphisms of algebraic surfaces with $G_m$-actions

Masuda

Miyanishi

2001

Math Ann

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Let X be an algebraic surface defined over the complex field C and endowed with an A 1 * -fibration. As such a surface we have a Platonic A 1 * -fiber space, a weighted hypersurface with its singular point deleted off and, more generally, an affine algebraic surface with an unmixed G m -action and its fixpoint deleted off. We consider anétale endomorphism ϕ : X → X and show that ϕ is an automorphism in most cases. Of particular interest is the case of a Platonic A 1 * -fiber space, for which ϕ being an automorphism is closely related to the Jacobian Problem for the affine plane A 2 . We also investigate the automorphism group of such surfaces.

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Section: A 1 * -Fibrations Given By G M -Actionsmentioning

confidence: 99%

Étale endomorphisms of algebraic surfaces with $G_m$-actions

Masuda

Miyanishi

2001

Math Ann

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“…Dans ce paragraphe onénoncera quelques conséquences du théorème principal, du théorème 7.1, et de la classification des surfaces quasi-projectives de dimension logarithmique de Kodaira −∞ (voir [13], [14]) sur la structure des surfaces Kählériènnes complètes, non compactes, de courbure de Ricci positive vérifiant la condition "C", ( les notations sont celles du paragraphe précédent). Donnons d'abord les deux définitions suivantes Définition 1.…”

Section: Conséquences Du Théorème Principalunclassified

Sur la dimension logarithmique de Kodaira des variétés Kählériennes complètes de courbure de Ricci positive

Anchouche

1998

Math Z

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IntroductionComme première approcheà l'étude des variétés Kählériennes complètes, non compactes de courbure de Ricci positive (analogue non compact des variétés de Fano), Mok aétabli le résultat suivant Theorème 1 (Mok [16]): Soit (X , g) une variété Kählérienne complète, non compacte, dim C X = n ≥ 2, de courbure de Ricci positive. Supposons que pour un point base x 0 de X , il existe deux constantes positives c 1 , c 2 telles que, pour tout (., .) désigne la distance géodésique, B (x 0 , r) la boule géodésique de centre x 0 et de rayon r, Vol g (B (x 0 , r)) désigne son volume, et Ric g la (1, 1) forme de Ricci associéeà la métrique g. Alors X est biholomorpheà une variété quasi-projectivē X \D, oùX est une variété projective lisse et D un diviseurà croisement normaux.Le cas dim C X = 1 aété traité par Blanc-Fiala [4];à savoir une surface de Riemann complète non compacte de courbure Gaussienne positive est biholomorpheà C. Il y a une erreure dans l'enoncé de ce théorème dans [16]; la formulation exacte se trouveà la page 381,Th 2.3 du même article. La compactification ci-dessus est obtenue en utilisant des sections holomorphes plurianticanoniques de classe L β , pour un certain β > 0 fixé, dont l'existence est une conséquence de la positivité de la courbure de Ricci et des estimations L 2 de Hormander. Plus précisément, il existe un entier q et un nombre fini de sections de classe L β , S 0 , ..., S N de (K −1 X ) ⊗q , où K −1 X est le fibré anticanonique de X telles que

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“…The above Theorem (2.3) can be reduced to the following Theorem (1.1) by taking a partial resolution of S. The crucial results used in the proof of Theorem (1.1) are those in [8,9] on affine-ruledness of open surfaces and in [11] or [12] on fundamental groups of some open rational surfaces. Here, if there are nodes in f ∆ + D * , we replace f by the composition of the minimal resolution of S and the blowing-up of nodes.…”

Section: Theorem (23) Letmentioning

confidence: 99%

“…Now we apply the log minimal model programme in [5] for the pair (X o , B o ). Since K Xo + B o is not nef, by the cone theorem [5], there are an extremal ray and a corresponding contraction σ 1 : [4] or [6], we see easily, in the notation of [8]…”

Section: Lemma (13) Let (S ∆) Be As In Theorem (11) With the Abomentioning

confidence: 99%

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Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts

Zhang

1996

Trans. Amer. Math. Soc.

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Abstract. Let (S, ∆) be a log surface with at worst log canonical singularities and reduced boundary ∆ such that −(K S + ∆) is nef and big. We shall prove that S o = S − SingS −∆ either has finite fundamental group or is affine-ruled. Moreover, π 1 (S o ) and the structure of S are determined in some sense when ∆ = 0. IntroductionLet S be a normal projective algebraic surface over the complex number field C with nef and big anti-canonical divisor −K S . Recently, in [11] and [15] (see also [12,13]) we have proved that the fundamental group of the smooth part S o = S − SingS is finite provided that S has at worst log-terminal singularities (see [5] or [6] for its definition).In this short note, we shall consider the case when S has at worst log canonical singularities (see (4) of Remark (2.4) below for background and see [4] or [6] for the complete list of log-canonical singularities). Then π 1 (S o ) is no longer finite. However, we shall describe precisely, in the case when π 1 (S o ) is infinite, this fundamental group and the structure of S.The following Lemma (2.1) is rather easy. Lemma (2.1). Let S be a normal projective algebraic surface with at worst log canonical singularities and with nef and big anti-canonical divisor −K S . Suppose that S has at least one non-rational singularity. Then we have:(1) S is obtained from a nonsingular elliptic ruled surface S by contracting a cross-section of the ruling to a simple elliptic singularity and several P 1 's in fibers into rational singularities.(2) We have the following surjective homomorphism, where S o = S − SingS is the smooth part, and hence the infiniteness ofThe following is one of our main theorems. It says that when π 1 (S o ) is infinite S has a finite Galois covering unramified over S o and with the covering surface described in Lemma (2.1). In Theorems (2.3) and (1.1) below, we refer to [5] or [6]

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Non-complete algebraic surfaces with logarithmic Kodaira dimension - ∞ and with non-connected boundaries at infinity

Cited by 46 publications

References 17 publications

Étale endomorphisms of algebraic surfaces with $G_m$-actions

Étale endomorphisms of algebraic surfaces with $G_m$-actions

Sur la dimension logarithmique de Kodaira des variétés Kählériennes complètes de courbure de Ricci positive

Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts

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