The complex structure of domains covering algebraic surfaces

Shabat, G. B.

doi:10.1007/bf01081892

Cited by 18 publications

(8 citation statements)

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“…In [14] and [15] Shabat studied the automorphism groups of universal covers of families of Riemann surfaces and proved a deep result which in the case of Kodaira fibrations amounts to the following theorem.…”

Section: Uniformization Of Kodaira Surfacesmentioning

confidence: 99%

The arithmeticity of a Kodaira fibration is determined by its universal cover

González-Diez¹,

Reyes-Carocca²

2015

Comment. Math. Helv.

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Section: Uniformization Of Kodaira Surfacesmentioning

confidence: 99%

The arithmeticity of a Kodaira fibration is determined by its universal cover

González-Diez¹,

Reyes-Carocca²

2015

Comment. Math. Helv.

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“…(This is essentially due to Poincaré; see [18,Chapter 5] for details.) While this sounds very general, there are few such examples known aside from bounded symmetric domains [22]. The results of Frankel [10], Vey [28] and Wong [29] show that under various additional restrictions, such a U is necessarily a bounded symmetric domain (see [15] for a survey of closely related results).…”

Section: Open Problemsmentioning

confidence: 99%

Algebraic varieties with semialgebraic universal cover

Kollár

Pardon

2012

Journal of Topology

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“…(c) (see e.g. [Sh1,2]). A smooth projective surface S is called a Kodaira surface if there is a smooth fibration π : S → B over a curve B, where both B and a generic fibre F of π are of genus ≥ 2 (usually π is supposed being a non-trivial deformation of F , but we don't need this assumption here).…”

Section: Examplesmentioning

confidence: 99%

Plane curves with a big fundamental group of the complement

Dethloff¹,

Orevkov²,

Zaidenberg³

1998

Voronezh Winter Mathematical Schools

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Let C ⊂ IP 2 be an irreducible plane curve whose dual C * ⊂ IP 2 * is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincaré group G = π 1 (IP 2 \ C) contains a free group with two generators. If the geometric genus g of C is at least 2, then a subgroup of G can be mapped epimorphically onto the fundamental group of the normalization of C, and the result follows. To handle the cases g = 0, 1, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Plücker curves. Such a curve C can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski-Lefschetz type arguments we deduce the result from 'the bigness' of the braid group B d, g , that is, of the group of d-string braids of a compact genus g Riemann surface. * Partually supported by Russian Foundation for Fundamental Research, N96-01-01218 paper) that the fundamental group of the complement of the dual of a nodal plane curve is big (with two evident exceptions). More precisely, we have 0.2. Theorem. Let C ⊂ IP 2 be an irreducible immersed curve 1 which is neither a line nor a conic nor a nodal cubic. Let C * ⊂ IP 2 * be the dual curve. Then the group π 1 (IP 2 * \ C * ) is big.Obviously, the statement does not hold for a line, nor for a conic. If C is a nodal cubic then C * is a three-cuspidal quartic and π 1 (IP 2 * \ C * ) is the metacyclic group of order 12 [Zar,.Let d be the degree and g be the geometric genus of C. For g ≥ 2 the proof of Theorem 0.2 is easy. Indeed, denote by ν :It is a smooth surface. Let µ 1 : R → C norm and µ 2 : R → IP 2 * be the canonical projections. Since C is immersed, it is easily seen that µ 2 is ramified exactly over C * . Denote by R 0 the part of R over IP 2 * \ C * . Then µ 1 : R 0 → C norm is a holomorphic surjection with connected fibres. It follows that (µ 1 ) * : π 1 (R 0 ) → π 1 (C norm ) is an epimorphism, and hence the group π 1 (R 0 ) is big as soon as g(C norm ) ≥ 2 (see e.g. 1.2(a) below). Since µ 2 : R 0 → IP 2 * \C * is a finite unramified covering, we have that (µ 2 ) * (π 1 (R 0 )) is a finite index subgroup of π 1 (IP 2 * \ C * ). Therefore, the group π 1 (IP 2 * \ C * ) is also big.Thus, the only non-trivial cases are g = 0 and g = 1. However, the proofs of most of the intermediate results needed for these two cases are valid for any g, some of them under the additional assumption that d ≥ 2g − 1 (which is automatically fulfilled for g = 0, 1). Therefore, we formulate everything for an arbitrary genus. This provides another proof of Theorem 0.2 for the case g ≥ 2, d ≥ 2g − 1.The paper is organized as follows. In Section 1 we provide some (mostly well known) examples of big groups. Besides, by several examples we illustrate a conjectural relation between bigness of the fundamental group and C-hyperbolicity. These include, in particular, the quasi-projective quotients of bounded symmetric domains and the complements of certain r...

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The complex structure of domains covering algebraic surfaces

Cited by 18 publications

References 5 publications

The arithmeticity of a Kodaira fibration is determined by its universal cover

The arithmeticity of a Kodaira fibration is determined by its universal cover

Algebraic varieties with semialgebraic universal cover

Plane curves with a big fundamental group of the complement

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