Holomorphic families of Riemann surfaces and Teichmüller spaces, II

Imayoshi, Yôichi

doi:10.2748/tmj/1178229731

Cited by 11 publications

(6 citation statements)

References 15 publications

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“…Since X c is a parabolic modular transformation, there is a hyperbolic Mobius transformation g in G such that (trace Θ(X n e (z)\ g) 2 -+ 4 as n-> ±°° for any ^e T(G), where Θ{φ, as z-+z A from the inside of c. For 7 = {7(ί) eD*; 0 ^ t < 1, 7(0) = 0, Inn,-! 7(t) = 0}, 7 = 7Γ -1 (7) = {τ(ί); 0 ^ ί < 1} is an open arc terminating at z A from the inside of c. Hence 4 = lim,^ (trace θ(Ψ(j(s)); g)f = lim^i (trace θ(Ψ r *(Ύ(s) m , w 0 ); g)f = lim,^ (trace θ(f(s); g)) 2 and we have the first assertion of Theorem 2.…”

Section: Pu (Z A(z)) ^ T S {ψ(Z) ψ(A(z))) = T S {ψ{Z) X C (ψ(Z))) ^mentioning

confidence: 99%

“…When 1 < dim T(S) < +<*>, the author does not know whether the cluster set of f(s) as s -> 1 consists of only regular 6-groups. Imayoshi [7] showed that if an arc 7 is contained in an angular domain {zeD*; θj. < arg z < θ 2 ^ θ 1 + 2π}, then /(s) converges to a regular δ-group as s -> 1.…”

Section: Pu (Z A(z)) ^ T S {ψ(Z) ψ(A(z))) = T S {ψ{Z) X C (ψ(Z))) ^mentioning

confidence: 99%

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Remarks on holomorphic families of Riemann surfaces

Shiga¹

1986

Tohoku Math. J. (2)

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Introduction.Let W be a domain in C or a Riemann surface and let S be a Riemann surface of type (g, n) with Sg -3 + n > 0, where g is the genus of S and n is the number of punctures of S. In this paper, we shall consider holomorphic families of S over W (see Section 1, Definition 1) or a locally holomorphic mapping of W to the Teichmϋller space of S, and study their boundary behavior.In Section 1, we shall state known results and set up our notations.In Section 2, we shall investigate a holomorphic family of S over the punctured disk 0 < \z\ < 1 and the behavior as z->0. Imayoshi [7] obtained a similar result. We shall show a uniqueness theorem of holomorphic families of S (Theorem 2).In Section 3, we shall discuss holomorphic families over a general domain or a Riemann surface, and consider the problems as in Section 2.In Section 4, we construct two examples of holomorphic families which might be of interest.Thanks are due to Professors Y. Imayoshi and M. Taniguchi for useful and stimulating conversation. The author expresses his thanks also to the referee for valuable suggestions.

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Section: Pu (Z A(z)) ^ T S {ψ(Z) ψ(A(z))) = T S {ψ{Z) X C (ψ(Z))) ^mentioning

confidence: 99%

Section: Pu (Z A(z)) ^ T S {ψ(Z) ψ(A(z))) = T S {ψ{Z) X C (ψ(Z))) ^mentioning

confidence: 99%

Remarks on holomorphic families of Riemann surfaces

Shiga¹

1986

Tohoku Math. J. (2)

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“…The genus of a fibered surface is defined as the genus of its general fiber. Although f is an unstable in general, since B is complex one-dimensional, the induced map from the complement of the critical locus B \ Σ f to M g extends to a holomorphic map from B to M g by [18] or by the valuative criterion. This extended map is again written as µ f : B → M g and is called the induced map.…”

Section: A Signature Divisor On M Gmentioning

confidence: 99%

Local signature of fibered complex surfaces via moduli and monodromy

Ashikaga¹

2010

Demonstratio Mathematica

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Abstract. The signature of fibered complex surfaces is sometimes localized at finite fiber germs. We analyze this phenomenon by using datum of local monodromy in the mapping class group and a certain rational divisor on the moduli space of stable curves. IntroductionLet f : S → B be a proper surjective holomorphic map from a compact complex surface S to a Riemann surface B such that the general fiber of f is a Riemann surface of genus g. Let Sign S be the signature of the intersection form on H 2 (S, Q). If there exist a finite number of fiber germs (f, F i ),and a local invariant σ(f, F i ) is geometrically well-defined such thatthen we say Sign S is localized and call σ(f, F i ) a local signature. Our problem here is to discuss how to formulate and also how to calculate the local signature. It should be formulated by depending on the nature of the general fiber of f (see §2.2).One of the motivation of this problem comes from the study of the topological structure of Lefschetz fibration, which also relates to symplectic geometry. The other motivation comes from the algebro-geometric study of complex surfaces, especially geography of surfaces of general type.The author already published two survey papers around these fields with Konno (1) In the case where f is an elliptic fibration, Matsumoto [24] formulated and calculated it by using Meyer function. Atiyah [7] essentially described it by the method of "signature defect" analogous to the Hirzebruch's signature defect of isolated singularities in some sense (see Remark 1.3.6).(2) Assume f is a genus 2 fibration. Horikawa [16]

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“…In order to prove this theorem, we need the following two theorems (cf. Imayoshi [5], Theorem 4 and Theorem 5): …”

Section: Holomorphci Sections (M π R)mentioning

confidence: 99%