Generalizations of de Franchis theorem

Imayoshi, Yôichi

doi:10.1215/s0012-7094-83-05017-2

Cited by 16 publications

(9 citation statements)

References 15 publications

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“…Therefore the problem is reduced to the case where N is a smooth algebraic curve of hyperbolic type. Then Theorem 6 of [10] implies our assertion. Q.E.D.…”

Section: Proof Let ~: R Xn~f\d Be As In (45) It Is First Noted Thatsupporting

confidence: 68%

Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces

Noguchi

1988

Invent Math

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IntroductionLet X be a connected Zariski open subset of a compact reduced complex space 3? such that X is complete hyperbolic and hyperbolically imbedded into X (cf. 1-12]). Let N be a Zariski open subset of a compact complex manifold b7 such that ON = ~7-N is empty or a hypersurface with only normal crossings. The purpose of this paper is to study the structure of the moduli space Hol(N, X) of all holomorphic mappings f: N~X of N into X. Especially interesting is the case where X is the quotient F\D of a symmetric bounded domain D by a torsion-free arithmetic subgroup F of the identity component Aut~ of the holomorphic transformation group Aut(D) of D. It is known that F\D is complete hyperbolic and hyperbolically imbedded into the Satake compactification F\D of F\D (cf. 1-17, 2, 14, 15]). We can deal with the case where F has a torsion if we are concerned only with holomorphic mappings which have liftings from the universal covering of N into D, compatible with homomorphisms p eHom(n 1 (N), F) (see Remarks (2.19) and (4.11)). Besides the interesting results of [30,31,28], the present work is motivated by Parshin-Arakelov's theorems for curves 1-26, 1] and for Abelian varieties I-5] (cf. also [23,11]). For example, let n: Y--*N be a fiber space over b7 which is smooth over N, such that the fibers Yx = n-1 (x) with x eN are g-dimensional Abelian varieties with principal polarization. Then the fiber space naturally induces a holomorphic mapping f: N-.Sp(g,Z)\S v where S~ denotes Siegel's generalized upper half space. Thus the deformation of n: Y--*N of those fiber spaces over N with degeneration at most over dN and the total space of those fiber spaces correspond respectively to the deformation of the holomorphic mapping f and the moduli space of holomorphic mappings from N into Sp(g, Z)\Sg (cf. Remark (4.11) for details). For the case of curves, see Remark (2.19). Thus it is quite natural to deal with the case where N and F\D are non-compact. Here we remark that the boundedness parts of [26,1,11] for curves and [5] for principally polarized Abelian varieties are covered by our arguments of this paper, but 16 J. Noguchi not their rigidity parts. We will however have several rigidity theorems from the viewpoint of the theory of holomorphic mappings. In the case where N is compact, there is an earlier work [20] on some fiber spaces of Abelian varieties.The natural topology of Hol(N, X) is the compact-open topology. In w 1 we first show the following extension and convergence theorem, where N may be non-compact and X is assumed only to be hyperbolically imbedded into a complex space )?: In N3 and 4 we deal with the case where X is the quotient F\D of a symmetric bounded domain D by a torsion-free discrete subgroup F of Aut (D). We assume that F is uniform or an arithmetic subgroup of Aut~In the case where F is uniform and N=N, the results of N 3 and 4 were already obtained in [30,31,28]. We are mainly interested in the case where F\D and N are non-compact, while our arguments work in the compact case. Applying the re...

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“…Therefore the problem is reduced to the case where N is a smooth algebraic curve of hyperbolic type. Then Theorem 6 of [10] implies our assertion. Q.E.D.…”

Section: Proof Let ~: R Xn~f\d Be As In (45) It Is First Noted Thatsupporting

confidence: 68%

Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces

Noguchi

1988

Invent Math

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“…Loosely speaking, we show that, in general, the manifold X 1 × · · · × X n splits into two factors, one of which is the product of those non-compact factors to which the Remmert-Stein method can be applied. The finiteness result that is essential to our proof is a result by Imayoshi [4]. This result, plus some other technical necessities are stated in Section 3.…”

Section: Introductionmentioning

confidence: 91%

Proper holomorphic mappings between hyperbolic product manifolds

Janardhanan

2014

Int. J. Math.

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“…De Franchis's theorem for Riemann surfaces of finite types and its generalization to the higher-dimensional case appears in [10,11].…”

Section: Introductionmentioning

confidence: 99%

“…The surface S 2 is defined by the equation u 2 = v(v 4 + av 2 + 1), while f (w, z) = (zw, z 2 ). By [20,21] for a = ± 10 3 we have Aut(S 2 ) = D 4 . For a = ± 10 3 we have | Aut(S 2 )| = 24.…”

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confidence: 99%

On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus

Mednykh

2012

Sib Math J

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We obtain an upper bound for the number of holomorphic mappings of a genus 3 Riemann surface onto a genus 2 Riemann surface in a series of cases. In particular, we establish that the number of holomorphic mappings of an arbitrary genus 3 Riemann surface onto an arbitrary genus 2 Riemann surface is at most 48. We show that this estimate is sharp and find pairs of Riemann surfaces for which it is attained.

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Generalizations of de Franchis theorem

Cited by 16 publications

References 15 publications

Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces

Moduli spaces of holomorphic mappings into hyperbolically imbedded complex spaces and locally symmetric spaces

Proper holomorphic mappings between hyperbolic product manifolds

On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus

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