On deformations of holomorphic maps

Horikawa, Eiji

doi:10.3792/pja/1195519751

Cited by 32 publications

(49 citation statements)

References 7 publications

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“…Therefore by Horikawa's theorem of completeness [Ho2] Theorem 5.2 there is a neighborhood S of s 0 ∈ T and a holomorphic mapping µ : S → N × H such that the restriction of (20) on S is isomorphic to the pull-back of (19) by µ. This proves the proposition.…”

Section: Is a Dolbeault Representative Of ρ( ∂ ∂τmentioning

confidence: 53%

“…We may replace T by h −1 (V ) and f ′ : X ′ → T by the corresponding restriction. We then obtain a deformation into the family q : Y → N as defined in [Ho2] § 5:…”

Section: Is a Dolbeault Representative Of ρ( ∂ ∂τmentioning

confidence: 99%

“…An easy calculation which uses [Ho2] Lemma 5.1 shows that the characteristic map τ : (19) is an isomorphism. Therefore by Horikawa's theorem of completeness [Ho2] Theorem 5.2 there is a neighborhood S of s 0 ∈ T and a holomorphic mapping µ : S → N × H such that the restriction of (20) on S is isomorphic to the pull-back of (19) by µ.…”

Section: Is a Dolbeault Representative Of ρ( ∂ ∂τmentioning

confidence: 99%

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Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

Kanev

2004

Ann. Mat. Pura Appl. IV. Ser.

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We prove that the moduli spaces A 3 (D) of polarized abelian threefolds with polarizations of types D = (1, 1, 2), (1, 2, 2), (1, 1, 3) or (1, 3, 3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space H 3,A (Y ) which parameterizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A −1 .

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Section: Is a Dolbeault Representative Of ρ( ∂ ∂τmentioning

confidence: 53%

“…We may replace T by h −1 (V ) and f ′ : X ′ → T by the corresponding restriction. We then obtain a deformation into the family q : Y → N as defined in [Ho2] § 5:…”

Section: Is a Dolbeault Representative Of ρ( ∂ ∂τmentioning

confidence: 99%

Section: Is a Dolbeault Representative Of ρ( ∂ ∂τmentioning

confidence: 99%

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Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

Kanev

2004

Ann. Mat. Pura Appl. IV. Ser.

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“…[9] and [10]) for the particular case we need to prove our results (i.e. for the case when the domain X of the map f is a curve); see also [2] With these preparations, we now prove the following results.…”

Section: Dimension Of Wd L (X) For General Branched Covering Of Irratmentioning

confidence: 83%

On multiple coverings of irrational curves

Ballico

Keem

1995

Arch. Math

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) 0. Introduction. In Section 1, we investigate the problem of the existence of base point free pencils of relatively low degree on a double covering X of genus 9 of a general curve C of genus q > 0. Such problem is classical and the picture is rather well known for the degree range close to the genus 9. On the other hand, by a simple application of the Castelnuovo-Severi inequality one can easily see that there does not exist a base point free pencil of degree less than or equal to g -2 q other than pull-backs from the base curve C; while for the degree beyond this range not many things have been known about the existence of such a pencil which is not composed with the given involution. Here we prove the following result. In the subsequent sections of this paper we consider the Brill-Noether theory for sufficiently general coverings of curves of genus q > 0. In Section 2 we use [12] to check the full Brill-Noether-Gieseker-Petri package for a general genus 9 covering of degree k of a general curve of genus q under very strong restrictions on 9, namely if kq<=g<=kq + (k-1).In Section 3 we consider the Brill-Noether theory for g~'s. Among the results we stress Theorem 3.1, Corollary 3.3, Theorem 3.4, and Corollary 3.5.In Section 4 we consider the Brill-Noether-Gieseker-Petri problem for 92 for a general curve Y in a "large" (i.e. of dimension at least g + 1) subvariety of ~0, when the g~ is base point free and induces a birational morphism from Y to IP 2. We prove the following result.

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“…By a theorem of Horikawa [10], it suffices to prove that # 2 (®Zo(~~'So)) = 0. But this follows from Lemma 1.9 of Campana [2] and a result of Kreusslerfll], since ZQ is Moishezon.…”

Section: Proposition 24 Let Zq Be Any Lebrun Twistor Space (Which Imentioning

confidence: 99%

Degenerate double solids as twister spaces

Honda¹

2002

Communications in Analysis and Geometry

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We show that there exist twistor spaces over 3CP 2 , the connected sum of three complex projective planes, whose half-anticanonical system gives a double covering map onto CP 3 branched along a quartic surface with not only ordinary double points, but other isolated singularities. We also show that if the quartic surface is in the most degenerate form, the twistor space admits a non-trivial enaction. Correspondingly, 3CP2 admits a self-dual metric of positive scalar curvature with [/(I)-symmetry, other than LeBrun's famous metrics. This leads us to a classification of twistor spaces over 3CP whose automorphism group is non-trivial (or equivalantly, self-dual metrics on 3CP 2 whose isometric group is non-trivial).

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On deformations of holomorphic maps

Cited by 32 publications

References 7 publications

Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of Abelian threefolds

On multiple coverings of irrational curves

Degenerate double solids as twister spaces

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