Un phénomène de Hartogs dans les variétés projectives

Dingoyan, Pascal

doi:10.1007/s00209-008-0381-y

Cited by 4 publications

(6 citation statements)

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“…By (6.7), Ñ À ¼ ´Å¼ Ä µ Ò , ½. Since Å ¼ is Andreottipseudoconcave, a theorem of Dingoyan [18,19] shows that there exist a branched covering Å Å with a section Ë on Å, a divisor À Å and an integer such that holomorphic sections of £ Ä over Ë´Å ¼ µ extends to a holomorphic section of £ Ä over Å Ò À or of Ä ª À℄ over Å. Thus, the restriction morphisms À ¼ ´ Å Ò À Ä µ À ¼ ´Ë´Å ¼ µ Ä µ and À ¼ ´ Å Ä ª À℄µ À ¼ ´Ë´Å ¼ µ Ä µ are isomorphisms.…”

Section: Examplesmentioning

confidence: 94%

“…2008-December 2010. We would like to thank P. Dingoyan for discussions about the extension of holomorphic sections [18]. We also thank the referee for many detailed remarks that have helped to improve the presentation.…”

Section: Acknowledgements the First-named Author Is Grateful To Bo Be...mentioning

confidence: 97%

“…If one knows that has a neighbourhood Å ¼ , which is a ´Ò ¾µ-convex-concave strip such that any meromorphic function on Å ¼ is rational, then [18,19] shows that there is no need to pass to a covering. Corollary 6.5.…”

Section: Examplesmentioning

confidence: 99%

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Szegö kernel asymptotics and Morse inequalities on CR manifolds

Hsiao

Marinescu

2011

Math. Z.

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ABSTRACT. Let be an abstract compact orientable CR manifold of dimension ¾Ò ½, Ò ¾, and let Ä be the -th tensor power of a CR complex line bundle Ä over . We assume that condition ´Õµ holds at each point of . In this paper we obtain a scaling upper-bound for the Szegö kernel on´¼ Õµ-forms with values in Ä , for large . After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convexconcave manifolds.

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Section: Examplesmentioning

confidence: 94%

Section: Acknowledgements the First-named Author Is Grateful To Bo Be...mentioning

confidence: 97%

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Szegö kernel asymptotics and Morse inequalities on CR manifolds

Hsiao

Marinescu

2011

Math. Z.

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“…Let us assume that β is a locally residual current of bidegree (q + p − 1, p − 1) in a linearly p−concave domain, union of p−planes. Then the Abel-Radon transform with respect to (p − 1)−planes gives a meromorphic q−form in a domain of the grassmannian G(p − 1, N ), which contains a IP p (the (p − 1)−planes contained in a IP p ), and so is concave in the sense of Andreotti; thus, q−form extends by [2] in a rational form; and by the preceding, β also extends on IP N as an algebraic current.…”

Section: Applicationsmentioning

confidence: 99%

On the Abel-Radon transform of locally residual currents

Fabre¹

2010

Preprint

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First we recall the definition of locally residual currents and their basic properties. We prove in this first section a trace theorem, that we use later. Then we define the Abel-Radon transform of a current R(α), on a projective variety X ⊂ IP N , for a family of p−cycles of incidence variety I ⊂ T × X, for which p1 : I → T is proper and p2 : I → X is submersive, and a domain U ⊂ T . Then we show the following theorem, for a family of sections of X with r−planes (which was proved for the family of lines of X = IP N by the author in [4] for p = 1, in [5] for R(α) = 0 and p−planes for any q > 0, and by Henkin and Passare in [7] for p−planes in IP N and integration currents α = ω ∧ [Y ], with a meromorphic q−form ω, and projective convexity on Ũ ):Let α be a locally residual current of bidegree (q + p, p) on U * , with U * := ∪t∈U Ht ⊂ X, where t×HtLet us assume that α is ∂−closed, and q > 0. If R(α) extends meromorphically (resp. holomorphically) to a greater domain Ũ , then α extends in a unique way as a locally residual current (resp. ∂−closed) to the greater domain Ũ * ⊂ X.In particular we recover the result of [5] without using [8]. We formulate another generalization, for complete intersections with respect to a fixed multidegree.

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“…Alors, P. Dingoyan a montré dans[5] le théorème suivant : Lemme 4.5. Soit U un domaine pseudoconcave au sens d'Andreotti d'une variété algébrique complexe X .…”

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Sur la transformation d'Abel-Radon des courants localement résiduels

Fabre¹

2009

Annali Scuola Normale Superiore - Classe Di Scienze

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After recalling the definitions of the Abel-Radon transformation of currents and of locally residual currents, we show that the Abel-Radon transform R(α) of a locally residual current α remains locally residual. Then a theorem of P. Griffiths, G. Henkin and M. Passare (cf. [7], [9] and [10]) can be formulated as follows : Let U be a domain of the Grassmannian variety G( p, N ) of complex p-planes in P N , U * :=∪ t∈U H t be the corresponding linearly p-concave domain of P N , and α be a locally residual current of bidegree (N , p). Suppose that the meromorphic n-form R(α) extends meromorphically to a greater domainŨ of G( p, N ). If α is of type ω∧[T ], with T an analytic subvariety of pure codimension p in U * , and ω a meromorphic (resp. regular) q-form (q > 0) on T , then α extends in a unique way as a locally residual current to the domainŨ * := ∪ t∈Ũ H t . In particular, if R(α) = 0, then α extends as a ∂-closed residual current on P N . We show in this note that this theorem remains valid for an arbitrary residual current of bidegree (N , p), in the particular case where p = 1.

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Un phénomène de Hartogs dans les variétés projectives

Cited by 4 publications

References 4 publications

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Szegö kernel asymptotics and Morse inequalities on CR manifolds

On the Abel-Radon transform of locally residual currents

Sur la transformation d'Abel-Radon des courants localement résiduels

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