Compact complex manifolds containing “global” spherical shells

Kato, Masahide

doi:10.3792/pja/1195517920

Cited by 68 publications

(51 citation statements)

References 2 publications

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“…In his remarkable article [Nak89] Nakamura, inspired by the previous work of Kato ([Kat77], [Kat78], [Kat83]), and Dloussky [Dlo88], stated a courageous conjecture, which would in principle solve the classification problem for class VII surfaces, as we explain below: THE GSS CONJECTURE. Any minimal class VII surface with b 2 > 0 contains a global spherical shell.…”

Section: Introductionmentioning

confidence: 99%

“…Minimal class VII surfaces which allow GSS's (which are usually called GSS surfaces, or Kato surfaces) are well understood; in particular it is known that any such surface is a degeneration of a holomorphic family of blown up primary Hopf surfaces, in particular it is diffeomorphic to such a blown up Hopf surface. Moreover, Kato showed [Kat77] that any GSS surface can be obtained by a very simple construction. D b belonging to the last exceptional curve of m, and then identifies holomorphically the two ends of the resulting manifold (which are both spherical shells).…”

Section: Introductionmentioning

confidence: 99%

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Instantons and curves on class VII surfaces

Teleman¹

2010

Ann. Math.

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We develop a general strategy, based on gauge theoretical methods, to prove existence of curves on class VII surfaces. We prove that, for b 2 D 2, every minimal class VII surface has a cycle of rational curves hence, by a result of Nakamura, is a global deformation of a one parameter family of blown up primary Hopf surfaces. The case b 2 D 1 was solved in a previous article. The fundamental object intervening in our strategy is the moduli space ᏹ pst .0; / of polystable bundles Ᏹ with c 2 .Ᏹ/ D 0, det.Ᏹ/ D . For large b 2 the geometry of this moduli space becomes very complicated. The case b 2 D 2 treated here in detail requires new ideas and difficult techniques of both complex geometric and gauge theoretical nature. We explain the substantial obstacles which must be overcome in order to extend our methods to the case b 2 3. Contents

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Instantons and curves on class VII surfaces

Teleman¹

2010

Ann. Math.

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“…-First, we can suppose that f is of the form (6). Moreover we can suppose that (ρ, h)| u=y=0 has lower triangular linear part, and that µ satisfies (9).…”

Section: Main Theoremmentioning

confidence: 99%

Contracting rigid germs in higher dimensions

Ruggiero¹

2013

Ann. inst. Fourier

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30 pages, 0 figuresInternational audienceFollowing Favre, we define a holomorphic germ f:(C^d,0) -> (C^d,0) to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of f and its linear action on the fundamental group of the complement of the critical set

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“…Kato in [9]. An important analytic object associated to this construction method was introduced and studied by Dloussky in [3].…”

Section: Kato Surfacesmentioning

confidence: 99%

On the Kähler Rank of Compact Complex Surfaces

Toma¹

2008

Bul. Soc. Math. France

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Abstract. -Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension (1, 1) for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions (C 2 , 0) → (C 2 , 0). Résumé (Sur le rang de Kähler des surfaces complexes compactes)Harvey et Lawson ont introduit et calculé le rang de Kähler en relation avec le cône des courants positifs fermés de bidimension (1, 1) pour beaucoup de classes de surfaces complexes compactes. Dans ce travail nous étendons ces calculs à la seule classe de surfaces connues et qui n'avait pas été considérée par eux, celle des surfaces de Kato. Notre outil principal est la réduction à la dynamique des contractions holomorphes (C 2 , 0) → (C 2 , 0) associées.Texte reçu le 23 avril

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Compact complex manifolds containing “global” spherical shells

Cited by 68 publications

References 2 publications

Instantons and curves on class VII surfaces

Instantons and curves on class VII surfaces

Contracting rigid germs in higher dimensions

On the Kähler Rank of Compact Complex Surfaces

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