On Locally Trivial Deformations

Flenner, Hubert; Kosarew, Siegmund

doi:10.2977/prims/1195176251

Cited by 31 publications

(21 citation statements)

References 8 publications

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“…Recall from [FK87] that the universal locally trivial deformation of X exists and that it is just the restriction of the universal deformation to the locally trivial locus Def lt (X) ⊂ Def(X) in the Kuranishi space, which is a closed subspace.…”

Section: Deformationsmentioning

confidence: 99%

Deformations of singular symplectic varieties and termination of the log minimal model program

Lehn¹,

Pacienza²

2016

Alg. Geom.

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We generalize Huybrechts' theorem on deformation equivalence of birational irreducible symplectic manifolds to the singular setting. More precisely, under suitable natural hypotheses, we show that two birational symplectic varieties are locally trivial deformations of each other. As an application we show the termination of any log minimal model program for a pair (X, ∆) of a projective irreducible symplectic manifold X and an effective R-divisor ∆. To prove this result we follow Shokurov's strategy and show that LSC and ACC for minimal log discrepancies hold for all the models appearing along any log MMP of the initial pair.

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Section: Deformationsmentioning

confidence: 99%

Deformations of singular symplectic varieties and termination of the log minimal model program

Lehn¹,

Pacienza²

2016

Alg. Geom.

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“…of Y is equivalent to that of families of locally stable holomorphic maps of compact complex manifolds, whose dimensions are smaller than that of Y, into Y. Then the theorem can be proved with'the use of the following three results: the result of Flenner concerning the existence of a semi-universal family for deformations of holomorphic maps between compact complex spaces [4], the result of Flenner and Kosarew concerning the existence bf the maximal family for locally trivial deformations of complex space germs [5], and the result of Douady concerning the existence of global moduli of compact closed complex subspaces of a complex manifold [2]. The Coo triviality of the family II: 3(Y) --* E(Y) follows from Theorem 1.8 in Sec.…”

Section: Furthermore the Underlying C ~176 Structure Of The Family Imentioning

confidence: 99%

Locally stable holomorphic maps and their application to a global moduli problem for some kinds of analytic subvarieties

Tsuboi¹

1996

J Math Sci

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“…In this case, we prove that the map p is proper. The existence of semi-universal locally trivial deformations is due to Flenner and Kosarew [4]. Hence, we establish the moduli space of hyperbolic compact complex spaces.…”

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confidence: 99%

The moduli space of hyperbolic compact complex spaces

Khalfallah

2006

Math. Z.

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In this paper, we construct the moduli space of reduced hyperbolic compact complex spaces. First, we prove an infinitesimal characterization of hyperbolicity using a family of Kobayashi-Royden pseudo-metrics introduced by Venturini and as a consequence we conclude that the property of Landau holds for complex spaces. Finally, we establish this moduli space in the case of locally trivial deformations, and in a more general situation, the case of equisingular deformations. IntroductionBrody [1] and Wright [26] constructed the moduli space of compact hyperbolic manifolds. In [11], the author generalized this result by establishing the moduli space of hyperbolically imbedded manifolds, which can be seen as a non compact version of the moduli space of Brody-Wright. To construct this moduli space, we use a general criterion to represent analytic functors by coarse moduli spaces. This criterion is due to Suchmacher [21] and Fujiki [6]. See also Kosarew and Okonek [17] for further generalizations.We notice that Fujiki [6] and Schumacher [20] used this criterion to construct the moduli space of polarized kähler manifolds.The paper is divided in three parts. In the first part, we prove that the property of Landau holds also for complex spaces and not only for manifolds [9] or for complex spaces with isolated singularities [23]. This can be achieved by a characterization of the hyperbolicity expressed with the help of the family {K k X }, the Kobayashi infinitesimal k-pseudo-metrics introduced by Venturini [25]. Royden [19] proved a similar characterization in the case of complex manifolds. A. Khalfallah (B) Institut Préparatoire aux études d'ingénieur, Rue Ibn El Jazzar,

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On Locally Trivial Deformations

Cited by 31 publications

References 8 publications

Deformations of singular symplectic varieties and termination of the log minimal model program

Deformations of singular symplectic varieties and termination of the log minimal model program

Locally stable holomorphic maps and their application to a global moduli problem for some kinds of analytic subvarieties

The moduli space of hyperbolic compact complex spaces

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