Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds

Laurent Meersseman

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 3, page 495-525
  • ISSN: 0012-9593

Abstract

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Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of 0 in ℂ p , for some p > 0 ) or differentiable (parametrized by an open neighborhood of 0 in ℝ p , for some p > 0 ) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point t of the parameter space, the fiber over t of the first family is biholomorphic to the fiber over t of the second family. Then, under which conditions are the two families locally isomorphic at 0? In this article, we give a sufficient condition in the case of holomorphic families. We show then that, surprisingly, this condition is not sufficient in the case of differentiable families. We also describe different types of counterexamples and give some elements of classification of the counterexamples. These results rely on a geometric study of the Kuranishi space of a compact complex manifold.

How to cite

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Meersseman, Laurent. "Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds." Annales scientifiques de l'École Normale Supérieure 44.3 (2011): 495-525. <http://eudml.org/doc/272230>.

@article{Meersseman2011,
abstract = {Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\mathbb \{C\}^p$, for some $p&gt;0$) or differentiable (parametrized by an open neighborhood of $0$ in $\mathbb \{R\}^p$, for some $p&gt;0$) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point $t$ of the parameter space, the fiber over $t$ of the first family is biholomorphic to the fiber over $t$ of the second family. Then, under which conditions are the two families locally isomorphic at 0? In this article, we give a sufficient condition in the case of holomorphic families. We show then that, surprisingly, this condition is not sufficient in the case of differentiable families. We also describe different types of counterexamples and give some elements of classification of the counterexamples. These results rely on a geometric study of the Kuranishi space of a compact complex manifold.},
author = {Meersseman, Laurent},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {deformations of complex manifolds; foliations; uniformization},
language = {eng},
number = {3},
pages = {495-525},
publisher = {Société mathématique de France},
title = {Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds},
url = {http://eudml.org/doc/272230},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Meersseman, Laurent
TI - Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 3
SP - 495
EP - 525
AB - Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\mathbb {C}^p$, for some $p&gt;0$) or differentiable (parametrized by an open neighborhood of $0$ in $\mathbb {R}^p$, for some $p&gt;0$) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point $t$ of the parameter space, the fiber over $t$ of the first family is biholomorphic to the fiber over $t$ of the second family. Then, under which conditions are the two families locally isomorphic at 0? In this article, we give a sufficient condition in the case of holomorphic families. We show then that, surprisingly, this condition is not sufficient in the case of differentiable families. We also describe different types of counterexamples and give some elements of classification of the counterexamples. These results rely on a geometric study of the Kuranishi space of a compact complex manifold.
LA - eng
KW - deformations of complex manifolds; foliations; uniformization
UR - http://eudml.org/doc/272230
ER -

References

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  10. [10] M. Kuranishi, A note on families of complex structures, in Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, 1969, 309–313. Zbl0211.10301MR254883
  11. [11] D. Mumford, Further pathologies in algebraic geometry, Amer. J. Math.84 (1962), 642–648. Zbl0114.13106MR148670
  12. [12] M. Namba, On deformations of automorphism groups of compact complex manifolds, Tôhoku Math. J.26 (1974), 237–283. Zbl0288.32019MR377115
  13. [13] J. J. Wavrik, Obstructions to the existence of a space of moduli, in Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, 1969, 403–414. Zbl0191.38003MR254882
  14. [14] J. J. Wavrik, Deforming cohomology classes, Trans. Amer. Math. Soc.181 (1973), 341–350. Zbl0238.32011MR326002
  15. [15] J. Wehler, Isomorphie von Familien kompakter komplexer Mannigfaltigkeiten, Math. Ann. 231 (1977/78), 77–90. Zbl0363.32016MR499327

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