Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds
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Bibliographic record
Abstract
L MEERSSEMAN A. -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. R. -Considrons le problme d'uniformisation suivant. Prenons deux familles de dformation holomorphes (paramtres par un ensemble analytique dfini dans un voisinage de 0 dans C p pour p > 0) ou diffrentiables (paramtres par un voisinage de 0 dans R p pour p > 0) de varits compactes complexes. Supposons-les ponctuellement isomorphes, c'est--dire que, pour tout point t de l'espace des paramtres, la fibre en t de la premire famille est biholomorphe la fibre en t de la deuxime famille. Sous quelle(s) condition(s) les deux familles sont-elles localement isomorphes en 0? Dans cet article, nous donnons une condition suffisante dans le cas de familles holomorphes. Nous montrons ensuite que, de faon surprenante, la condition n'est pas suffisante dans le cas des familles diffrentiables. Nous dcrivons galement plusieurs types de contre-exemples et donnons quelques lments de classifications de ces contre-exemples. Ces rsultats reposent sur une tude gomtrique de l'espace de Kuranishi d'une varit compacte complexe.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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