# Structures différentielles en géométrie complexe et presque complexe

Abstract : We give a generalization, in the context of coherent analytic sheaves, of a classical result of Koszul-Malgrange concerning the integrability of connections of type $(0,1)$ over a $(\cal C)^(\infty)$ complex vector bundle over a complex manifold. We introduce the notion of $\bar(\partial)$-coherent sheaf, which is a $(\cal C)^(\infty)$ notion, and we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of $\bar(\partial)$-coherent sheaves. The main application of this caracterisation concerns a method which allows to find analytic structures which are obtained by smooth deformations of other ones. In a second part we conjecture, as in the complex case, that the plurisubharmonicity of a function $u$ over an almost complex manifold is equivalent to the positivity of the $(1,1)$-current $i\partial\bar(\partial)u$. We prove the necessity of the positivity of this current. We prove also the sufficiency of the positivity in the particular case of an upper semi-continuous function which is continuous in the complement of the locus in which it takes the value $-\infty$.
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Theses
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https://tel.archives-ouvertes.fr/tel-00007104
Contributor : Arlette Guttin-Lombard <>
Submitted on : Thursday, October 14, 2004 - 8:43:08 AM
Last modification on : Wednesday, November 4, 2020 - 2:05:30 PM
Long-term archiving on: : Thursday, September 13, 2012 - 12:15:17 PM

### Identifiers

• HAL Id : tel-00007104, version 1

### Citation

Nefton Pali. Structures différentielles en géométrie complexe et presque complexe. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2004. Français. ⟨tel-00007104⟩

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