% Session SMF l'Etat de la Recherche, Grenoble, 25-26-27 Novembre 1994
% Texte pour Panoramas et Synth\`eses
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% J. Bertin, J.P. Demailly, Ch. Peters, L. Illusie
% Abstract

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Each of the three chapters collected in this book is concerned with 
various aspects -- important ones in several respects -- 
of Hodge theory. The text is an expanded version, including
substantial additions, of lectures presented on the occasion of the
meeting ``l'Etat de la Recherche'' devoted to Hodge theory, that has been
held at Universit\'e Joseph Fourier in Grenoble from Friday November 25, 
1994 till Sunday November 27, under the auspices of the SMF (Soci\'et\'e
Math\'ematique de France). The authors wishes would be fulfilled if, 
in accordance with the general goals of sessions ``l'Etat de la
Recherche'', this book could help the nonexpert reader to get a
precise idea of the current status of Hodge theory.

The three main subjects developed here ($L^2$ Hodge theory and
vanishing theorems, Frobenius and Hodge degeneration, Variations of
Hodge structures and mirror symmetry) cover a wide range of
techniques: elliptic PDE theory, complex differential geometry,
algebraic geometry in characteristic $p$, cohomological and
sheaf-theoretic methods, deformation theory of complex varieties,
Calabi-Yau manifolds, a few aspects of singularity theory~$\ldots$~ 
This accumulation of tools arising from various fields probably makes 
the access to the theory rather uneasy for newcomers.  We hope that 
the present book will greatly facilitate this access: a special effort 
has been made to approach various themes by their most natural
starting point, each of the three chapters being supplemented with a
detailed introduction and numerous references. The reader will find
precise statements of quite a number of open problems which have been
the subject of active research in the last years.

The authors are grateful to SMF and MESR (Minist\`ere de l'Enseignement 
Sup\'erieur et de la Recherche) for their decisive action -- both
psychological and financial -- without which the Grenoble session
``Hodge theory'' would probably never have taken place. They address
special thanks to the Scientific Committee of Sessions l'Etat de la 
Recherche, in behalf of its two successive directors Pierre Schapira 
and Colette M{\oe}glin, as well as to Mich\`ele Audin, Editor in Chief 
of the Journal ``Panoramas et Synth\`eses'', for her strong encouragement
to publish the present manuscript. Finally, they express their gratitude
to the referee for his careful reading of the manuscript and a large
number of invaluable suggestions.
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\noindent
November 27, 1995

\noindent
Jos\'e Bertin${}^{\star}$, Jean-Pierre Demailly${}^{\star}$,
Luc Illusie${}^{\star\star}$, Chris Peters${}^{\star}$

\noindent
$\star\phantom{\star}$ Universit\'e de Grenoble I, Institut
Fourier,\hfill\break
\phantom{$\star\star$} BP74, 38402 Saint-Martin d'H\`eres, France

\noindent
$\star\star$ Universit\'e de Paris-Sud, D\'epartement de
Math\'ematiques,\hfill\break
\phantom{$\star\star$} B\^atiment 425, 91405 Orsay, France

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