Abstract
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 . . . 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 Mœglin, 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.
November 27, 1995
Jos´ e Bertin
⋆ , Jean-Pierre Demailly
⋆ , Luc Illusie
⋆⋆ , Chris Peters
⋆
⋆ Universit´ e de Grenoble I, Institut Fourier,
BP74, 38402 Saint-Martin d’H` eres, France
⋆⋆ Universit´ e de Paris-Sud, D´ epartement de Math´ ematiques,
Bˆ atiment 425, 91405 Orsay, France
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 . . . 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 Mœglin, 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.
November 27, 1995
Jos´ e Bertin
⋆ , Jean-Pierre Demailly
⋆ , Luc Illusie
⋆⋆ , Chris Peters
⋆
⋆ Universit´ e de Grenoble I, Institut Fourier,
BP74, 38402 Saint-Martin d’H` eres, France
⋆⋆ Universit´ e de Paris-Sud, D´ epartement de Math´ ematiques,
Bˆ atiment 425, 91405 Orsay, France