Complex Geometry An Introduction - Daniel Huybrechts.pdf

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Daniel Huybrechts
Complex Geometry
An Introduction
4u Springer
Daniel Huybrechts
Universite Paris VII Denis Diderot
Institut de Mathematiques
2, place Jussieu
75251 Paris Cedex 05
France
e-mail:
huybrech@math.jussieu.fr
Mathematics Subject Classification (2000): 14J32,14J60,14J81,32Q15,32Q20,32Q25
Cover figure is taken from page 120.
Library of Congress Control Number: 2004108312
ISBN 3-540-21290-6 Springer Berlin Heidelberg New York
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Preface
Complex geometry is a highly attractive branch of modern mathematics that
has witnessed many years of active and successful research and that has re-
cently obtained new impetus from physicists' interest in questions related to
mirror symmetry. Due to its interactions with various other fields (differential,
algebraic, and arithmetic geometry, but also string theory and conformal field
theory), it has become an area with many facets. Also, there are a number of
challenging open problems which contribute to the subject's attraction. The
most famous among them is the Hodge conjecture, one of the seven one-million
dollar millennium problems of the Clay Mathematics Institute. So, it seems
likely that this area will fascinate new generations for many years to come.
Complex geometry, as presented in this book, studies the geometry of
(mostly compact) complex manifolds. A complex manifold is a differentiable
manifold endowed with the additional datum of a complex structure which is
much more rigid than the geometrical structures in differential geometry. Due
to this rigidity, one is often able to describe the geometry of complex manifolds
in very explicit terms. E.g. the important class of projective manifolds can, in
principle, be described as zero sets of polynomials.
Yet, a complete classification of all compact complex manifolds is too
much to be hoped for. Complex curves can be classified in some sense (in-
volving moduli spaces etc.), but already the classification of complex surfaces
is tremendously complicated and partly incomplete.
In this book we will concentrate on more restrictive types of complex
manifolds for which a rather complete theory is in store and which are also
relevant in the applications. A prominent example are Calabi-Yau manifolds,
which play a central role in questions related to mirror symmetry. Often,
interesting complex manifolds are distinguished by the presence of special
Riemannian metrics. This will be one of the central themes throughout this
text. The idea is to study cases where the Riemannian and complex geometry
on a differentiable manifold are not totally unrelated. This inevitably leads to

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