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Providing a timely description of the present state of the art of
moduli spaces of curves and their geometry, this volume is written
in a way which will make it extremely useful both for young people
who want to approach this important field, and also for established
researchers, who will find references, problems, original
expositions, new viewpoints, etc. The book collects the lecture
notes of a number of leading algebraic geometers and in particular
specialists in the field of moduli spaces of curves and their
geometry. This is an important subject in algebraic geometry and
complex analysis which has seen spectacular developments in recent
decades, with important applications to other parts of mathematics
such as birational geometry and enumerative geometry, and to other
sciences, including physics. The themes treated are classical but
with a constant look to modern developments (see Cascini, Debarre,
Farkas, and Sernesi's contributions), and include very new
material, such as Bridgeland stability (see Macri's lecture notes)
and tropical geometry (see Chan's lecture notes).
In this book, Claire Voisin provides an introduction to
algebraic cycles on complex algebraic varieties, to the major
conjectures relating them to cohomology, and even more precisely to
Hodge structures on cohomology. The volume is intended for both
students and researchers, and not only presents a survey of the
geometric methods developed in the last thirty years to understand
the famous Bloch-Beilinson conjectures, but also examines recent
work by Voisin. The book focuses on two central objects: the
diagonal of a variety--and the partial Bloch-Srinivas type
decompositions it may have depending on the size of Chow groups--as
well as its small diagonal, which is the right object to consider
in order to understand the ring structure on Chow groups and
cohomology. An exploration of a sampling of recent works by Voisin
looks at the relation, conjectured in general by Bloch and
Beilinson, between the coniveau of general complete intersections
and their Chow groups and a very particular property satisfied by
the Chow ring of K3 surfaces and conjecturally by hyper-Kahler
manifolds. In particular, the book delves into arguments
originating in Nori's work that have been further developed by
others."
The main goal of the CIME Summer School on "Algebraic Cycles and
Hodge Theory" has been to gather the most active mathematicians in
this area to make the point on the present state of the art. Thus
the papers included in the proceedings are surveys and notes on the
most important topics of this area of research. They include
infinitesimal methods in Hodge theory; algebraic cycles and
algebraic aspects of cohomology and k-theory, transcendental
methods in the study of algebraic cycles.
In this book, Claire Voisin provides an introduction to
algebraic cycles on complex algebraic varieties, to the major
conjectures relating them to cohomology, and even more precisely to
Hodge structures on cohomology. The volume is intended for both
students and researchers, and not only presents a survey of the
geometric methods developed in the last thirty years to understand
the famous Bloch-Beilinson conjectures, but also examines recent
work by Voisin. The book focuses on two central objects: the
diagonal of a variety--and the partial Bloch-Srinivas type
decompositions it may have depending on the size of Chow groups--as
well as its small diagonal, which is the right object to consider
in order to understand the ring structure on Chow groups and
cohomology. An exploration of a sampling of recent works by Voisin
looks at the relation, conjectured in general by Bloch and
Beilinson, between the coniveau of general complete intersections
and their Chow groups and a very particular property satisfied by
the Chow ring of K3 surfaces and conjecturally by hyper-Kahler
manifolds. In particular, the book delves into arguments
originating in Nori's work that have been further developed by
others."
The 2003 second volume of this account of Kaehlerian geometry and
Hodge theory starts with the topology of families of algebraic
varieties. Proofs of the Lefschetz theorem on hyperplane sections,
the Picard-Lefschetz study of Lefschetz pencils, and Deligne
theorems on the degeneration of the Leray spectral sequence and the
global invariant cycles follow. The main results of the second part
are the generalized Noether-Lefschetz theorems, the generic
triviality of the Abel-Jacobi maps, and most importantly Nori's
connectivity theorem, which generalizes the above. The last part of
the book is devoted to the relationships between Hodge theory and
algebraic cycles. The book concludes with the example of cycles on
abelian varieties, where some results of Bloch and Beauville, for
example, are expounded. The text is complemented by exercises
giving useful results in complex algebraic geometry. It will be
welcomed by researchers in both algebraic and differential
geometry.
The first of two volumes offering a modern introduction to
Kaehlerian geometry and Hodge structure. The book starts with basic
material on complex variables, complex manifolds, holomorphic
vector bundles, sheaves and cohomology theory, the latter being
treated in a more theoretical way than is usual in geometry. The
author then proves the Kaehler identities, which leads to the hard
Lefschetz theorem and the Hodge index theorem. The book culminates
with the Hodge decomposition theorem. The meanings of these results
are investigated in several directions. Completely self-contained,
the book is ideal for students, while its content gives an account
of Hodge theory and complex algebraic geometry as has been
developed by P. Griffiths and his school, by P. Deligne, and by S.
Bloch. The text is complemented by exercises which provide useful
results in complex algebraic geometry.
The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. The main results are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly, Nori's connectivity theorem, which generalizes the above. The last part deals with the relationships between Hodge theory and algebraic cycles. The text is complemented by exercises offering useful results in complex algebraic geometry. Also available: Volume I 0-521-80260-1 Hardback $60.00 C
This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions.
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