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The text of this book has its origins more than twenty- ve years
ago. In the seminar of the Dutch Singularity Theory project in 1982
and 1983, the second-named author gave a series of lectures on
Mixed Hodge Structures and Singularities, accompanied by a set of
hand-written notes. The publication of these notes was prevented by
a revolution in the subject due to Morihiko Saito: the introduction
of the theory of Mixed Hodge Modules around 1985. Understanding
this theory was at the same time of great importance and very hard,
due to the fact that it uni es many di erent theories which are
quite complicated themselves: algebraic D-modules and perverse
sheaves. The present book intends to provide a comprehensive text
about Mixed Hodge Theory with a view towards Mixed Hodge Modules.
The approach to Hodge theory for singular spaces is due to Navarro
and his collaborators, whose results provide stronger vanishing
results than Deligne s original theory. Navarro and Guill en also
lled a gap in the proof that the weight ltration on the nearby
cohomology is the right one. In that sense the present book
corrects and completes the second-named author s thesis."
This is the first comprehensive basic monograph on mixed Hodge
structures. Starting with a summary of classic Hodge theory from a
modern vantage point the book goes on to explain Deligne's mixed
Hodge theory. Here proofs are given using cubical schemes rather
than simplicial schemes. Next come Hain's and Morgan's results on
mixed Hodge structures related to homotopy theory. Steenbrink's
approach of the limit mixed Hodge structure is then explained using
the language of nearby and vanishing cycle functors bridging the
passage to Saito's theory of mixed Hodge modules which is the
subject of the last chapter. Since here D-modules are essential,
these are briefly introduced in a previous chapter. At various
stages applications are given, ranging from the Hodge conjecture to
singularities. The book ends with three large appendices, each one
in itself a resourceful summary of tools and results not easily
found in one place in the existing literature (homological algebra,
algebraic and differential topology, stratified spaces and
singularities). The book is intended for advanced graduate
students, researchers in complex algebraic geometry as well as
interested researchers in nearby fields (algebraic geometry,
mathematical physics
The theory of motives was created by Grothendieck in the 1960s as
he searched for a universal cohomology theory for algebraic
varieties. The theory of pure motives is well established as far as
the construction is concerned. Pure motives are expected to have a
number of additional properties predicted by Grothendieck's
standard conjectures, but these conjectures remain wide open. The
theory for mixed motives is still incomplete. This book deals
primarily with the theory of pure motives. The exposition begins
with the fundamentals: Grothendieck's construction of the category
of pure motives and examples. Next, the standard conjectures and
the famous theorem of Jannsen on the category of the numerical
motives are discussed. Following this, the important theory of
finite dimensionality is covered. The concept of Chow-Künneth
decomposition is introduced, with discussion of the known results
and the related conjectures, in particular the conjectures of
Bloch-Beilinson type. We finish with a chapter on relative motives
and a chapter giving a short introduction to Voevodsky's theory of
mixed motives.
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