The book aims in part to understand the geometric basis and the limitations of Spencer Bloch's beautiful formula for the tangent space to Chow groups. Bloch's formula is motivated by algebraic K-theory and involves differentials over Q. The theory developed here is characterized by the appearance of arithmetic considerations even in the local infinitesimal theory of algebraic cycles. The map from the tangent space to the Hilbert scheme to the tangent space to algebraic cycles passes through a variant of an interesting construction in commutative algebra due to Angéniol and Lejeune-Jalabert. The link between the theory given here and Bloch's formula arises from an interpretation of the Cousin flasque resolution of differentials over Q as the tangent sequence to the Gersten resolution in algebraic K-theory. The case of 0-cycles on a surface is used for illustrative purposes to avoid undue technical complications.
The first volume appeared 1985 as vol. 267 of the same series.
and representation theory, two of the most active and important areas
in contemporary mathematics. The underlying theme is the use of complex
geometry to understand the two subjects and their relationships to one
another--an approach that is complementary to what is in the
literature. Finite-dimensional representation theory and complex
geometry enter via the concept of Hodge representations and Hodge
domains. Infinite-dimensional representation theory, specifically the
discrete series and their limits, enters through the realization of
these representations through complex geometry as pioneered by Schmid,
and in the subsequent description of automorphic cohomology. For the
latter topic, of particular importance is the recent work of Carayol
that potentially introduces a new perspective in arithmetic
automorphic representation theory.
The present work gives a
treatment of Carayol's work, and some extensions of it, set in a
general complex geometric framework. Additional subjects include a
description of the relationship between limiting mixed Hodge structures
and the boundary orbit structure of Hodge domains, a general treatment
of the correspondence spaces that are used to construct Penrose
transforms and selected other topics from the recent literature.
A co-publication of the AMS and CBMS.
New to the Second Edition:
*Fully-revised appendices including an expanded discussion of the Hirsch lemma
*Presentation of a natural proof of a Serre spectral sequence result
*Updated content throughout the book, reflecting advances in the area of homotopy theory
With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.