Tata Institute of Fundamental Research 2010; 141 pp; softcover ISBN10: 8184870124 ISBN13: 9788184870121 List Price: US$40 Member Price: US$32 Order Code: TIFR/15
 These notes are based on a series of lectures given at the Tata Institute of Fundamental Research, Mumbai, in 2007, on the theme of Hodge theoretic motives associated to various geometric objects. Starting with the topological setting, the notes go on to Hodge theory and mixed Hodge theory on the cohomology of varieties. Degenerations, limiting mixed Hodge structures and the relation to singularities are addressed next. The original proof of Bittner's theorem on the Grothendieck group of varieties, with some applications, is presented as an appendix to one of the chapters. The situation of relative varieties is addressed next using the machinery of mixed Hodge modules. Chern classes for singular varieties are explained in the motivic setting using Bittner's approach, and their full functorial meaning is made apparent using mixed Hodge modules. An appendix explains the treatment of Hodge characteristic in relation with motivic integration and string theory. Throughout these notes, emphasis is placed on explaining concepts and giving examples. A publication of the Tata Institute of Fundamental Research. Distributed worldwide except in India, Bangladesh, Bhutan, Maldavis, Nepal, Pakistan, and Sri Lanka. Readership Graduate students and research mathematicians interested in Hodge theory. Table of Contents  Motives and topology
 The Hodge characteristic makes its appearance
 The Hodge characteristic: examples
 Hodge theory revisited
 Mixed Hodge theory
 Motivic Hodge theory
 Motivic aspects of degenerations
 Motivic nearby fibre: examples
 Motivic aspects of degenerations: Applications
 Motives in the relative setting: Topological aspects
 Variations of Hodge structure
 Hodge modules
 Motives in the relative setting: Mixed Hodge modules
 The motivic Chern class transformation
 Bibliography
 Index
