Motivic Aspects of Hodge Theory is based on a series of lectures given at the Tata Institute of Fundamental Research, Mumbai, 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 the Hodge characteristic in relation with motivic integration and string theory. Throughout these notes, emphasis is placed on explaining concepts and giving examples.

Introductory Remarks / Motives and Topology / The Hodge Characteristic makes it Appearance / Appendix A: A Proof of Bittner’s Theorem and some Applications / 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 / Appendix B: Motivic Integration / Bibliography / Index.

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