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A bibliography on the subject of Hodge Theory is presented including the articles "Compactification of Arithmetic Quotients of Bounded Symmetric Domains," by L. Baily and A. Borel, "Corners and arithmetic groups," by A. Borel, and "Resolution of singularities of an algebraic variety over a field of characteristic zero," by H. Hironaka.

A subject index for the June 2009 issue of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented.

The article lists the symbols that appeared in the articles within the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui.

An appendix is presented of the positive directions of local monodromy.

Chapter Twelve of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It offers some examples and open problems related to Siegel upper half spaces, and describes some points at infinity of Siegel upper half spaces. It considers their behaviors in the map.

Chapter Eleven of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It summarizes Theorems to the case of polarized logarithmic Hodge structures with coefficients. It shows that the functo from the category of A-PLH on X to the category of PLH on X is fully faithful.

Chapter Ten of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It considers the case of the standard SL(2)-orbit of the upper half plane. It indicates that homeomorphism is something like a real analytic local coordinate of DSL(2).

Chapter Nine of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It considers the relationship between the theory and the theory of Cattani and Kaplan (CK1). It discusses the extended period maps, including the canonical map and the composite map.

Chapter Two of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It illustrates basic facts about logarithmic structures and introduces the introduce the notion of the polarized logarithmic Hodge structure. It observes the relationship of polarized logarithmic Hodge structures and nilpotent orbits and interprets the nilpotent orbit theorem of Schmid in the language of polarized logarithmic Hodge structures.

Chapter Six of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It reviews the theory of the associated SL(2)-orbit in one variable, and the theory of R-split mixed Hodge structures. It also outlines the associated SL (2)-orbit in several variables.

Chapter Five of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It discusses the main results of the book. It constructs the fundamental diagram which gives an illustration of various enlargements of D. It discusses the theory of Borel-Serre spaces which is closely related to the theory of Iwasawa decomposition.

Chapter Four of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It discusses the main results of the book. It outlines the Griffiths' period map and the period map associated with a variation of polarized Hodge structure over a boundary.

Chapter Eight of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It studies moduli of pairs (X, a polarized Hodge structure on X) comparing it with (X, a local system on Xlog). It discusses the relationship among categories of generalized analytic spaces.

Chapter Seven of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It recalls the notion of proper action and some related results to prove the integral theorem A. It outlines the category of topological spaces and the category of logarithmic manifolds.

Chapter One of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It introduces the Hodge structures and polarized hodge structures, as well as the classifying spaces of hodge structures. It elaborates the space of nilpotent orbits D∑ and the space of nilpotent i-orbit D∑# in the directions of ∑ as enlargements of D.

Chapter Three of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It introduces several categories of various kinds of logarithmic generalized analytic spaces and consider relations between them. It considers logarithmic modifications of objects of a generalization of the modification in the toric geometry arising from subdivision of a fan which we will need to extend period maps to the boundary.

Chapter Zero of the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented. It introduces the basic idea of logarithmic Hodge theory, and reviews classifying spaces (degenerating) D of Griffiths as the moduli spaces of polarized Hodge structures. It describes the toroidal partial compactifications of the classifying spaces of Griffiths and the other seven enlargements of D in the fundamental diagram.

An introduction to the book "Classifying Spaces of Degenerating Polarized Hodge Structures," by Kazuya Kato and Sampei Usui is presented.

Chapter 10 of the book "The Hypoelliptic Laplacian and Ray-Singer Metrics," by Jean-Michel Bismut and Gilles Lebeau is presented. It offers information on the calculation of the asymptotics of the generalized metrics. It mentions that the direct proof of Hodge theorem has been discussed in the chapter.

Chapter 2 of the book "The Hypoelliptic Laplacian and Ray-Singer Metrics," by Jean-Michel Bismut and Gilles Lebeau is presented. The chapter focuses on the result of hypoelliptic Laplacian obtained by mathematician, Jean-Michel Bismut and presents information on the families version of Laplacian operators. It mentions that new Hodge theory is obtained form the deformation of classical Hodge theory.

Chapter 1 of the book "The Hypoelliptic Laplacian and Ray-Singer Metrics," by Jean-Michel Bismut and Gilles Lebeau is presented. It offers information on some of the basic results related to the standard Hodge theory. The chapter focuses on the results related to the forms of elliptic analytic torsion obtained by two mathematicians including Jean-Michel Bismut and John Lott.