## Classifying Spaces of Degenerating Polarized Hodge - download pdf or read online

By Kazuya Kato, Sampei Usui

ISBN-10: 0691138214

ISBN-13: 9780691138213

ISBN-10: 0691138222

ISBN-13: 9780691138220

In 1970, Phillip Griffiths predicted that issues at infinity should be further to the classifying house D of polarized Hodge constructions. during this ebook, Kazuya Kato and Sampei Usui observe this dream through making a logarithmic Hodge conception. They use the logarithmic constructions all started via Fontaine-Illusie to restore nilpotent orbits as a logarithmic Hodge constitution. The booklet makes a speciality of primary issues. First, Kato and Usui build the high-quality moduli house of polarized logarithmic Hodge buildings with extra buildings. Even for a Hermitian symmetric area D, the current conception is a refinement of the toroidal compactifications via Mumford et al. For common D, high quality moduli areas can have slits because of Griffiths transversality on the boundary and be not in the neighborhood compact. moment, Kato and Usui build 8 enlargements of D and describe their kinfolk by way of a basic diagram, the place 4 of those enlargements dwell within the Hodge theoretic quarter and the opposite 4 dwell within the algebra-group theoretic region. those parts are hooked up by way of a continuing map given via the SL(2)-orbit theorem of Cattani-Kaplan-Schmid. This diagram is used for the development within the first subject.

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**Extra resources for Classifying Spaces of Degenerating Polarized Hodge Structures**

**Example text**

C 1 (X, X) are used as well. c (greatest one). c. cp-spaces are similar to lp-spaces. c 1(X, Y). c(X, Y). The spaces L 2 (lR n , lR m ) and lR nm are isometric. Chapter 1. Basic Elements of Functional Analysis 24 Note that for sake of similarity the space of bounded linear operators should be denoted by £00 instead of £. However, we follow the traditional notation £ for this space reserving £00 for the space of compact linear operators. We will also use the following space given in Rozovskii [84].

IIA I / 2 1I L2= trA. (e) If A E £1(X) and A > 0, then dimX < 00. (f) A linear operator A from the product of Xl, ... , X n E H to the product of YI , ... 3), is a Hilbert-Schmidt operator if and only if A ij E £2(X j , Yi), i = 1, ... , m, j = 1, ... , nj moreover, IIAII~ = 2 2 Li,j II AijIIL2' For fixed u E X and v E Y, define the operator u 0 v by the formula (u0v)h=u(v,h), hEY. 26. Suppose that X,Y,Z,H E H, and B E £(Y, H). Then (a) (u0v) E £1(Y,X); (b) (u 0 v)* = (v 0 u); (c) Ilu 0 vile!

Two Hilbert spaces are isometric if and only if they have bases of the same cardinality. In particular, for fixed kEN U {oo}, all k-dimensional separable Hilbert spaces are isometric. 9. But, in general, it is convenient to use distinct symbols for isometric infinite dimensional Hilbert spaces. The next theorem presents another isometry. 21 (Riesz). Any Hilbert space H and its dual H* are isometric under the isometry H* 3 h* +-+ Jh* = h E H: h*x = (h, x), x E H. We will identify the Hilbert spaces Hand H*.

### Classifying Spaces of Degenerating Polarized Hodge Structures by Kazuya Kato, Sampei Usui

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