**Translations of Mathematical Monographs
Iwanami Series in Modern Mathematics**

2002; 300 pp; Softcover

MSC: Primary 54; 14; Secondary 46; 20

**Print ISBN: 978-0-8218-2156-5**

Product Code: MMONO/206

Product Code: MMONO/206

List Price: $70.00

AMS Member Price: $56.00

MAA member Price: $63.00

# Advances in Moduli Theory

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*Yuji Shimizu; Kenji Ueno*

The word “moduli” in the sense of this book first appeared in the
epoch-making paper of B. Riemann, Theorie der Abel'schen Funktionen,
published in 1857. Riemann defined a Riemann surface of an algebraic function
field as a branched covering of a one-dimensional complex projective space, and
found out that Riemann surfaces have parameters. This work gave birth to the
theory of moduli.

However, the viewpoint regarding a Riemann surface as an algebraic curve became
the mainstream, and the moduli meant the parameters for the figures (graphs)
defined by equations.

In 1913, H. Weyl defined a Riemann surface as a complex manifold of dimension
one. Moreover, Teichmüller's theory of quasiconformal mappings and
Teichmüller spaces made a start for new development of the theory of moduli,
making possible a complex analytic approach toward the theory of moduli of
Riemann surfaces. This theory was then investigated and made complete by
Ahlfors, Bers, Rauch, and others. However, the theory of Teichmüller
spaces utilized the special nature of complex dimension one, and it was
difficult to generalize it to an arbitrary dimension in a direct way.

It was Kodaira-Spencer's deformation theory of complex manifolds that allowed
one to study arbitrary dimensional complex manifolds. Initial motivation in
Kodaira-Spencer's discussion was the need to clarify what one should mean by
number of moduli. Their results, together with further work by Kuranishi,
provided this notion with intrinsic meaning.

This book begins by presenting the Kodaira-Spencer theory in its original naive
form in Chapter 1 and introduces readers to moduli theory from the viewpoint of
complex analytic geometry. Chapter 2 briefly outlines the theory of period
mapping and Jacobian variety for compact Riemann surfaces, with the Torelli
theorem as a goal. The theory of period mappings for compact Riemann surfaces
can be generalized to the theory of period mappings in terms of Hodge
structures for compact Kähler manifolds. In Chapter 3, the authors state
the theory of Hodge structures, focusing briefly on period mappings. Chapter 4
explains conformal field theory as an application of moduli theory.

This is the English translation of a book originally published in Japanese.
Other books by Kenji Ueno published in this AMS series, Translations of
Mathematical Monographs, include An Introduction to Algebraic
Geometry, Volume 166, Algebraic Geometry 1: From Algebraic Varieties
to Schemes, Volume 185, and Algebraic Geometry 2: Sheaves and
Cohomology, Volume 197.

#### Readership

Graduate students and research mathematicians interested in topology and algebraic geometry.

#### Reviews & Endorsements

This physically compact book thus provides a good pocket guide to a subject of increasing importance.

-- Bulletin of the LMS