An Introduction to Families, Deformations and Moduli

Broadly speaking, Moduli Theory involves the discovery of rich geometric structures on the set of isomorphism (or suitable equivalence) classes of familiar objects from Geometry, such as manifolds, varieties and bundles over them. This book is a humble and basic introduction to the fundamental notions of family and deformation, together with their role in the construction of moduli spaces. The attempt is to capture the genesis of the main denitions and results of the theory from heuristic ideas. This is done by fostering a mathematical intuition that is grounded in some basic (though nontrivial) motivating examples. These examples serve not only to point to the kind of results one can expect, but also to pathologies that commonly arise in the construction of moduli spaces.
The goals of this book are:
(1) to explain the notion of a family and construct nontrivial examples of families;
(2) to explain the notion of an infinitesimal deformation of complex structure;
(3) to compute all the infinitesimal deformations of any complex torus (of arbitrary dimension) and of any holomorphic line bundle of degree zero on a compact Riemann surface;
(4) to explain the definitions and the major results of the Theory of Local Moduli of Kodaira-Spencer and apply these results to justify Riemann's formula for the number of moduli of a compact Riemann surface;
(5) to explain Kuranishi's Theorem and to give some of its applications;
(6) to explain the notion of a local moduli space and give some examples;
(7) to explain the notion of a moduli problem, of a coarse moduli space and of a fine moduli space;
(8) to construct the fine moduli space from local moduli spaces for the problem of moduli of holomorphic line bundles of degree zero on a compact Riemann surface;
(9) to construct the coarse moduli space from local moduli spaces for the problem of moduli of elliptic curves over the field of complex numbers;
(10) to present the Theory of Moduli as a topic whose study draws from and enriches diverse areas of Mathematics.


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