CHAPTER I

VARIATION OF HODGE STRUCTURE

This chapter contains the definition and facts needed for the proof of

the local Torelli theorems. To keep the article short, proofs are supplied

only for those propositions for which no comparable proofs are readily

available in the literature. For more examples, background materials, and

references, the reader is referred to [26].

§1. The period map.

We first state a well-known proposition which allows us to identify in

a family of smooth algebraic varieties the differentiable structures of the

fibers. Let X and B be differentiable manifolds. A differentiable map

f: X -» - B has maximal rank at a point p in X if its differential f^:

T X -» • T,.,

V

B is surjective. We say that f has maximal rank if it has

P f(p) J J

maximal rank at every point of X.

Proposition 1.1. A proper differentiable map of maximal rank f : X - B

is locally differentiably trivial.

Now let X and B be smooth complex algebraic varieties and f: X-*-B

a proper holomorphic map of maximal rank. By the proposition, f is a

differentiable fiber bundle over B. However, it need not be a holomorphic

fiber bundle as the fibers do not necessarily have the same complex

structures. So we may view the fiber bundle f : X - * B as a family of

varying complex structures on a fixed differentiable manifold. One way of

measuring the variation of the complex structure in the fibers is the period

map, which we now describe.

Received by the editors March 19, 1981 and, in revised form December 30,

1981. Research partially supported by NSF Grant MCS80-02272.

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