A Theorem Concerning the Differential Equations Satisfied by Normal Functions Associated to Algebraic Cycles

Abstract

In this paper we shall give a general discussion of normal functions culminating in a result characterizing those normal functions arising from algebraic cycles as being solutions to a family of ordinary differential equations parametrized by the hypersurfaces of large degree passing through the given cycle or through one homologous to it. In Section 1 the theorem will be informally discussed for the case of curves on a surface where a minimum of technical machinery is necessary. Along the way we give proofs of the main classical results in the theory of normal functions and find some new information on the Hodge bundles arising from a Lefschetz pencil of curves. Then in Section 2 we turn to higher dimensions. Following a discussion of the definition and basic properties of normal functions we analyze the Hodge bundles arising from the cotangent spaces to the intermediate Jacobians in a Lefschetz pencil (c.f. (2.13c) and (2.14c)), and then shall reprove the result (c.f. (2.9) for the statement) characterizing the fundamental classes of normal functions by their Hodge type. Finally, after some general observations on Picard-Fuchs equations we formulate and prove our main result Theorem 2.2, and then conclude the paper with some observations concerning the problem of constructing algebraic cycles. Unless otherwise specified, homology will be with Z-coefflcients and cohomology with C-coefficients. Hopefully the other notations and terminology are standard.