Periods of families of curves in threefolds

TL;DR

This paper investigates the properties of rational curves in generic quintic threefolds, linking their existence to the vanishing of certain periods, and employs advanced Hodge theory techniques.

Contribution

It generalizes Max Noether's theorem approach to study periods of rational curves in threefolds, providing new insights into Clemens' conjecture.

Findings

Certain periods of rational curves must vanish if infinitely many exist.

The method extends infinitesimal variation of Hodge structures to this context.

Provides a new approach to understanding rational curves in threefolds.

Abstract

Clemens' conjecture states that the the number of rational curve in a generic quintic threefold is finite. If it is false we prove that certain periods of rational curves in such a quintic threefold must vanish. Our method is based on a generalization of a proof of Max Noether's theorem using infinitesimal variation of Hodge structures and its reformulation in terms of integrals and Gauss-Manin connection.