Clemens' conjecture: part I

Abstract

This is a series of two papers in which we solve the Clemens conjecture: there are only finitely many smooth rational curves of each degree in a generic quintic threefold. In this first paper, we deal with a family of smooth Calabi-Yau threefolds f_εfor a small complex number ε. We give an geometric obstruction, deviated quasi-regular deformations B_b of c_ε, to a deformation of the rational curve c_εin a Calabi-Yau threefold f_ε.