Singularities of the infinitesimal invariant of normal functions

Abstract

Normal functions $\nu$ provide a method for studying algebraic cycles $Z_t\subset X_t$ varying in a family of smooth projective varieties. Associated with $\nu$ is an infinitesimal invariant $\delta \nu$ that reflects the first-order variation of pairs $(X_t,Z_t)$. Over the years, $\delta \nu$ has been widely used in the study of various geometric questions. We note that whereas $\nu$ is a transcendental invariant, like periods of algebraic integrals, $\delta \nu$ has a natural filtration whose associated graded gives algebraic sections of coherent sheaves. In a number of interesting cases, these sections have had geometric interpretations. In this paper, we will discuss an identification between the singularities $\nu$ and $\delta \nu$. The formal proof of this result will be given in a separate work.