Topology of Algebraic Varieties

In the theory of algebraic cycles, the archimedean height is a real number attached to null-homologous non-intersecting algebraic cycles on a smooth projective variety \(X\) whose dimensions add up to one less than the dimension of \(X\). For example, to every pair of non-intersecting degree \(0\) divisors on a curve, one can associate a real number. As the name suggests the archimedean height is the archimedean component of a globally defined height pairing. Hain interpreted this height in Hodge theory as a metric on a certain bundle attached to pairs of normal functions. I will try to explain all of this along with the connection between the asymptotics of this metric and the singularities of the normal functions defining the line bundle. The last part is joint work with G. Pearlstein.