Junior Geometry Seminar – Zhaoyang Liu

Abstract: In this talk we will discuss intermediate Jacobians and Abel–Jacobi maps for smooth projective varieties over C. We begin with a brief review of the classical Jacobian of a smooth projective curve, viewed as a complex torus built from H^1 and parametrizing degree-zero divisor classes.

We then introduce intermediate Jacobians associated to the cohomology groups H^{2k−1} and explain how they are constructed using Hodge structures. After that, we define the Abel–Jacobi map from homologically trivial algebraic cycles to the corresponding intermediate Jacobian, and interpret it as a measure of the gap between homological and rational equivalence.

If time permits, we will also introduce the general version Abel-Jacobi map from higher Chow groups to Deligne cohomology. This construction has interesting applications to the problem of the non-rationality of special values of the Riemann \zeta-function.