Ph.D Thesis Defense: "Limits and Singularities of Normal Functions"

Abstract: On a projective complex variety $X$, the higher Chow groups $CH^p(X,n)$ have a natural bigraded product structure. When a higher cycle in $CH^p(X,1)$ is not in the image of the usual algebraic cycles in $CH^{p-1}(X)$, it is called indecomposable. Constructing indecomposable cycles is an interesting question toward the Hodge conjecture, motives, and other arithmetic applications. A standard method to determine whether a given higher cycle is indecomposable or not is to consider it as a general fiber of a degenerate family of higher cycles, and observe the asymptotic behaviors of the associated higher normal functions.

In this talk, we introduce two examples of construction of indecomposable: On Jacobians of smooth projective curves by A. Collino and N. Fakhruddin, and on the general polarized $K3$ surfaces by X. Chen and J.Lewis (which shows the Hodge-$\mathcal{D}$-conjecture in this case). Then we show a more explicit construction of higher cycles on certain types of $K3$ surfaces. We prove its indecomposability by using the limit invariant and singularity invariant of higher normal functions. As an application, we also construct new examples of non trivial elements in the Griffiths groups on a certain Calabi-Yau threefold, which is a general fiber of a Tyurin degeneration arising from two reflexive polytopes. Since these Calabi-Yau manifolds and (higher or usual algebraic) cycles are totally derived from the combinatorial geometry of these polytopes, we expect that their dual polytopes encodes the “mirror” objects via mirror symmetry.

Host: Matt Kerr