## Arts & Sciences Electronic Theses and Dissertations

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#### Title

Limits and Singularities of Normal Functions.

Spring 5-15-2019

#### Author's School

Graduate School of Arts and Sciences

Mathematics

#### Degree Name

Doctor of Philosophy (PhD)

Dissertation

#### Abstract

On a projective complex variety $X$, constructing indecomposable higher Chow 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 thesis, we introduce some known examples of indecomposable cycles and a new method to detect the linearly independence of $\mathbb{R}$-regulator indecomposable $K_1$-cycles which is based on the singularities and limits of admissible normal functions with real coefficients. We also construct a collection of higher Chow cycles on certain surfaces in $\mathbb{P}^3$ of degree $d \ge 4$ which degenerate to an arrangement of $d$ planes in general position. By applying our method, we show that these higher Chow cycles are enough to show the surjectivity of the real regulator map when $d = 4$. Hence our construction gives a new explicit proof of the Hodge-$\mathcal{D}$-Conjecture for a certain type of $K3$ surfaces. 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.

English (en)

Matt Kerr

#### Committee Members

Neithalath M. Kumar, Ravindra Girivaru, Escobar V. Laura, Martha Precup,