Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1908.09965

Periods of CY $n$-folds and mixed Tate motives, a numerical study

Wenzhe Yang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the mirror symmetry of Calabi-Yau threefolds, the instanton expansion of the prepotential has a constant term that is a rational multiple of $ζ(3)/(2 πi)^3$, the motivic origin of which has been carefully studied in the author's paper with M. Kim. The Gamma conjecture claims that higher zeta values in fact appear in the theory of Calabi-Yau $n$-folds for $n \geq 4$. Moreover, a natural question is whether they also have a motivic origin. In this paper, we will study the limit mixed Hodge structure (MHS) of the Fermat pencil of Calabi-Yau $n$-folds at the large complex structure limit, which is actually a mixed Hodge-Tate structure. We will compute the period matrix of this limit MHS for the cases where $n=4,5,6,7,8,9,10,11,12$ by numerical method, and our computations have shown the occurrence of higher zeta values, which provide evidence to the Gamma conjecture. Furthermore, we will also provide a motivic explanation to our numerical results.

Explore related subjects

math.AGhep-thmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Wenzhe Yang. 2019-10-28. Periods of CY $n$-folds and mixed Tate motives, a numerical study. https://arxiv.org/abs/1908.09965

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Brauer-Manin obstruction for stacky curves

We show that the Brauer-Manin obstruction is the only obstruction to strong approximation for all stacky curves over global fields with finite abelian fundamental groups. This includes all stacky curves of genus $g = \frac{1}{2}$, thus explaining a recent counterexample to the Hasse principle of Bhargava-Poonen. We will furthermore show that the elementary obstruction is the only obstruction to the integral Hasse principle for smooth proper integral models of stacky curves of genus $g < 1$. We then compute the Brauer-Manin obstruction for smooth proper integral models of stacky curves of genus $\frac{1}{2}$.

math.AG↗
arXiv2026

Tropicalization of super Gromov-Witten invariants

We show that genus-0, $n$ Neveu-Schwarz marked, super Gromov-Witten invariants of a convex, toric variety $X$ can be defined and computed using tropical geometry. When $X$ is a point, the tropical, super Gromov-Witten invariants of $X$ are descendant invariants on the moduli space of tropical curves. When $X$ is a general convex, toric variety, we define a procedure that computes the tropical, inverse Euler class of the SUSY normal bundle $\overline{N}_{n, β} \rightarrow \overline{\mathcal{M}}_{0,n}(X, β)$, under the assumption that $\overline{N}_{n, β}$ is in some sense locally tropicalizable. We define the tropical, genus-0, $n$ Neveu-Schwarz marked, super Gromov-Witten invariants of $X$, and show that the definition recovers the tropical, super Gromov-Witten invariants of a point. We compute a tropical, super Gromov-Witten invariant of $\mathbb{P}^1$.

math.AG↗
arXiv2026

Optimal bounds for local volumes of threefold singularities

We establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Specifically, we show that a klt threefold singularity with local volume at least $9$ is either a hypersurface singularity or a quotient singularity. As applications, we obtain new restrictions on the singularities of members in K-moduli spaces of Fano threefolds, and we establish a sharp inequality between local volumes and minimal log discrepancies for threefold singularities.

math.AG↗