We give a Hodge-theoretic criterion for a Calabi--Yau variety to have finite Weil--Petersson distance on higher dimensional bases up to a set of codimension $\geq 2$. The main tool is variation of Hodge structures and variation of mixed Hodge structures.
We also give a description on the codimension 2 locus for the moduli space of Calabi--Yau threefolds. We prove that the points lying on exactly one finite and one infinite divisor have infinite Weil--Petersson distance along angular slices. Finally, by giving a classification of the dominant term of the candidates of the Weil--Petersson potential, we prove that the points on the intersection of exact two infinite divisors have infinite distance measured by the metric induced from the dominant terms of the candidates of the Weil--Petersson potential.
Based on the uniformization theorems of gravitation instantons by Chen--Chen arXiv:1505.01790, Chen--Viaclovsky arXiv:2110.06498, Collins--Jacob--Lin arXiv:2111.09260, and Hein--Sun--Viaclovsky--Zhang arXiv:2111.09287, we prove that the period maps for the ALH*, ALG, and ALG* gravitational instantons are surjective.
It has been conjectured that the hemisphere partition function arXiv:1308.2217, arXiv:1308.2438 in a gauged linear sigma model (GLSM) computes the central charge arXiv:math/0212237 of an object in the bounded derived category of coherent sheaves for Calabi--Yau (CY) manifolds. There is also evidence in arXiv:alg-geom/ 9511001, arXiv:hep-th/0007071. On the other hand, non-commutative resolutions of singular CY varieties have been studied in the context of abelian GLSMs arXiv:0709.3855. In this paper, we study an analogous construction of abelian GLSMs for non-commutative resolutions and propose they can be used to study a class of recently discovered mirror pairs of singular CY varieties. Our main result shows that the hemisphere partition functions (a.k.a.~$A$-periods) in the new GLSM are in fact period integrals (a.k.a.~$B$-periods) of the singular CY varieties. We conjecture that the two are completely equivalent: $B$-periods are the same as $A$-periods. We give some examples to support this conjecture and formulate some expected homological mirror symmetry (HMS) relation between the GLSM theory and the CY. As shown in arXiv:2003.07148, the $B$-periods in this case are precisely given by a certain fractional version of the $B$-series of arXiv:alg-geom/9511001. Since a hemisphere partition function is defined as a contour integral in a cone in the complexified secondary fan (or FI-theta parameter space) arXiv:1308.2438, it can be reduced to a sum of residues (by theorems of Passare-Tsikh-Zhdanov and Tsikh-Zhdanov). Our conjecture shows that this residue sum may now be amenable to computations in terms of the $B$-series.
The Boolean logic is a tool to model the algebra of truth values. It is an essential subject in Computer Programming and Digital Electronics. While teaching the Boolean logic, teacher usually gives examples to elaborate the Boolean logic formulation. However, the physical meaning of the Boolean logic is difficult for students to imagine without the realization to real-world case. The classical Pac-Man game which has clear Boolean logic game rules and simple game scenario is chosen as the teaching aid. Accordingly, two learning activities, the Boolean logic realization activity and game scenario formulation activity, are proposed. Our idea is that through game rule tuning in the learning activities, the students can comprehend how the changing of game rule (Boolean logic) affects the game scenario. To manipulate the game scenario for teaching specific learning objectives of Boolean logic, the Game Rule of the original Pac-Man game is defined and seven versions of modified game scenarios provided as illustrations are appropriately organized for students to practice one Boolean logic concept each time. The experimental results showed the proposed activity can have higher learning achievements compared to the traditional lecturing.
The presented paper is a continuation of the series of papers [$\href{https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnz259/30788308/rnz259.pdf}{17}$, $\href{https://dx.doi.org/10.4310/CNTP.2020.v14.n4.a2}{18}$]. In this paper, utilizing Batyrev and Borisov's duality construction on nef‑partitions, we generalize the recipe in [$\href{https://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnz259/30788308/rnz259.pdf}{17}$, $\href{https://dx.doi.org/10.4310/CNTP.2020.v14.n4.a2}{18}$] to construct a pair of singular double cover Calabi–Yau varieties $(Y,Y^\vee)$ over toric manifolds and compute their topological Euler characteristics and Hodge numbers. In the $3$-dimensional cases, we show that $(Y, Y^\vee)$ forms a topological mirror pair, i.e., $h^{p,q} (Y ) = h^{3-p,q} (Y^\vee)$ for all $p, q$.
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