We show that the periodic $\eta$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to existence of cobordisms with positive scalar curvature metrics between manifolds dimensions $4$ and $6$. The proof combines a relative version Schoen--Yau minimal surface technique an end-periodic index theorem for Dirac operator. As result, we bordism groups $\Omega^{spin,+}...

A content analysis of 1387 commercials broadcast in China and Hong Kong in 1993 and 1998 was conducted using Cheng and Schweitzer’s framework of cultural values. The five cultural values that dominated Chinese television advertising were ‘modernity’, ‘family’, ‘tradition’, ‘technology’ and ‘collectivism’. The five dominant cultural values in Hong Kong television advertising were ‘quality’, ‘eff...

We investigate a method of construction of Calabi–Yau manifolds, that is, by smoothing normal crossing varieties. We develop some theories for calculating the Picard groups of the Calabi–Yau manifolds obtained in this method. As an application, we construct more than two hundred new families of Calabi–Yau 3-folds with Picard number one that have different Hodge numbers (h’s). We also exhibit a ...

Abstract We study the Li–Yau inequality for heat equation corresponding to Dunkl harmonic oscillator, which is a nonlocal Schrödinger operator parameterized by reflections and multiplicity functions. In particular case when reflection group isomorphic ℤ 2 d {\mathbb{Z}...

In this article, we construct simply connected symplectic Calabi-Yau 6-manifold by applying Gompf’s symplectic fiber sum operation along T. Using our method, we also construct symplectic nonKähler Calabi-Yau 6-manifolds with fundamental group Z. We also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery on non si...

In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the rigidity of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieties are rigid, for examples., Lefschetz pencils of Calabi-Yau varieties, strongly de...

A Calabi-Yau threefold is a complex projective threefold X (possibly with some suitable class of singularities, say terminal or canonical) with ω X ∼ = O X and h 1 (O X) = h 2 (O X) = 0. One of the fundamental gaps in the classification of algebraic threefolds is the lack of understanding of Calabi-Yau threefolds. Here I will try to set forth a program to bring the morass of thousands of exampl...

One of the most fruitful sources of Calabi–Yau threefolds is hypersurfaces, or more generally complete intersections, in toric varieties. This is especially true since there is a proposal for the mirror of any such Calabi–Yau threefold. Usually the toric varieties associated to convex lattice polytopes are singular, causing the Calabi–Yau threefolds in them also to be singular, so that to get s...