A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles

Borcea-Voisin Calabi-Yau threefolds and invertible potentials

We prove that the Borcea-Voisin mirror pairs of Calabi-Yau threefolds admit projective birational models that satisfy the Berglund-Hübsch-Chiodo-Ruan transposition rule. This shows that the two mirror constructions provide the same mirror pairs, as soon as both can be defined.

论极大非阶乘节点 Fano 三维簇

On maximally non-factorial nodal Fano threefolds

本文研究非阶乘节点 Fano 三维簇，完成了对恰含 1 个节点且类群秩为 2 情形的分类。

Bogomolov--Sommese type vanishing for globally $F$-regular threefolds

In this paper, we show that every invertible subsheaf of the cotangent bundle of a smooth globally $F$-regular threefold of characteristic $p>3$ has Iitaka dimension less than or equal to one.

Maximally-graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik--Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles. The purpose of this paper is to prove the algebraic counterpart of the Orlik--Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund--Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we s…

八平面排列所定义的双八次卡拉比-约当三重流形的分类 II

Classification of double octic Calabi-Yau threefolds defined by an arrangement of eight planes II

本文通过研究八平面排列，完整分类了双八次卡拉比-约当三重流形，并揭示了其几何结构和模性新实例。

A version of the Berglund-Hübsch-Henningson duality with non-abelian groups

A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a "non-abelian" generalization of the statement about the equivariant Saito duality property for invertible polynomials. It turns out that the statement holds only under a special condition on the action of the subgroup of the permutation group called here PC ("parity condition"). An inspect…