Complex multiplication, rationality and mirror symmetry for abelian varieties

Landau-Ginzburg模型的纤维与有理性

Fibers of Landau-Ginzburg models and rationality

本文通过镜像Landau-Ginzburg模型的纤维单值性刻画Fano三维流形的有理性，证明有理当且仅当所有可约纤维的单值性为幺幂。

Some Mirror partners with Complex multiplication

In this note we provide examples of families of Calabi-Yau 3-manifolds over Shimura varieties, whose mirror families contain subfamilies over Shimura varieties. Therefore these original families and subfamilies on the mirror side contain dense sets of complex multiplication fibers. In view of the work of S. Gukov and C. Vafa this is of special interest in theoretical physics.

Rationality of motivic zeta-functions for curves with finite abelian group actions

Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,σ)$, where $X$ is a variety over $k$ and $σ$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of isomorphism classes in the category $\mathfrak{Var}_k^G$ modulo a cutting and pasting relation. The multiplication in this ring is defined by the fiber product of varieties. This allows for motivic zeta-functions for varieties with group actions to be defined. This is a formal power series $\sum_{n=0}^{\infty}[\text{Sym}^n (X,σ)]t^n$ with coefficients in the Grothendieck rin…

On some Hermitian variations of Hodge structure of Calabi-Yau type with real multiplication

We prove that, for every totally real number field E_0, there exists a weight three variation of Hodge structure of Calabi-Yau type defined over the rational numbers with associated endomorphism algebra E_0 such that the unique irreducible factor of Calabi-Yau type of the corresponding real variation of Hodge structure is the canonical real VHS of CY type over the Hermitian symmetric domain II_6, associated to the real group SO^*(12). The main point is a rationality result for the half spin representations of a form of the group SO^*(4m) defined over a number field.

弓型簇上稳定包络的霍尔代数乘法

Hall algebra multiplication for stable envelopes on bow varieties

针对椭圆稳定包络这一量子群表示几何实现的基本组件，给出A型切尔基斯弓型簇上椭圆稳定包络的霍尔代数乘积公式，并结合3维镜像对称性质推导出同行列和01矩阵对的theta函数恒等式。

On genus one mirror symmetry in higher dimensions and the BCOV conjectures

The mathematical physicists Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed, in a seminal article from '94, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck-Riemann-Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann-Roch theorem of Gillet-Soulé and of our previous results on the BCOV invariant, we establish this conjecture for Calabi-Yau hypersurfaces in projective spaces. Our contribution takes place on the $B$-side, and together with …