Mirror symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces

TL;DR

This paper explores mirror symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces, establishing a relationship between orbifold Euler numbers and stringy Euler numbers, and identifying special mirror points with symmetries.

Contribution

It proves a formula linking orbifold and stringy Euler numbers for Calabi-Yau hypersurfaces and constructs explicit mirror examples with symmetry properties.

Findings

Orbifold Euler number equals stringy Euler number for these hypersurfaces.

Existence of special mirror points with symmetry groups in the moduli space.

Mirror Calabi-Yau varieties are birational to quotients of Fermat hypersurfaces.

Abstract

We consider a $d$-dimensional well-formed weighted projective space $\mathbb{P}(\overline{w})$ as a toric variety associated with a fan $\Sigma(\overline{w})$ in $N_{\overline{w}} \otimes \mathbb{N}$ whose $1$-dimensional cones are spanned by primitive vectors $v_0, v_1, \ldots, v_d \in N_{\overline{w}}$ generating a lattice $N_{\overline{w}}$ and satisfying the linear relation $\sum_i w_i v_i =0$. For any fixed dimension $d$, there exist only finitely many weight vectors $\overline{w} = (w_0, \ldots, w_d)$ such that $\mathbb{P}(\overline{w})$ contains a quasi-smooth Calabi-Yau hypersurface $X_w$ defined by a transverse weighted homogeneous polynomial $W$ of degree $w = \sum_{i=0}^d w_i$. Using a formula of Vafa for the orbifold Euler number $\chi_{\rm orb}(X_w)$, we show that for any quasi-smooth Calabi-Yau hypersurface $X_w$ the number $(-1)^{d-1}\chi_{\rm orb}(X_w)$ equals the stringy Euler number $\chi_{\rm str}(X_{\overline{w}}^*)$ of Calabi-Yau compactifications $X_{\overline{w}}^*$ of affine toric hypersurfaces $Z_{\overline{w}}$ defined by non-degenerate Laurent polynomials $f_{\overline{w}} \in \mathbb{C}[N_{\overline{w}}]$ with Newton polytope $\text{conv}(\{v_0, \ldots, v_d\})$. In the moduli space of Laurent polynomials $f_{\overline{w}}$ there always exists a special point $f_{\overline{w}}^0$ defining a mirror $X_{\overline{w}}^*$ with a $\mathbb{Z}/w\mathbb{Z}$-symmetry group such that $X_{\overline{w}}^*$ is birational to a quotient of a Fermat hypersurface via a Shioda map.