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

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.