We consider a $d$-dimensional well-formed weighted projective space $\mathbb{P}\left(\overline{w}\right)$ as a toric variety associated with a fan $\Sigma \left(\overline{w}\right)$ in $N_{\overline{w}}\otimes \mathbb{R}$ whose 1-dimensional cones are spanned by primitive vectors $v_0,v_1,\dots ,v_d\in N_{\overline{w}}$ 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 $\overline{w}=(w_0,\dots ,w_d)$ such that $\mathbb{P}\left(\overline{w}\right)$ 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 ${\chi }_{\mathrm{orb}}\left(X_w\right)$, we show that for any quasi-smooth Calabi–Yau hypersurface $X_w$ the number ${(-1)}^{d-1}{\chi }_{\mathrm{orb}}\left(X_w\right)$ equals the stringy Euler number ${\chi }_{\mathrm{str}}\left(X_{\overline{w}}^{\ast }\right)$ of Calabi–Yau compactifications $X_{\overline{w}}^{\ast }$ of affine toric hypersurfaces $Z_{\overline{w}}$ defined by non-degenerate Laurent polynomials $f_{\overline{w}}\in ℂ\left[N_{\overline{w}}\right]$ with Newton polytope $\mathrm{conv}\left(\{v_0,\dots ,v_d\}\right)$. In the moduli space of Laurent polynomials $f_{\overline{w}}$ there always exists a special point $f_{\overline{w}}^0$ defining a mirror $X_{\overline{w}}^{\ast }$ with a $\mathbb{Z}∕w\mathbb{Z}$-symmetry group such that $X_{\overline{w}}^{\ast }$ is birational to a quotient of a Fermat hypersurface via a Shioda map.

我们考虑一个 $d$维格式良好的加权投影空间 $\mathbb{P}（\overline{w}）$ 作为与风扇相关的复曲面 $\Sigma （\overline{w}）$ 在 ${ñ}_{\overline{w}}\otimes \mathbb{[R}$ 其一维视锥由原始向量跨越 $v_0，v_{\mathrm{1个}}，\dots ，v_d\in {ñ}_{\overline{w}}$ 产生晶格 ${ñ}_{\overline{w}}$ 并满足线性关系 ${\sum }_{\mathrm{一世}}{w}_{\mathrm{一世}}{v}_{\mathrm{一世}}=0$。对于任何固定尺寸$d$，仅存在有限的多个权重向量 $\overline{w}=（w_0，\dots ，w_d）$ 这样 $\mathbb{P}（\overline{w}）$ 包含准光滑的Calabi–Yau超曲面 $X_w$ 由横向加权齐次多项式定义 $\mathrm{w\; ^}$ 度 $w={\sum }_{\mathrm{一世}=0}^d{w}_{\mathrm{一世}}$。使用Vafa公式计算单向Euler数${\chi }_{\mathrm{宝珠}}（X_w）$，我们证明，对于任何准光滑的Calabi–Yau超曲面 $X_w$ 号码 ${（-\mathrm{1个}）}^{d-\mathrm{1个}}{\chi }_{\mathrm{宝珠}}（X_w）$ 等于严格的欧拉数 ${\chi }_{\mathrm{力量}}（X_{\overline{w}}^{\ast }）$ 卡拉比丘油压实 $X_{\overline{w}}^{\ast }$ 仿射复曲面的曲面 ${ž}_{\overline{w}}$ 由非退化的Laurent多项式定义 $F_{\overline{w}}\in ℂ\left[{ñ}_{\overline{w}}\right]$ 牛顿多聚体 $\mathrm{转换}（\{v_0，\dots ，v_d\}）$。在Laurent多项式的模空间中$F_{\overline{w}}$ 总有一个特别的地方 $F_{\overline{w}}^0$ 定义镜子 $X_{\overline{w}}^{\ast }$ 与一个 $\mathbb{ž}∕w\mathbb{ž}$-对称组 $X_{\overline{w}}^{\ast }$ 通过Shioda贴图与费马超曲面的商成正比。