论文标题
在加权投影空间中的准平滑calabi-yau hypersurfaces的镜子对称性
Mirror symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces
论文作者
论文摘要
我们将$ d $二维的加权射影空间$ \ mathbb {p}（\ overline {w}）$作为与fan $σ（\ overline {w}）$相关的折叠品种，$ n _ {\ n_ {\ otline {w}} \ otime {原始矢量$ v_0，v_1，\ ldots，v_d \ in n _ {\ overline {w}} $生成一个晶格$ n _ {\ overline {w}} $，并满足线性关系$ \ sum_i w_i w_i v_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 $。使用VAFA的公式对Orbifold Euler编号$χ_ {\ rm orb}（x_w）$，我们表明，对于任何准静态的calabi-yau hypersurface $ x_w $ x_w $ $χ_ {\ rm str}（x _ {\ overline {w}}}^*） $ f _ {\ overline {w}} \ in \ mathbb {c} [n _ {\ overline {w}}] $带有牛顿polytope $ \ text {cons}（\ {v_0，\ ldots，\ ldots，v_d \}）$。在laurent多项式的模块空间中，$ f _ {\ edropline {w}} $总是存在一个特殊的点$ f _ {\ overline {w}}^0 $定义镜像$ x _ {\ edimane {w} $ x _ {\ overline {w}}^*$通过shioda地图对fermat hypersurface的商而言。
We consider a $d$-dimensional well-formed weighted projective space $\mathbb{P}(\overline{w})$ as a toric variety associated with a fan $Σ(\overline{w})$ 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 $χ_{\rm orb}(X_w)$, we show that for any quasi-smooth Calabi-Yau hypersurface $X_w$ the number $(-1)^{d-1}χ_{\rm orb}(X_w)$ equals the stringy Euler number $χ_{\rm str}(X_{\overline{w}}^*)$ 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.
