AI 中文总结
本文研究具有$[3,-1]$块的gCICY三折叠的镜像对称性，通过VGIT将其约化为Batyrev-Borisov对偶，并构造显式镜像及计算周期系统。
AI 中文摘要
广义完全交Calabi-Yau（gCICY）三折叠的配置矩阵包含负项，因此其定义线丛不是nef的，Batyrev-Borisov构造不能直接适用。我们研究$\nmathbb{P}^1\times B$中具有$[3,-1]$块的gCICY三折叠，即由$O(3)\boxtimes L$和$O(-1)\boxtimes(L+R)$的截面所截出，其中$B$是光滑射影四折叠且$2L+R=-K_B$。辅助变量将广义截面转化为相对GIT变化，其相反相位是$\mathbb{P}^2\times B$上秩3丛的零轨迹；对于一般数据，两个相位同构或通过$\int_B c_1(L)^4$个不相交的Atiyah flop相关联。在射影空间乘积上，恰好有18个具有$[3,-1]$块的Calabi-Yau三折叠配置。除伴随论文中研究的非Gorenstein情形外，所有这些配置都具有Gorenstein环面nef相位。因此，对于该分类中的每一个其他配置，镜像对称性在GIT变化后归结为普通的Batyrev-Borisov对偶性，这给出了广义相位本身的镜像Hodge数。我们详细分析了四个具有$[3,-1]$块的例子和一个此类之外的例子。对于每个例子，我们构造了一个显式镜像并计算其基本周期和低阶Picard-Fuchs系统；在最大幺正边界点处的指数代数与A模型相位的有理偶上同调环同构。
英文摘要
The configuration matrices of generalized complete intersection Calabi-Yau (gCICY) threefolds contain negative entries, so their defining line bundles are not nef and the Batyrev-Borisov construction does not apply directly. We study gCICY threefolds in $\mathbb{P}^1\times B$ with a $[3,-1]$ block, that is, cut out by sections of $O(3)\boxtimes L$ and $O(-1)\boxtimes(L+R)$, where $B$ is a smooth projective fourfold and $2L+R=-K_B$. Auxiliary variables turn the generalized section into a relative variation of GIT, whose opposite phase is the zero locus of a rank-3 bundle on $\mathbb{P}^2\times B$; for general data the two phases are isomorphic or related by $\int_B c_1(L)^4$ disjoint Atiyah flops. Over products of projective spaces there are exactly 18 Calabi-Yau threefold configurations with a $[3,-1]$ block. All of them except the non-Gorenstein case studied in the companion paper have Gorenstein toric nef phases. Thus, for every other configuration in this classification, mirror symmetry is reduced after variation of GIT to ordinary Batyrev-Borisov duality, which gives the mirror Hodge numbers of the generalized phase itself. We analyze four examples with a $[3,-1]$ block and one outside this class in detail. For each we construct an explicit mirror and compute its fundamental period and low-order Picard-Fuchs system; the indicial algebras at the maximally unipotent boundary points are isomorphic to the rational even cohomology rings of the A-model phases.
