Mirror symmetry for gCICY threefolds (II): $[3,-1]$ blocks, VGIT, and toric phases

Source abstract

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.