Mirror symmetry for gCICY threefolds (II): [3,−1] blocks, VGIT, and toric phases
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 ℙ1× 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=−KB. 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 ℙ2× B; for general data the two phases are isomorphic or related by ∫B c1(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.
Related
- Mirror symmetry for gCICY threefolds (I): a [3,−1] block and a non-Gorenstein toric phase
- Doubly isolated Batyrev mirror pairs and non-smoothable Calabi-Yau threefolds
- New inhomogeneous Bismut Hermitian-Einstein threefolds with large symmetry group
- All genus open mirror symmetry for footballs
- Local Homological Mirror Symmetry
BibTeX
@misc{kanazawa2026mirror,
title = {{Mirror symmetry for gCICY threefolds (II): $[3,-1]$ blocks, VGIT, and toric phases}},
author = {Atsushi Kanazawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03688},
eprint = {2610.03688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03688},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mirror symmetry for gCICY threefolds (II): [3,−1] blocks, VGIT, and toric phases
AU - Atsushi Kanazawa
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03688
UR - https://arxiv.org/abs/2610.03688
AB - 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 ℙ^1× 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 ℙ^2× B; for general data the two phases are isomorphic or related by ∫_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.
ER -