On intrinsic homological mirror symmetry for toric degenerations
Abstract
This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross--Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi--Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber Xt contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of Xt. We verify both of these assumptions for certain Batyrev--Borisov toric degenerations, as well as some toric degenerations of Calabi--Yau threefolds coming from the Gross--Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi--Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.
Related
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BibTeX
@misc{bai2026intrinsic,
title = {{On intrinsic homological mirror symmetry for toric degenerations}},
author = {Shaoyun Bai and Sebastian Haney},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21720},
eprint = {2609.21720},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21720},
}
RIS
TY - UNPB
TI - On intrinsic homological mirror symmetry for toric degenerations
AU - Shaoyun Bai
AU - Sebastian Haney
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21720
UR - https://arxiv.org/abs/2609.21720
AB - This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross--Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi--Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber X_t contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of X_t. We verify both of these assumptions for certain Batyrev--Borisov toric degenerations, as well as some toric degenerations of Calabi--Yau threefolds coming from the Gross--Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi--Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.
ER -