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PreprintarXiv licencearXiv

On the hyperplane conjecture

  • Tsung-Ju Lee
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Abstract

We prove the hyperplane conjecture, proposed by Hosono, Lian, and Yau in arXiv:alg-geom/9511001, for Batyrev's mirror pairs of Calabi--Yau hypersurfaces and its orbifold generalization, without assuming that the lattice points in the defining reflexive polytope generate the ambient lattice. We then identify the pushforward of the extended hypersurface I-function with a generating series of normalized compact periods of the mirror, providing a period interpretation of the hypergeometric series governing ambient genus-zero orbifold Gromov--Witten theory. As an application of our method, we also identify the residue of the Gauss--Manin connection and the cup product morphism on the A-side.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026hyperplane,
  title = {{On the hyperplane conjecture}},
  author = {Tsung-Ju Lee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26186},
  eprint = {2610.26186},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26186},
}
RIS
TY  - UNPB
TI  - On the hyperplane conjecture
AU  - Tsung-Ju Lee
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.26186
UR  - https://arxiv.org/abs/2610.26186
AB  - We prove the hyperplane conjecture, proposed by Hosono, Lian, and Yau in arXiv:alg-geom/9511001, for Batyrev's mirror pairs of Calabi--Yau hypersurfaces and its orbifold generalization, without assuming that the lattice points in the defining reflexive polytope generate the ambient lattice. We then identify the pushforward of the extended hypersurface I-function with a generating series of normalized compact periods of the mirror, providing a period interpretation of the hypergeometric series governing ambient genus-zero orbifold Gromov--Witten theory. As an application of our method, we also identify the residue of the Gauss--Manin connection and the cup product morphism on the A-side.
ER  -