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PreprintarXiv licencearXiv

Motivic Mirror Symmetry for Sink-recurrent Cluster Varieties

  • Haoyang Liu
  • Tianle Liu
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Abstract

We prove motivic mirror symmetry for full-rank sink-recurrent cluster varieties. We allow skew-symmetrizable principal parts and invert all frozen variables. The rational geometric motive of such a variety is isomorphic to the additive orbifold motive of the mirror stack of Hu and Shende. This gives comparisons of cohomology, higher Chow groups, and algebraic K-theory with rational coefficients. The proof combines quantum deformation with the splitting of these motives into Tate summands. For really full-rank skew-symmetric seeds, a finite étale morphism realizes the comparison. We also study this geometric comparison in families over the projective line.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026motivic,
  title = {{Motivic Mirror Symmetry for Sink-recurrent Cluster Varieties}},
  author = {Haoyang Liu and Tianle Liu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.12863},
  eprint = {2611.12863},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.12863},
}
RIS
TY  - UNPB
TI  - Motivic Mirror Symmetry for Sink-recurrent Cluster Varieties
AU  - Haoyang Liu
AU  - Tianle Liu
PY  - 2026
DA  - 2026/11/09
PB  - arXiv
DO  - 10.48550/arxiv.2611.12863
UR  - https://arxiv.org/abs/2611.12863
AB  - We prove motivic mirror symmetry for full-rank sink-recurrent cluster varieties. We allow skew-symmetrizable principal parts and invert all frozen variables. The rational geometric motive of such a variety is isomorphic to the additive orbifold motive of the mirror stack of Hu and Shende. This gives comparisons of cohomology, higher Chow groups, and algebraic K-theory with rational coefficients. The proof combines quantum deformation with the splitting of these motives into Tate summands. For really full-rank skew-symmetric seeds, a finite étale morphism realizes the comparison. We also study this geometric comparison in families over the projective line.
ER  -