Motivic Vanishing Cycles at Infinity and Toric Landau-Ginzburg Mirror Symmetry

@misc{wang2026motivic,
  title = {{Motivic Vanishing Cycles at Infinity and Toric Landau-Ginzburg Mirror Symmetry}},
  author = {Haoxu Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00289},
  eprint = {2610.00289},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00289},
}

TY  - UNPB
TI  - Motivic Vanishing Cycles at Infinity and Toric Landau-Ginzburg Mirror Symmetry
AU  - Haoxu Wang
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00289
UR  - https://arxiv.org/abs/2610.00289
AB  - We compare motivic nearby cycles at infinity with classes computed directly on rational compactifications. Let f be a regular function on a smooth variety over a field of characteristic zero, and let X be a smooth proper compactification. Suppose that a reduced divisor containing the boundary and the support of div_X(f) has simple normal crossings near the polar locus. We prove that the open-pair class defined by 1/f agrees with Raibaut's compactification-independent motivic nearby cycle at infinity. The key step is a local cancellation under blow-ups of transverse zero-pole intersections. For Laurent polynomials nondegenerate at infinity, this comparison yields a direct toric proof of Raibaut's essential-face formula. Equivariant Hodge realization and weighted Ehrhart theory give Newton formulas retaining cohomological degree and the limit-weight filtration, and determine irregular Hodge numbers through the limiting-irregular comparison. The weight-forgetting specialization computes the zero-potential stringy invariant of toric stacky fans with the same radial support, yielding a motivic-Hodge derivation of the corresponding Fano-Landau-Ginzburg mirror identity. In the full-dimensional weak Fano case, the refined formulas identify monodromy eigenvalues and Jordan-block sizes with inertia ages and anticanonical Lefschetz strings, giving a noncanonical comparison over ℂ.
ER  -