Year: 2026
Type: preprint
Abstract: 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 $\operatorname{div}_X(f)$ 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 $\mathbb C$.
Source: arXiv (Cornell University)
Author: Wang, Haoxu
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215708256