Logarithmic Lefschetz fixed point formulae and resonant boundary indices
Abstract
We study boundary contributions to the holomorphic Lefschetz formula for strict self-maps of compact complex manifolds with a simple normal crossings divisor. At a normally non-resonant boundary fixed point, the local term reduces to the fixed-point problem on the boundary stratum and vanishes in the de Rham specialisation. For resonant points satisfying a normal admissibility condition, finite flat rescaling and relative duality recover the local trace as the coefficient at the excess normal order of a holomorphic family of relative residue traces; in the balanced case, this coefficient is the trace of a class on the special fibre. The numerical coefficient is independent of the admissible presentation. In a coupled family, the de Rham boundary index is the product of the normal contact order and the tangential complete-intersection multiplicity. Boundary contributions to the fixed-point formula for the complement are supported on the resonant fixed points.
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BibTeX
@misc{shahsavaripour2026logarithmic,
title = {{Logarithmic Lefschetz fixed point formulae and resonant boundary indices}},
author = {Elaheh Shahsavaripour},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.28288},
eprint = {2607.28288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.28288},
}
RIS
TY - UNPB
TI - Logarithmic Lefschetz fixed point formulae and resonant boundary indices
AU - Elaheh Shahsavaripour
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2607.28288
UR - https://arxiv.org/abs/2607.28288
AB - We study boundary contributions to the holomorphic Lefschetz formula for strict self-maps of compact complex manifolds with a simple normal crossings divisor. At a normally non-resonant boundary fixed point, the local term reduces to the fixed-point problem on the boundary stratum and vanishes in the de Rham specialisation. For resonant points satisfying a normal admissibility condition, finite flat rescaling and relative duality recover the local trace as the coefficient at the excess normal order of a holomorphic family of relative residue traces; in the balanced case, this coefficient is the trace of a class on the special fibre. The numerical coefficient is independent of the admissible presentation. In a coupled family, the de Rham boundary index is the product of the normal contact order and the tangential complete-intersection multiplicity. Boundary contributions to the fixed-point formula for the complement are supported on the resonant fixed points.
ER -