Twisted logarithmic de Rham complexes and the rational hit conjecture

Abstract

We study the top degree concentration property for the cohomology of twisted logarithmic de Rham complexes of multiarrangements of hyperplanes. Our main tool is a short exact sequence of twisted logarithmic de Rham complexes based on the deletion-restriction exact sequences for logarithmic differential forms found recently by Abe and Denham. We prove a number of general results that can be used to establish the top degree cohomology concentration property, and also establish that property for some families of multiarrangements. As a consequence of our results, we prove the rational hit conjecture of R.~M.~W.~Wood about the coinvariants of the rational Steenrod algebra acting on the polynomial ring. We also make progress on the deformed version of that conjecture proposed by Hivert and Thiéry, prove generalizations of the rational hit conjecture for all complex reflection groups G(m,p,n), thus answering questions of Bergeron, Borie and Thiéry, and generalize the rational hit conjecture to all standard subalgebras of the rational Steenrod algebra.

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BibTeX RIS

BibTeX

@misc{dotsenko2026twisted,
  title = {{Twisted logarithmic de Rham complexes and the rational hit conjecture}},
  author = {Vladimir Dotsenko},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26028},
  eprint = {2610.26028},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26028},
}

RIS

TY  - UNPB
TI  - Twisted logarithmic de Rham complexes and the rational hit conjecture
AU  - Vladimir Dotsenko
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26028
UR  - https://arxiv.org/abs/2610.26028
AB  - We study the top degree concentration property for the cohomology of twisted logarithmic de Rham complexes of multiarrangements of hyperplanes. Our main tool is a short exact sequence of twisted logarithmic de Rham complexes based on the deletion-restriction exact sequences for logarithmic differential forms found recently by Abe and Denham. We prove a number of general results that can be used to establish the top degree cohomology concentration property, and also establish that property for some families of multiarrangements. As a consequence of our results, we prove the rational hit conjecture of R.~M.~W.~Wood about the coinvariants of the rational Steenrod algebra acting on the polynomial ring. We also make progress on the deformed version of that conjecture proposed by Hivert and Thiéry, prove generalizations of the rational hit conjecture for all complex reflection groups G(m,p,n), thus answering questions of Bergeron, Borie and Thiéry, and generalize the rational hit conjecture to all standard subalgebras of the rational Steenrod algebra.
ER  -

Abstract

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