We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen’s -conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field . These results provide another self-contained proof of the -conjecture for homology spheres in characteristic .
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Keywords: Simplicial homology spheres, pseudo-manifolds, Stanley-Reisner rings, anisotropy, Hard Lefschetz theorem, $g$-conjecture
Karu, Kalle 1; Xiao, Elizabeth 1
CC-BY 4.0
@article{ALCO_2023__6_5_1313_0,
author = {Karu, Kalle and Xiao, Elizabeth},
title = {On the anisotropy theorem of {Papadakis} and {Petrotou}},
journal = {Algebraic Combinatorics},
pages = {1313--1330},
publisher = {The Combinatorics Consortium},
volume = {6},
number = {5},
year = {2023},
doi = {10.5802/alco.298},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.298/}
}
TY - JOUR AU - Karu, Kalle AU - Xiao, Elizabeth TI - On the anisotropy theorem of Papadakis and Petrotou JO - Algebraic Combinatorics PY - 2023 SP - 1313 EP - 1330 VL - 6 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.298/ DO - 10.5802/alco.298 LA - en ID - ALCO_2023__6_5_1313_0 ER -
%0 Journal Article %A Karu, Kalle %A Xiao, Elizabeth %T On the anisotropy theorem of Papadakis and Petrotou %J Algebraic Combinatorics %D 2023 %P 1313-1330 %V 6 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.298/ %R 10.5802/alco.298 %G en %F ALCO_2023__6_5_1313_0
Karu, Kalle; Xiao, Elizabeth. On the anisotropy theorem of Papadakis and Petrotou. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1313-1330. doi : 10.5802/alco.298. https://alco.centre-mersenne.org/articles/10.5802/alco.298/
[1] Combinatorial Lefschetz theorems beyond positivity (2018) | arXiv
[2] Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles (2021) | arXiv
[3] Combinatorial duality and intersection product: a direct approach, Tohoku Math. J. (2), Volume 57 (2005) no. 2, pp. 273-292 | MR | Zbl
[4] The algebra of continuous piecewise polynomials, Adv. Math., Volume 76 (1989) no. 2, pp. 170-183 | DOI | MR | Zbl
[5] A proof of the sufficiency of McMullen’s conditions for -vectors of simplicial convex polytopes, J. Combin. Theory Ser. A, Volume 31 (1981) no. 3, pp. 237-255 | DOI | MR | Zbl
[6] Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes, Indiana Univ. Math. J., Volume 54 (2005) no. 1, pp. 263-307 | DOI | MR | Zbl
[7] The structure of the polytope algebra, Tohoku Math. J. (2), Volume 49 (1997) no. 1, pp. 1-32 | MR | Zbl
[8] Vector fields and Chern numbers, Math. Ann., Volume 225 (1977) no. 3, pp. 263-273 | DOI | MR | Zbl
[9] Hard Lefschetz theorem for nonrational polytopes, Invent. Math., Volume 157 (2004) no. 2, pp. 419-447 | DOI | MR | Zbl
[10] The numbers of faces of simplicial polytopes, Israel J. Math., Volume 9 (1971), pp. 559-570 | DOI | MR | Zbl
[11] The characteristic 2 anisotropicity of simplicial spheres (2020) | arXiv
[12] Lie groups. An approach through invariants and representations, Universitext, Springer, New York, 2007, xxiv+596 pages | MR
[13] The number of faces of a simplicial convex polytope, Adv. in Math., Volume 35 (1980) no. 3, pp. 236-238 | DOI | MR | Zbl
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