Quaternionic LVM Manifolds
Abstract
An admissible configuration Λ⊂Hhm≃Rr4m defines a compact intersection of quadrics ZHh(Λ) and a free diagonal Sp(1) quotient NHh(Λ). We realize this quotient as the leaf space of a real dilation action on an open subset of quaternionic projective space. The total space is a quaternionic moment-angle polyhedral product. Every quotient is $2$-connected; when there are no indispensable coordinates, it is $3$-connected and its fourth integral cohomology is generated by the Euler class e of the associated oriented rank-four bundle. We compute p1(TNHh)=2(n−2)e and prove that no LVMQ manifold admits a quaternionic Kähler metric. An explicit infinite family consists of products of spheres admitting both homogeneous positive Einstein metrics and quaternionic toric structures in the sense of Gentili--Gori--Sarfatti. We also describe the trace foliations of the real action in the Poincaré domain. For the separate rank-one distribution XHh, an invertible quaternionic-linear field is integrable only in the real scalar case. A nonlinear field with positive real scalar derivative at the origin and involutive XHh has exactly the standard quaternionic Hopf foliation on every sufficiently small centered sphere.
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BibTeX
@misc{verjovsky2026quaternionic,
title = {{Quaternionic LVM Manifolds}},
author = {Alberto Verjovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.26269},
eprint = {2609.26269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26269},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quaternionic LVM Manifolds
AU - Alberto Verjovsky
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.26269
UR - https://arxiv.org/abs/2609.26269
AB - An admissible configuration Λ⊂Hh^m≃Rr^4m defines a compact intersection of quadrics Z_Hh(Λ) and a free diagonal Sp(1) quotient N_Hh(Λ). We realize this quotient as the leaf space of a real dilation action on an open subset of quaternionic projective space. The total space is a quaternionic moment-angle polyhedral product. Every quotient is $2$-connected; when there are no indispensable coordinates, it is $3$-connected and its fourth integral cohomology is generated by the Euler class e of the associated oriented rank-four bundle. We compute p_1(TN_Hh)=2(n−2)e and prove that no LVMQ manifold admits a quaternionic Kähler metric. An explicit infinite family consists of products of spheres admitting both homogeneous positive Einstein metrics and quaternionic toric structures in the sense of Gentili--Gori--Sarfatti. We also describe the trace foliations of the real action in the Poincaré domain. For the separate rank-one distribution XHh, an invertible quaternionic-linear field is integrable only in the real scalar case. A nonlinear field with positive real scalar derivative at the origin and involutive XHh has exactly the standard quaternionic Hopf foliation on every sufficiently small centered sphere.
ER -