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Quaternionic LVM Manifolds
Differential Geometry · Submitted 14 Aug 2026
abstract compiled · 2026-09-24 · no language-model review
The authors' main claim Every LVMQ manifold is $2$-connected; in the absence of indispensable coordinates it is $3$-connected and \[ H^4(N_{\mathbb H}(\Lambda);\Zz)\cong\Zz, \qquad p_1(TN_{{\mathbb H}}(\Lambda))=2(n-2)e.
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The abstract, sentence by sentence
We study a quaternionic counterpart of the quotient construction underlying LVM manifolds. From an admissible configuration \[ \Lambda=(\lambda_1,\ldots,\lambda_n)\subset{\mathbb H}^m\simeq{\mathbb R}r^{4m}, \qquad n>4m, \] we obtain a smooth compact intersection of real quadrics $Z_{\mathbb H}(\Lambda)\subset{\mathbb H}^n$ whose $\text{Sp}(1)^n$ orbit space is a simple convex polytope. A canonical action by positive real coordinate dilations has $Z_{\mathbb H}(\Lambda)$ as a global normalized transversal and, after projectivization, gives the compact quotient \[ N_{\mathbb H}(\Lambda)=Z_{{\mathbb H}}(\Lambda)/\text{Sp}(1), \] which we call an LVMQ manifold. The total space is the quaternionic moment-angle polyhedral product $(D^4,{\mathbb S}^{3})^{K_{P_\Lambda}}$. Every LVMQ manifold is $2$-connected; in the absence of indispensable coordinates it is $3$-connected and \[ H^4(N_{\mathbb H}(\Lambda);\Zz)\cong\Zz, \qquad p_1(TN_{{\mathbb H}}(\Lambda))=2(n-2)e. \] Here $p_1$ denotes the first Pontryagin class and $e$ is the Euler class of the principal $\Sp(1)$-bundle $Z_{{\mathbb H}}(\Lambda)\to N_{{\mathbb H}}(\Lambda)$. For an explicit infinite sphere-product family, the ambient Kraines form restricts to a $4$-plectic form and the resulting LVMQ manifolds are quaternionic toric in the sense of Gentili--Gori--Sarfatti. We also study the complementary Poincar\'e domain and its trace foliations. For the quaternionic rank-one problem, Frobenius integrability of an invertible linear field forces the scalar Hopf model, while small nonlinear integrable perturbations give Hopf-like ${\mathbb S}^{3}$-fibrations over smooth manifolds homotopy equivalent to quaternionic projective space.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study a quaternionic counterpart of the quotient construction underlying LVM manifolds. | contribution 0.97 | 0.88 | 0.06 | asserted 0.90 |
| s1 | From an admissible configuration \[ \Lambda=(\lambda_1,\ldots,\lambda_n)\subset{\mathbb H}^m\simeq{\mathbb R}r^{4m}, \qquad n>4m, \] we obtain a smooth compact intersection of real quadrics $Z_{\mathbb H}(\Lambda)\subset{\mathbb H}^n$ whose $\text{Sp}(1)^n$ orbit space is a simple convex polytope. | contribution 0.43 | 0.89 | 0.13 | asserted 1.00 |
| s2 | A canonical action by positive real coordinate dilations has $Z_{\mathbb H}(\Lambda)$ as a global normalized transversal and, after projectivization, gives the compact quotient \[ N_{\mathbb H}(\Lambda)=Z_{{\mathbb H}}(\Lambda)/\text{Sp}(1), \] which we call an LVMQ manifold. | contribution 0.79 | 0.91 | 0.20 | asserted 1.00 |
| s3 | The total space is the quaternionic moment-angle polyhedral product $(D^4,{\mathbb S}^{3})^{K_{P_\Lambda}}$. | result 0.80 | 0.77 | 0.10 | asserted 1.00 |
| s4 | Every LVMQ manifold is $2$-connected; in the absence of indispensable coordinates it is $3$-connected and \[ H^4(N_{\mathbb H}(\Lambda);\Zz)\cong\Zz, \qquad p_1(TN_{{\mathbb H}}(\Lambda))=2(n-2)e. | result 1.00 | 0.89 | 0.29 | asserted 1.00 |
| s5 | \] Here $p_1$ denotes the first Pontryagin class and $e$ is the Euler class of the principal $\Sp(1)$-bundle $Z_{{\mathbb H}}(\Lambda)\to N_{{\mathbb H}}(\Lambda)$. | interpretation 0.54 | 0.35 | 0.06 | asserted 0.62 |
| s6 | For an explicit infinite sphere-product family, the ambient Kraines form restricts to a $4$-plectic form and the resulting LVMQ manifolds are quaternionic toric in the sense of Gentili--Gori--Sarfatti. | result 0.94 | 0.85 | 0.13 | asserted 1.00 |
| s7 | We also study the complementary Poincar\'e domain and its trace foliations. | contribution 1.00 | 0.88 | 0.03 | asserted 0.91 |
| s8 | For the quaternionic rank-one problem, Frobenius integrability of an invertible linear field forces the scalar Hopf model, while small nonlinear integrable perturbations give Hopf-like ${\mathbb S}^{3}$-fibrations over smooth manifolds homotopy equivalent to quaternionic projective space. | result 0.84 | 0.72 | 0.09 | asserted 1.00 |
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Original abstract
An admissible configuration $\Lambda\subset\Hh^m\simeq\Rr^{4m}$ defines a compact intersection of quadrics $Z_{\Hh}(\Lambda)$ and a free diagonal $\Sp(1)$ quotient $N_{\Hh}(\Lambda)$. 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 K\"ahler 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 Poincar\'e domain. For the separate rank-one distribution $X\Hh$, 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 $X\Hh$ has exactly the standard quaternionic Hopf foliation on every sufficiently small centered sphere.
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