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p-lck with potential structure on LVMB manifolds

Bastien Faucard

TL;DR

The work studies p-lck with potential structures on LVMB manifolds, clarifying that lck with potential is rare and, aside from diagonal Hopf manifolds, typically does not occur on LVMBs. It develops a discrete $oldsymbol{Z}^m$-presentation, identifies a rationality condition $(K)$ that yields a complex torus inside $ ext{Aut}(N,J)$, and establishes obstructions to lck metrics, implying that lck-with-potential is essentially restricted to diagonal Hopf cases. A novel condition $(H)$ is introduced to produce non-compact, non-Kähler lck covers of rank $oldsymbol{m-1}$, with corollaries for multiple covers; these results are shown to be deformation-stable and open. The paper also provides concrete LVMB examples, including non-LVM instances, to refute prior conjectures on the minimal covering rank $p$, and it proposes new conjectures supported by computational experiments. Overall, the work significantly advances understanding of when lck (and lck with potential) structures occur on LVMB manifolds and offers a framework for constructing and obstructing such structures via discrete automorphisms and covering theory.

Abstract

In this paper, I present a natural generalization of all the results from [6] to LVMB manifolds: to summarize, very few LVMB manifolds are lck, and none are lck with potential except for diagonal Hopf manifolds. Moreover, if $N$ is an LVMB manifold with a sufficient number of indispensable coordinates, and under a certain assumption $(H)$ (which may be artificial, as I conjecture) on the localization of the configuration $Λ$, there exists a non-compact and non-Kählerian $\mathbb{Z}^p$-lck with potential cover of $N$ with $p = m-1$. Furthermore, I show that the conjecture stated at the end of [5] (that $p$ is bounded below by $m-1$) is false, by exhibiting examples of LVMB manifolds that are $1$-lck with potential when $m\geq 3$ and $n>2m+1$. This leads to the formulation of a new conjecture: if assumption $(H)$ holds, then $N$ is $1$-lck with potential. Moreover, assumption $(H)$ seems entirely artificial. This conjecture is supported by several examples.

p-lck with potential structure on LVMB manifolds

TL;DR

The work studies p-lck with potential structures on LVMB manifolds, clarifying that lck with potential is rare and, aside from diagonal Hopf manifolds, typically does not occur on LVMBs. It develops a discrete -presentation, identifies a rationality condition that yields a complex torus inside , and establishes obstructions to lck metrics, implying that lck-with-potential is essentially restricted to diagonal Hopf cases. A novel condition is introduced to produce non-compact, non-Kähler lck covers of rank , with corollaries for multiple covers; these results are shown to be deformation-stable and open. The paper also provides concrete LVMB examples, including non-LVM instances, to refute prior conjectures on the minimal covering rank , and it proposes new conjectures supported by computational experiments. Overall, the work significantly advances understanding of when lck (and lck with potential) structures occur on LVMB manifolds and offers a framework for constructing and obstructing such structures via discrete automorphisms and covering theory.

Abstract

In this paper, I present a natural generalization of all the results from [6] to LVMB manifolds: to summarize, very few LVMB manifolds are lck, and none are lck with potential except for diagonal Hopf manifolds. Moreover, if is an LVMB manifold with a sufficient number of indispensable coordinates, and under a certain assumption (which may be artificial, as I conjecture) on the localization of the configuration , there exists a non-compact and non-Kählerian -lck with potential cover of with . Furthermore, I show that the conjecture stated at the end of [5] (that is bounded below by ) is false, by exhibiting examples of LVMB manifolds that are -lck with potential when and . This leads to the formulation of a new conjecture: if assumption holds, then is -lck with potential. Moreover, assumption seems entirely artificial. This conjecture is supported by several examples.
Paper Structure (14 sections, 17 theorems, 73 equations, 1 figure, 2 tables)

This paper contains 14 sections, 17 theorems, 73 equations, 1 figure, 2 tables.

Table of Contents

  1. lck/lck with potential, LVM and Hopf manifolds
  2. LVMB manifolds
  3. From continous to discret
  4. Diagonal Hopf manifolds ans LVMB manifolds
  5. Existence of a complex torus in $\mathrm{Aut}_J(N^\epsilon_\Lambda)$
  6. lck metrics and LVMB manifolds
  7. Some LVMB manifolds are $(m-1)$-lck with potential
  8. Examples and conjecture
  9. Python scripts used for the search of examples
  10. Studyability of $(\epsilon,\Lambda)$
  11. Imbrication condition
  12. Is $(\epsilon,\Lambda)$ the good system of an LVM manifold ?
  13. Condition $(H)$
  14. Randomly generate configurations $\Lambda$

Key Result

Theorem 1.3

The quotient $N_\Lambda = S_\Lambda / a_\Lambda$ is called an LVM manifold. It is a compact complex manifold of complex dimension $n - m - 1$. If $n > 2m + 1$, then $N_\Lambda$ is non-symplectic (and hence non-Kählerian). If $n = 2m + 1$, then it is a complex torus $\mathbb{T}^m$ of complex dimensio

Figures (1)

  • Figure 1: $\Lambda_{i_1}, \Lambda_{i_2}$ and the line passing through $\Lambda_1$ and $\Lambda_2$.

Theorems & Definitions (47)

  • Conjecture 1
  • Definition 1.1
  • Definition 1.2
  • Example 1
  • Theorem 1.3: and definition meersseman1998procede
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • ...and 37 more