New irreducible components of the space of foliations associated to the affine Lie algebra
type: Item
instance of: scholarly article
title: New irreducible components of the space of foliations associated to the affine Lie algebra
zbMATH Open document ID: 1400.37054
author: Ruben Lizarbe
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2017-12-08T00:00:00Z
review text: A foliation of degree \(k\) on \(\mathbb P^n\) may be defined by an integrable \(1\)-form on \(\mathbb C^{n+1}\) whose coefficients are homogeneous polynomials of degree \(k+1\). Denote by \(\mathrm{Fol}(k, n)\) the space of all such foliations. In general this space will consist of many irreducible components and much is known about the components of \(\mathrm{Fol}(k, n)\) for \(k \leq 2\), including the number of such components and what the general foliation in each such component is, see, for example [\textit{J. P. Jouanolou}, Equations de Pfaff algébriques. Lecture Notes in Mathematics. 708. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0477.58002)] for \(k = 0, 1\) and [\textit{D. Cerveau} and \textit{A. Lins Neto}, Ann. Math. (2) 143, No. 3, 577--612 (1996; Zbl 0855.32015)] for \(k = 2\). In higher degrees this classification problem is largely open. The main object of this paper is to produce examples of components of \(\mathrm{Fol}(k, 3)\) for \(k \geq 3\) which have not previously appeared in the literature. We now briefly summarize the construction of these components. We first define a distinguished collection of foliations on \(\mathbb{P}^3\), first considered in [\textit{O. Calvo-Andrade} et al., Ergodic Theory Dyn. Syst. 24, No. 4, 987--1014 (2004; Zbl 1068.32022)]. Let \(1 \leq l_0 \leq l_1 \leq l_2\) be integers with \(\text{gcd}(l_0, l_1, l_2) = 1\). Consider the vector field on \(\mathbb{C}^3\) \[ S = l_0x\frac{\partial}{\partial x}+l_1y\frac{\partial}{\partial y}+l_2z\frac{\partial}{\partial z} \] and let \(X\) be another vector field on \(\mathbb{C}^3\) with polynomial coefficients, not collinear with \(S\) and such that \([S, X] = \lambda X\) for some \(\lambda \in \mathbb{Z}\). Thus \(S\) and \(X\) define a foliation of some degree \(\nu\) which is denoted \(\mathcal{F}(S, V)\). We define \[ \mathrm{Fol}((l_0, l_1, l_2), \lambda, \nu) := \{\mathcal{F} \in \mathrm{Fol}(\nu, 3) | \mathcal{F} = \mathcal{F}(S, X) \text{ in some affine cha…
reviewed by: Calum Spicer
Mathematics Subject Classification ID: 37F75
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 6816644
zbMATH Keywords: holomorphic foliation
zbMATH Keywords: weighted projective plane
Wikidata QID: Q115385749
MaRDI profile type: Publication
full work available at URL: s00574-017-0029-9
OpenAlex ID: W2583481192
cites work: Intersection Theory on Abelian-Quotient V-Surfaces and Q-Resolutions
cites work: On the Hodge structure of projective hypersurfaces in toric varieties
cites work: Q4832193
cites work: Positivity, vanishing theorems and rigidity of codimension one holomorphic foliations
cites work: Irreducible components of the space of foliations associated to the affine Lie algebra
cites work: Q3336052
cites work: Irreducible components of the space of holomorphic foliations of degree two in \(\mathbb{C} P(n)\), \(n\geq 3\)
cites work: A note on Poincaré's problem for quasi-homogeneous foliations
cites work: Q4324883
cites work: Stability of holomorphic foliations with split tangent sheaf
cites work: Singularities of logarithmic foliations
cites work: Q3663431
cites work: Q4846449
cites work: Equations de Pfaff algébriques
cites work: Foliations with trivial canonical bundle on Fano 3-folds
DOI: 10.1007/S00574-017-0029-9
Recommended article: Irreducible components of the space of foliations associated to the affine Lie algebra
Recommended article: Foliations on projective spaces associated to the affine Lie algebra
Recommended article: Q3777090
Recommended article: Lie algebras of vector fields and codimension one foliations
Recommended article: Q4772430
Recommended article: Q3799272
Recommended article: On Lie algebras of vector fields related to Riemannian foliations
Recommended article: Lie algebras of vector fields and generalized foliations
Recommended article: A new family of irreducible, integrable modules for affine Lie algebras
Recommended article: Q4880971
description: scientific article; zbMATH DE number 6816644