New irreducible components of the space of foliations associated to the affine Lie algebra
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 \[ \mathrm{Fol}((l_0, l_1, l_2), \lambda, \nu) := \{\mathcal{F} \in \mathrm{Fol}(\nu, 3) | \mathcal{F} = \mathcal{F}(S, X) \text{ in some affine chart}\}, \] where we vary over choices of \(S\), \(X\) and \(\lambda\) as above. With this notation the main result of the article can be stated: let \(l_0 >l_1\), \(\text{gcd}(l_0, l_1) = 1\), \(l_0 \geq 3\) and \(q \geq 1\). Then \(\overline{\mathrm{Fol}((l_0, l_1, l_1), l_1(ql_0-1), ql_0+1)}\) is an irreducible component of \(\mathrm{Fol}(ql_0+1, 3)\) and \(\overline{\mathrm{Fol}((l_0, l_1, l_1), l_1l_0q, ql_0+2)}\) is an irreducible component of \(\mathrm{Fol}(ql_0+2, 3)\).
scientific article; zbMATH DE number 6816644
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