On Homogenous Minimal Involutive Varieties

Abstract

AbstractЅ(2n,k) be the variety of homogeneous polynomials of degree k in 2n variables. The authors of this paper give a computer-assisted proof that there is an analytic open set Ω of Ѕ(4,3) such that the surface F = 0 is a minimal homogeneous involutive variety of ℂ4 for all F ∈ Ω. As part of the proof, they give an explicit example of a polynomial with rational coefficients that belongs to Ω.

The characteristic variety of a generic foliation

“…If the singular set of F is non-empty but of dimension zero then the fibers over it, and some subvarieties of these fibers, are also left invariant by F (1) . We will say that ch(F ) is a quasi-minimal characteristic variety if (a) F has isolated singularities; and (b) every irreducible homogeneous ( on the fibers of ch(F ) → X ) subvariety of ch(F ) left invariant by F (1) is either the zero section, or a subvariety of a fiber over the singular set of F , or the whole ch(F ). Theorem 1.…”

“…, x n ) where F is induced by the vector field ξ = ∂ x1 . Hence F (1) is still induced by ∂ x1 now seen as a vector field on the total space of N * F . If ω is any of the 1-forms {ω i } i∈k then ω = adx 2 + bdx 3 for suitable holomorphic functions a, b.…”

“…Clearly ch(F ) is a hypersurface of E(T * X). If ω is the non-degenerate 2-form which induces the canonical symplectic structure on T * X then its restriction to ch(F ) induces a one-dimensional foliation F (1) on (the smooth locus of) ch(F ) which will be called the first prolongation of F .…”

“…In this work we are interested in the subvarieties of ch(F ) invariant by F (1) when F is sufficiently general. For no matter which F there is always at least one subvariety of ch(F ) invariant by F (1) : the zero section of T * X.…”

Families of minimal involutive surfaces in projective space

“…The first concrete example of a homogeneous polynomial that defines a minimal involutive homogeneous hypersurface of A 4 seems to have been the rather unwieldy one presented in [1]. In this paper we describe an algorithm capable of determining that a given homogeneous polynomial of degree 3 or 4 gives rise to a minimal involutive homogeneous hypersurface of A 4 .…”