Local classification of singular hexagonal 3-webs with holomorphic Chern connection and infinitesimal symmetries

Flat 3-webs via semi-simple Frobenius 3-manifolds

“…Proof: It is sufficient to check that the vector fields v 1 , v 2 ∈ T S commute. Holds true θ 1 (e 1 ) = 0, θ 2 (e 1 ) = −1, where the forms θ i are defined by (1). Setting e…”

“…Theorem 5 [1] Suppose ODE (12) admits an infinitesimal symmetry X vanishing at the point (0, 0) on the discriminant curve ∆ and the germ of the Chern connection form is exact γ = d(f ), where f is some function germ. Then the equation germ and the symmetry are biholomorphic to one of the following normal forms:…”

Frobenius 3-Folds via Singular Flat 3-Webs

“…Note that the existence of a symmetry at a singular point is not a trivial condition for a flat 3-web: even though a flat 3-web has 3-dimensional symmetry algebra at regular points (see [7]), not all symmetries survive at singular ones (see the discussion in [3]). For a booklet 3-web the symmetry is generated by the flow of the Euler field E.…”

“…The analyticity condition of the Chern connection at singular points is also rather restrictive: a flat 3-web can have closed but not holomorphic connection form. Generically it has a pole on the discriminant curve (see [3] for examples).…”

“…For studying our webs we use implicit ODEs; this approach has proved its efficiency in both non-singular (see for instance [11]) and singular cases [3,4]. Let us consider singular 3-webs having the first three of the above properties, and suppose that the web directions are well defined (probably, with multiplicities).…”