Polynomial bounds for automorphisms groups of foliations

Abstract

Let (X, F ) be a foliated surface and G a finite group of automorphisms of X that preserves F . We investigate invariant loci for G and obtain upper bounds for its order that depends polynomially on the Chern numbers of X and F . As a consequence, we estimate the order of the automorphism group of some foliations under mild restrictions. We obtain an optimal bound for foliations on the projective plane which is attained by the automorphism groups of the Jouanolou's foliations.

Analytic varieties invariant by foliations and Pfaff systems

“…Such result is also related with the problem of bounding the order of the automorphism group of foliations of general type, see [35,37,85,103]. In the recent paper [67] the authors prove a bound for other birational invariant of a foliation, called the slope, which is related to the Poincaré problem.…”

“…Since ξ 1 , • • • , ξ m are linearly independent over M (X), there exist g 1 , • • • , g m ∈ M (M ) such that dR j = m i=1 g i ξ i . By (35) we get…”

Foliations on the projective plane with finite group of symmetries

“…if the response is yes, what is the minimal function f ? In [8], Corrêa and the first author of this paper show that the function f (d) = 3(d 2 + d + 1) bounds the order of Aut(F ) for a generic class of foliations F of degree d. The maximal value 3(d 2 + d + 1) in this class is attained by the Jouanolou Foliation [11]. In the present work, we show that the bound 3 d 2 + d + 1 does not work for all foliations but we prove that the function f indeed exists.…”