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.
We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector µ ∈ [0, 1] 5 . The moduli space corresponding to the central weight µ c = ( 1 2 , . . . , 1 2 ) is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the C * -action we obtain a proof of Simpson's foliation conjecture in this case. For each n ≥ 5, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus n points is false.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.