Density of monodromy actions on non-abelian cohomology

Abstract

In this paper we study the monodromy action on the first Betti and de Rham nonabelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski dense monodromy orbit. In particular we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.(ii) In the case of M B (X o , n), considered with its natural structure of an affine algebraic variety, there exists a point x B ∈ M B (X o , n) so that the orbitis Zariski dense in M B (X o , n); or in the case of M DR (X/B, n) → B there is a leaf of the foliation defined by ∇ n DR which is Zariski dense in the algebraic Zariski topology.This theorem suggests that for families f : X → B with a "large enough" geometric monodromy one should expect Zariski dense monodromy actions on non-abelian cohomology. Geometrically families with large monodromy naturally arise from hyperplane sections and Lefschetz fibrations. In this context we prove the following Theorem B Let Z be a smooth projective surface with b 1 (Z) = 0. Let O Z (1) be an ample line bundle on Z and let n > 1 be a fixed odd integer. Then there exists a positive integer ℓ (depending only on Z and O Z (1)), such that for every k ≥ ℓ and for every Lefschetz fibration f : Z → P 1 in the linear system |O Z (k)| we have:(i) There is no meromorphic function on M B (Z o , n) an which is invariant under the action of mon n B (π 1 (P 1 \ {p 1 , . . . , p µ }, o)) (equivalently there is no meromorphic function on M DR ( Z/P 1 \ {p 1 , . . . , p µ }, n) an which is ∇ n DR -invariant);(ii) In the case of M B (Z o , n), there exist a point x B ∈ M B (Z o , n) so that the orbitis Zariski dense in M B (Z o , n); or in the case of the space M DR ( Z/P 1 \ {p 1 , . . . , p µ }, n) the foliation defined by ∇ n DR has a Zariski dense leaf.Here as usual Z is the blow-up of Z at the base points of the pencil and p 1 , . . . p µ ∈ P 1 are the points where the map Z → P 1 is not submersive. Preliminary reductionsWe start with some general results about linear group actions on algebraic varieties, which will allow us to localize at a point the Zariski density property of an action.

Polynomials and Vanishing Cycles

“…In the literature, examples of nongeneric pencils occured sporadically; more recently they got into light, e.g. [KPS,Ok2], precisely because one can use nongeneric pencils towards more efficient computations. For instance, M. Oka uses special pencils tangent to flex points of projective curves in order to compute the fundamental group of the complement, see e.g.…”

Painlevé equations and complex reflections

“…One of the themes of this article is the study of monodromy actions in nonabelian cohomology. In [14] Katzarkov-Pantev-Simpson searched for fundamental group representations having dense orbits under the monodromy action, whereas we will consider the opposite extreme of finite orbits, in the simplest non-trivial case. Namely we will look for finite orbits of the monodromy action on the set of SL 2 (C) representations of the fundamental group of the four-punctured sphere, or equivalently for finite branching solutions of the sixth Painlevé equation.…”