Orbifold K{ä}hler Groups related to arithmetic complex hyperbolic lattices

Abstract

We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the Albanese mapping on an étale covering space generates jets on the interior, if the lattice is small enough. We also explore some specific examples of Picard-Eisenstein type.

Orbifold Kähler Groups related to Mapping Class groups

“…, i m−1 , j m ) satisfies the conditions P (k, m). This amounts to one parity obstruction (19) i m + i m+1 + j m ≡ 0 (mod 2), along with the following two sets of inequalities:…”

“…It remains to observe that there exist solutions satisfying the parity condition above. In fact both endpoints of the interval specified by inequality (20) are compatible with the parity obstruction (19), while the endpoints of the interval given by ( 21) are both congruent to i m + i m+1 (mod 2). It follows that there exists a solution j m satisfying the parity obstruction.…”

“…Constructing stacky compactifications of such objects can be used an intermediate step to construct interesting Kähler groups [18,19] and in the present case, to recast some of the basic ideas of Teichmüller level structures [7,9,42].…”

“…Conventions. As advocated by [41], we follow the conventions of [19] and view an orbifold as a smooth Deligne-Mumford stack with trivial generic isotropy groups relative to the category of complex analytic spaces with Hausdorff topology. There is an analytification 2functor from algebraic Deligne-Mumford stacks over C to complex analytic Deligne-Mumford stacks and an underlying topological stack functor from complex analytic Deligne-Mumford stacks to topological Deligne-Mumford stacks [45,46,3].…”

The fundamental group of partial compactifications of the complement of a real line arrangement

“…• If #π 1 (X) = +∞, there is a representation ρ : π 1 (X) → GL N (C) with N ∈ N * , such that #ρ(π 1 (X)) = +∞. (See ( [Eys18]) for motivation and related questions for Khler groups).…”