Fundamental groups of strata of abelian differentials of low codimension
Nick Salter
TL;DR
This work proves that for strata of projectivized Abelian differentials with a sufficiently large number of simple zeros, the inclusion into the moduli space of pointed curves induces an injection on orbifold fundamental groups, aligning with a Kontsevich–Zorich aspiration for strata to be aspherical and closely tied to mapping class groups. The method combines Shimada's fiber-bundle–like morphism approach with a detailed Abel–Jacobi analysis of linear systems and a canonical specialization via the Shimada–Severi strategy, establishing short exact sequences that describe the fundamental groups in terms of simple braid groups and framed mapping class groups. A key technical contribution is showing that, in the low-codimension regime, the global fundamental group of the stratum fits into a sequence with base Mod_g and a kernel governed by configuration-space braids, and that for canonical projections the monodromy becomes trivial under genericity hypotheses. The results provide a concrete, algebro-geometric realization of the Kontsevich–Zorich conjecture in low codimension, with implications for the topology of strata and their connections to plane-curve braid groups and Severi-type moduli problems.
Abstract
We show that for a stratum of projectivized abelian differentials with sufficiently many simple zeroes, the inclusion into the appropriate moduli space of pointed curves induces an injection at the level of orbifold fundamental group, thereby confirming part of a 1997 conjecture of Kontsevich-Zorich. Combined with previous work of the author with Calderon, this describes the orbifold fundamental groups of these strata as framed mapping class groups. Our approach is algebro-geometric in nature, and is based on Shimada's techniques for computing fundamental groups via morphisms of varieties that behave like fiber bundles away from loci of high codimension.