Cécile Gachet

In algebraic topology, the Galois correspondence allows to view the fundamental group \pi_1(X) of a topological manifold X as parametrizing unramified covers of X, via its subgroups. In algebraic geometry, it is however common to deal with finite covers that are not étale. A notion of orbifold fundamental group for pairs (X, D), which has been circulating in the algebro-geometric literature since the 1990ies, and has been revived in many recent papers on the subject, exhibits a similar Galois correspondence for algebraic covers with allowed ramification. Indeed, for a complex projective variety X with relatively mild singularities, and an effective divisor D on X with rational coefficients in [0,1], the normal finite index subgroups of the orbifold fundamental group \pi_1(X, D) parametrize finite Galois covers of X whose ramification is controlled, both geometrically and numerically, by the divisor D.

In this talk, we impose that X is a complex projective curve or surface. In this set-up, we explain how mild positivity conditions on the curvature of the pair (X, D) and on its singularities forces the group \pi_1(X, D) to be rather reasonable. We provide many examples where X is a curve and (X, D) is a log canonical Calabi—Yau pair, and the group \pi_1(X, D) can be explicitly computed. We then state our main result: If X is a surface and (X, D) is a log canonical Calabi—Yau pair, then the orbifold fundamental group \pi_1(X, D) admits a normal subgroup of index at most 7200 that is either abelian of rank at most 4, or part of a very explicit list of nilpotent groups of length 2. If time permits, we finally give some examples of the extreme cases in this result.