Evento
Descrizione:
Abstract: 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.
This talk reports on joint work with J. Moraga and Zh. Liu on pairs in dimension 1 and 2. We explain how positivity conditions on the curvature of a pair (X, D) and on its singularities forces the group \pi_1(X, D) to be rather small. We provide many examples where 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. We also explain to what extent our proof relies on birational geometry, and on the MMP for surface pairs. If time permits, we also present some criteria for pairs to have a large orbifold fundamental group.