Global and Local fundamental groups in Algebraic Geometry

Abstract

In the first part of this thesis, we study the local fundamental group of low-dimensional log canonical singularities. In dimensions 2 and 3 we establish some constraints on the possible local fundamental groups, while in dimensions 3 and 4 we construct examples of interesting groups that can appear. In dimension 2, by classifying all the possible singularities, we can prove that the local fundamental groups are virtually solvable. Moreover, we give a bound on the number of generators and relations of the group, along with identifying the circumstances under which this maximum is achieved. In dimension 3, we show that free groups with 2 or more generators do not appear as local fundamental groups of log canonical isolated singularities. In dimensions 3 and 4, we construct the fundamental groups of 2-dimensional closed manifolds and some special 3-dimensional manifolds, respectively, as the local fundamental groups of isolated log canonical singularities. One notable example includes the connected sum of copies of S1 × S2, leading to free groups appearing in dimension 4. In the second part, we study the orbifold fundamental groups of the smooth locus of Calabi-Yau type pairs of low coregularity. Here, we establish that the fundamental group of pairs with low coregularity exhibits similar behavior to log Calabi-Yau pairs of low dimension, specifically being virtually abelian of bounded rank. Furthermore, we prove in the case of virtually nilpotency, there are effective bounds on the index and length depending only on the dimension and coregularity.