Take a pair (X, D), where X is a complex projective variety, and D is a sum of curves on X with rational coefficients in [0,1]. There is a notion of orbifold fundamental group of the pair (X, D), that satisfies a Galois-style correspondence. Roughly speaking, this orbifold fundamental group encodes properties of finite Galois covers of X with ramification allowed above the singular locus of X and above D, and of degree controlled by the coefficients appearing in D.
In this talk, we impose that X has (complex) dimension 2. We explain how mild positivity conditions on the curvature of the pair (X, D) and on its singularities then force the corresponding orbifold fundamental group to be quite reasonable. More precisely, it admits a normal subgroup of index at most 7200 that is abelian, or nilpotent of
length at most 2, and that has rank at most 4. We give some ideas of which non virtually abelian groups may appear in this result, and of the pairs that give rise to infinite fundamental groups in general. Finally, we explain how the (sharp) constant 7200 that appears in this result is the Jordan constant of the complex Cremona group in dimension 2, computed by E. Yasinsky.
This talk is a report on joint work with J. Moraga and Z. Liu.
