Abstract
We prove a conjecture of Kollár stating that the local fundamental group of a klt singularity x is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of x is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair.
Highlights
We work over the field C of complex numbers
A log pair is an algebraic variety X together with a boundary divisor 0 ≤ Δ < 1 of the form Δ = Δ + Δ, with 0 ≤ Δ and Δ = (1 − 1/mi )Δi is a sum of prime divisors Δi, whose coefficients satisfy mi ∈ Z>1
The local fundamental group of a normal singularity x ∈ X is π1loc(X, x) := π1(B\x) = π1(Link(x)), where B is the intersection of X with a small euclidean ball around x and the link Link(x) is the boundary ∂ B
Summary
We work over the field C of complex numbers. A log pair is an algebraic variety X together with a boundary divisor 0 ≤ Δ < 1 of the form Δ = Δ + Δ , with 0 ≤ Δ and Δ = (1 − 1/mi )Δi is a sum of prime divisors Δi , whose coefficients satisfy mi ∈ Z>1. The proof of connectedness of weakly Fano varieties X of Takayama [56] manages to avoid the L2-index theorem and instead relies on the so called Γ -reduction or Shafarevich map, independently constructed by Campana and Kollár in [14] and [42] for compact Kähler manifolds and normal proper varieties respectively Said, it parameterizes maximal subvarieties of X with finite fundamental group. Consider a log resolution f : X → Y of the n-dimensional weakly Fano pair (Y, D + D ) with exceptional prime divisors Ei. a very small loop γi around a general point ei of Ei can be pushed forward to Ysm and there it lies in the smooth locus of a very small neighbourhood of the image of ei , which is a klt singularity.
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