On Seshadri constants of varieties with large fundamental group

Let X be a smooth variety and let L be an ample line bundle on X. If π^alg ₁ (X) is large, we show that the Seshadri constant ϵ(p^∗L) can be made arbitrarily large by passing to a finite étale cover p : X^' → X. This result answers affirmatively to a conjecture of J.-M. Hwang. Moreover, we prove an analogous result when π₁(X) is large and residually finite. Finally, under the same topological assumptions, we appropriately generalize these results to the case of big and nef line bundles. More precisely, given a big and nef line bundle L on X and a positive number N > 0, we show that there exists a finite étale cover p : X^' → X such that the Seshadri constant ϵ(p^∗L; x) ≥ N for any x ϵ p^-1B₊(L) = B+(p^∗L), where B₊(L) is the augmented base locus of L.