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 ${\pi }_1^{alg}(X)$ is large, we show that the Seshadri constant $ϵ(p^{\ast }L)$ can be made arbitrarily large by passing to a finite 'etale cover $p:X^{\mathrm{\prime }}\to X$. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an analogous result when ${\pi }_1(X)$ 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 'etale cover $p: X’\rightarrow X$suchthattheSeshadriconstant$\epsilon(p^{}L; x)\geq N$forany$x\notin p^{}$\mathbf{\text{B}}${+}(L)=$\mathbf{\text{B}}${+}(p^{*}L)$,where$$\mathbf{\text{B}}$_{+}(L)$istheaugmentedbaselocusof$L$.