On Seshadri constants of varieties with large fundamental group

Abstract

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 $\epsilon (p^{\ast }L)$ can be made arbitrarily large by passing to a finite étale cover $p:X^{\prime }\to X$. 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 étale cover $p:X^{\prime }\to X$ such that the Seshadri constant $\epsilon (p^{\ast }L;x)\ge N$ for any $x\notin p^{\ast }{\text{B}}_{+}(L)={\text{B}}_{+}(p^{\ast }L)$, where ${\text{B}}_{+}(L)$ is the augmented base locus of $L$.

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