On Seshadri constants of varieties with large fundamental group

TL;DR

The paper proves that for varieties with large fundamental groups, the Seshadri constants of ample and big nef line bundles can be made arbitrarily large via finite étale covers, confirming a conjecture of Hwang.

Contribution

It establishes that large fundamental groups imply unbounded Seshadri constants on finite covers, extending to big and nef line bundles under topological assumptions.

Findings

Seshadri constants can be arbitrarily increased on finite covers for varieties with large fundamental groups.

The results affirm a conjecture of J.-M. Hwang regarding Seshadri constants.

Generalization to big and nef line bundles under residual finiteness of the fundamental group.

Abstract

Let $X$ be a smooth variety and let $L$ be an ample line bundle on $X$. If $\pi^{alg}_{1}(X)$ is large, we show that the Seshadri constant $\epsilon(p^{*}L)$ can be made arbitrarily large by passing to a finite \'etale cover $p:X'\rightarrow X$. 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 $p: X'\rightarrow X$ such that the Seshadri constant $\epsilon(p^{*}L; x)\geq N$ for any $x\notin p^{*}\textbf{B}_{+}(L)=\textbf{B}_{+}(p^{*}L)$, where $\textbf{B}_{+}(L)$ is the augmented base locus of $L$.