度量半正线丛的非阿基米德体积
Non-Archimedean volumes of metrized nef line bundles
AI总结：
本文将 ample 线丛上非阿基米德体积与能量一致性及可微性的已知结果推广至 nef 情形，基于对赋值环上有限表示挠模的研究证明了半正度量的相关性质。
摘要：
设 $L$ 是定义在具有非平凡赋值的非阿基米德域 $K$ 上的固有几何既约概型 $X$ 的线丛。粗略地说，$L$ 的 Berkovich 解析化上连续度量的非阿基米德体积衡量了 $L$ 的张量幂的小截面空间的渐近增长。对于 Zhang 意义下 $L$ 上的连续半正度量，我们首先证明非阿基米德体积与能量一致。此类半正度量的存在性导致 $L$ 是 nef 的。第二个结果是，非阿基米德体积在任意半正连续度量处是可微的。这些结果在 $L$ 是 ample 时是已知的，本文旨在将其推广至 nef 情形。该方法基于对 $K$ 的（可能非诺特）赋值环上有限表示挠模的内容和体积的详细研究。
英文摘要：
Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a non-trivially valued non-Archimedean field $K$. Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of $L$ measures the asymptotic growth of the space of small sections of tensor powers of $L$. For a continuous semipositive metric on $L$ in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that $L$ is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when $L$ is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of $K$.
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