时间：2026年4月1日（周三）14:00-16:00
地点：E14-116
主讲人：Antoine Sedillot, Universität Regensburg
讲座主题：(Arithmetic) Volumes in (arithmetic) geometry
讲座摘要: In algebraic geometry, the volume of a line bundle is a fundamental birational invariant that detects if tensor powers of the line bundle have "enough" global sections. This construction defines a volume function on the space of $\mathbb{R}$-divisors that enjoys nice properties (e.g., continuity, differentiability).
In this talk, we will deal with an arithmetic analogue of this volume function in the setting of adelic curves/globally valued fields. Roughly speaking, a globally valued field is a field equipped with a global height, namely, a measure of "arithmetic complexity." It is known that any such height (over a countable field $K$) can be represented by an adelic curve, namely, a measure on the space of absolute values on $K$. In this very general framework, we are able to define analogues of usual algebro-geometric tools (e.g., adelic line bundles, Arakelov degree, arithmetic volumes...) and height functions are obtained in analogy with the process used to define heights over a function field.
The problem of understanding the differentiability property of the arithmetic volume function in this context is intimately linked to the equidistribution question and is thus fundamental, although it is a hard problem. A possible approach is to obtain a so-called arithmetic Siu inequality, namely an inequality relating the arithmetic volume of a difference of ample adelic line bundles and some arithmetic intersection product.
In the first part of the talk, we will focus on the volume function in algebraic geometry and its counterpart over number fields. In the second part, we will give some progress towards an arithmetic Siu inequality.
This is joint work with Nuno Hultberg and Michal Swachniewicz.
