Essential Minimum and Pseudo-effectivity in Arakelov Geometry
发布时间：2023年03月07日
浏览次数：2704
发布者: Biao Ma
主讲人： 曲炳钢（BICMR）
活动时间： 从 2023-03-13 15:00 到 16:30
场地： 北京国际数学研究中心，全斋全29教室
Let $\mathcal{X}/\mathbb{Z}$ be an arithmetic variety with generic fiber $X/\mathbb{Q}$, and let $\overline{\mathcal{L}}$ on $\mathcal{X}$ be a Hermitian line bundle. Then $\overline{\mathcal{L}}$ induces an Arakelov height function $h_{\overline{\mathcal{L}}}: X(\overline{\mathbb{Q}}) \longrightarrow \mathbb{R}$. It is natural to believe that the positivity of $h_{\overline{\mathcal{L}}}$ is correlated with the positivity of $\overline{\mathcal{L}}$. In fact, Yuan made a conjecture that $\operatorname{ess}(h_{\overline{\mathcal{L}}}) \geq 0$ $\Longleftrightarrow$ $\overline{\mathcal{L}}$ pseudo-effective. Joint with Yin Hang, we develop a new arithmetic Bertini-type theorem which we call ``arithmetic Demailly approximation''. As an application, we prove half of the above conjecture that $\operatorname{ess}(h_{\overline{\mathcal{L}}}) \geq 0$ $\Longrightarrow$ $\overline{\mathcal{L}}$ pseudo-effective.
