Riemann θ函数和Zhang-Kawazumi不变量的退化及其在一致Bogomolov猜想中的应用
Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture
摘要：
本文研究了穿孔复单位圆上一类主极化阿贝尔簇的Riemann $θ$ -函数范数在相应的极化实环面上的退化行为.作为应用，我们得到了一类Riemann曲面$M_t$的Zhang-Kawazumi不变量$φ(M_t)$根据相应度量化约简图$Γ$的Zhang不变量$φ(Γ)$的退化行为.这使我们能够推导出一个一致的下界的本质最小值的Néron-Tate高度上的重言式循环的任何雅可比簇在数域上。
英文摘要：
In this paper we study the degeneration behavior of the norm of the Riemann $θ$-function in a family of principally polarized abelian varieties over the punctured complex unit disc in terms of the associated polarized real torus. As an application, we obtain the degeneration behavior of the Zhang--Kawazumi invariant $φ(M_t)$ of a family of Riemann surfaces $M_t$ in terms of Zhang's invariant $φ(Γ)$ of the associated metrized reduction graph $Γ$. This allows us to deduce a uniform lower bound for the essential minimum of the Néron-Tate height on the tautological cycles of any Jacobian variety over a number field.
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