论文标题
在代数纤维空间的相对反典型分裂的渐近基基因座上
On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces
论文作者
论文摘要
In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $ϕ: X \to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers.本文包含三个主要结果：第一个结果表明，上述所有基础基因座都位于水平方向上，除非它们为空。第二个结果是campana的代数证明 - -Cao-Matsumura在Hacon上的平等 - $ \ rm {m^c} $ kernan的问题，其原始证明取决于分析方法。第三个结果部分解决了一个问题，该问题询问具有半样本相对反典型分裂的代数纤维空间实际上是通过基本变化的产品结构，其适当的有限étale覆盖率为$ y $。我们的证明是基于代数和分析方法，用于直接图像滑轮的阳性。
In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $ϕ: X \to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana--Cao--Matsumura's equality on Hacon--$\rm{M^c}$Kernan's question, whose original proof depends on analytics methods. The third result partially solves the question which asks whether algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of $Y$. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.
