论文标题
关于形态算术程度的可靠性
On the realizability of arithmetic degrees of morphisms
论文作者
论文摘要
Kawaguchi-Silverman的猜想将圆形内态性的两个不同不变性（动力学和算术程度）相关联。由于野牛 - 伊尔弗曼的猜想只有在形态具有zariski密集轨道时才有意义，因此它没有阳性kodaira尺寸的品种的满足。在正kodaira尺寸中有意义的Kawaguchi-Silverman猜想的概括是所谓的沙子猜想，它涉及一组“小”算术程度。 Kawaguchi和Silverman表明，一个小的算术度是$ f^*\ colon n^1（x）\ rightarrow n^1（x）$的特征值的模量。在本文中，我们研究了哪些可能的特征值作为算术程度。我们表明，阿贝尔品种的滤清性内态可能具有算术学的特征值。相反，我们表明，使用最小模型程序的曲折化型内态性质的每个特征值都是算术程度。最后，我们研究了如何将最小模型程序应用于接受插入内态的品种的这个可实现性问题。
The Kawaguchi-Silverman conjecture relates two different invariants of a surjective endomorphism, the dynamical and arithmetic degrees. As the Kawaguchi-Silverman conjecture is only meaningful when a morphism has a Zariski dense orbit, it has no content for varieties with positive Kodaira dimension. A generalization of the Kawaguchi-Silverman conjecture which is meaningful in positive Kodaira dimension is the so called sAND conjecture, which involves the set of "small" arithmetic degrees. Kawaguchi and Silverman showed that a small arithmetic degree is the modulus of an eigenvalue of $f^*\colon N^1(X)\rightarrow N^1(X)$. In this article we investigate which possible eigenvalues arise as an arithmetic degree. We show that surjective endomorphisms of abelian varieties may have eigenvalues which are not arithmetic degrees. Conversely, we show that every eigenvalue of a surjective endomorphism of a toric variety is an arithmetic degree using the minimal model program. Finally, we investigate how the minimal model program may be applied to study this realizability question for varieties that admit an int-amplified endomorphism.
