论文标题
一个猜想加强了动态学位的扎里斯基密集轨道问题
A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one
论文作者
论文摘要
我们为动态学位的生育图制定了Zariski致密轨道猜想的加强。因此，考虑到在代数封闭的$ k $ $ 0 $ 0 $上定义的quasiproigntive品种$ x $，并带有一个男子式的自动图$ neftraction self-triendation selftical $ 1 $ $ 1 $，我们期望是否存在非稳定理性函数$ f：x \ dashrightArrow \ dashrightArrow \ Mathbb \ Mathbb {p} $ f。子变量$ y \ subset x $带有该属性，对于任何不变的subvariety $ z \ subset x $，我们都有$ z \ subseteq y $。我们证明了我们对自动形态的猜想$ 1 $ 1 $的semiabelian品种$ x $。 Also, we prove a related result for regular dominant self-maps $ϕ$ of semiabelian varieties $X$: assuming $ϕ$ does not preserve a non-constant rational function, we have that the dynamical degree of $ϕ$ is larger than $1$ if and only if the union of all $ϕ$-invariant proper subvarieties of $X$ is Zariski dense.我们将结果应用于表示与Abelian品种相关的扭曲均匀坐标环的表示理论问题。
We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a birational self-map $ϕ$ of dynamical degree $1$, we expect that either there exists a non-constant rational function $f:X\dashrightarrow \mathbb{P}^1$ such that $f\circ ϕ=f$, or there exists a proper subvariety $Y\subset X$ with the property that for any invariant proper subvariety $Z\subset X$, we have that $Z\subseteq Y$. We prove our conjecture for automorphisms $ϕ$ of dynamical degree $1$ of semiabelian varieties $X$. Also, we prove a related result for regular dominant self-maps $ϕ$ of semiabelian varieties $X$: assuming $ϕ$ does not preserve a non-constant rational function, we have that the dynamical degree of $ϕ$ is larger than $1$ if and only if the union of all $ϕ$-invariant proper subvarieties of $X$ is Zariski dense. We give applications of our results to representation theoretic questions about twisted homogeneous coordinate rings associated to abelian varieties.
