Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism
Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism
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DOI：
10.4310/mrl.2020.v27.n2.a8
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发表时间：
2018-06
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影响因子：
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通讯作者：
Sheng Meng;De-Qi Zhang
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文献类型：
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作者：
Sheng Meng;De-Qi Zhang
假设$X$是一个具有极化（或者更一般地说，是被放大的）自同态的投影变体。我们证明：只有有限个可收缩的极值射线；当$X$为$\mathbb{Q}$-阶乘正态时，每个最小模型程序相对于所有满射自同态的一元$SEnd(X)$是等变的，直至有限指标。进一步，当$X$理性连接且光滑时，我们证明了$SEnd(X)$的有限指数子拟元$G$使得$G$通过回拉作为Neron-Severi群上的对角矩阵（因此是交换矩阵）；满自同构群$Aut(X)$具有有限多个连通分量；每一个放大的自同态都是内放大的。
Let $X$ be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $SEnd(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth, we show: there is a finite-index submonoid $G$ of $SEnd(X)$ such that $G$ acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group $Aut(X)$ has finitely many connected components; and every amplified endomorphism is int-amplified.