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arXiv 1702.07883math.AGmath.DS
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环面簇的极化自同态刻画

Characterizations of Toric Varieties via Polarized Endomorphisms

Sheng Meng, De-Qi Zhang

备注：
Mathematische Zeitschrift (to appear); minor changes
发表信息：
Mathematische Zeitschrift, August 2019, Volume 292, Issue 3 - 4, pp 1223 - 1231
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摘要：

设$X$是正规投射簇，$f:X\to X$是非同构的极化自同态.我们给出了$X$是复曲面簇的两个刻画。首先证明了若$X$是$\mathbb{Q}$ -阶乘的且$G$ -几乎齐次的，且$f$是$G$ -等变的，则$X$是环面簇.其次，我们给出了一个几何刻画：若$X$是Fano型的，余维2光滑的，且存在一个$f^{-1}$不变的约化因子$D$使得$f|_{X\backslash D}$是拟tale的，$K_X+D$是$\mathbb{Q}$的，则$X$允许一个拟tale覆盖$\widetilde{X}$使得$\widetilde{X}$是复曲面簇，$f$提升到$\widetilde{X}$ .特别地，如果进一步假设$X$是光滑的，则$X$是复曲面簇。

英文摘要：

Let $X$ be a normal projective variety and $f:X\to X$ a non-isomorphic polarized endomorphism. We give two characterizations for $X$ to be a toric variety. First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for some linear algebraic group $G$ such that $f$ is $G$-equivariant, then $X$ is a toric variety. Next we give a geometric characterization: if $X$ is of Fano type and smooth in codimension 2 and if there is an $f^{-1}$-invariant reduced divisor $D$ such that $f|_{X\backslash D}$ is quasi-étale and $K_X+D$ is $\mathbb{Q}$-Cartier, then $X$ admits a quasi-étale cover $\widetilde{X}$ such that $\widetilde{X}$ is a toric variety and $f$ lifts to $\widetilde{X}$. In particular, if $X$ is further assumed to be smooth, then $X$ is a toric variety.

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