Some constraints on positive entropy automorphisms of smooth threefolds
Some constraints on positive entropy automorphisms of smooth threefolds
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DOI：
10.24033/asens.2380
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发表时间：
2015-03
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通讯作者：
John Lesieutre
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作者：
John Lesieutre
假设$X$是在$\mathbb C$上的光滑投影三倍体，并且$\phi: X \到X$是正熵的自同构。我们证明，在用迭代替换$\phi$后，下列条件之一必须成立：i) $X$的标准类在数值上是平凡的；Ii) $\phi$是非基元的；Iii) $\phi$不是动态最小的。因此，我们证明了如果一个光滑的三重$M$不承认正熵的原始自同构，那么由$M$的光滑膨胀序列构造的任何一个变量都不承认正熵的原始自同构。为了解释为什么该方法不适用于具有终端奇点的三倍，我们展示了一个非规则的，终端三倍$X$在$NE(X)$上具有无限多个$K_X$负极值射线。
Suppose that $X$ is a smooth, projective threefold over $\mathbb C$ and that $\phi : X \to X$ is an automorphism of positive entropy. We show that one of the following must hold, after replacing $\phi$ by an iterate: i) the canonical class of $X$ is numerically trivial; ii) $\phi$ is imprimitive; iii) $\phi$ is not dynamically minimal. As a consequence, we show that if a smooth threefold $M$ does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of $M$ can admit a primitive automorphism of positive entropy. In explaining why the method does not apply to threefolds with terminal singularities, we exhibit a non-uniruled, terminal threefold $X$ with infinitely many $K_X$-negative extremal rays on $NE(X)$.