Polarized endomorphisms of normal projective threefolds in arbitrary characteristic
Polarized endomorphisms of normal projective threefolds in arbitrary characteristic
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DOI：
10.1007/s00208-019-01877-6
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发表时间：
2017-10
影响因子：
1.4
通讯作者：
P. Cascini;Sheng Meng;De-Qi Zhang
中科院分区：
文献类型：
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作者：
P. Cascini;Sheng Meng;De-Qi Zhang
设 X 是具有任意特征的代数闭域 k 上的射影簇。如果对于某个充足的卡地亚除数手整数，则 Xisq 极化的满射自同态。假设 f 是可分离的，并且 Xis-Gorenstein 和正规的。我们证明反正则除数在数值上等价于有效卡地亚除数，稍微强化了 Boucksom、de Fernex 和 Favre 的结论（Duke Math J 161(8):1455–1520, 2012, Theorem C），并且还涵盖了任意特征的代数闭域上的奇异簇。假设 f 可分离且 X 是正规的。我们证明X的Albanese态射是一个代数纤维空间，并发现X的Albanese态射和Picard变态上的偏振自同态，并且是伪有效的和-Cartier意味着是一个挠因子。让伽罗瓦闭包关闭。我们证明，如果 和 互质 ，则可以在将 f 替换为正幂之后，对温和奇异三重 X 运行最小模型程序 (MMP)f - 等变，并达到具有扭转正则除数的簇 Y（并且当 Y 是阿贝尔簇的准 eta 商时）。在此过程中，我们证明，当 X 是一个光滑且有理链连接的维度最多为 3 的射影簇时，幂 f 表现为 X（模扭转）的 Neron-Severi 群上的标量乘法。
LetXbe a projective variety over an algebraically closed fieldkof arbitrary characteristic. A surjective endomorphismfofXisq-polarized iffor some ample Cartier divisorHand integer. Supposefis separable andXis-Gorenstein and normal. We show that the anti-canonical divisoris numerically equivalent to an effective-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Duke Math J 161(8):1455–1520, 2012, Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. Supposefis separable andXis normal. We show that the Albanese morphism ofXis an algebraic fibre space andfinduces polarized endomorphisms on the Albanese and also the Picard variety ofX, andbeing pseudo-effective and-Cartier means being a torsion-divisor. Letbe the Galois closure off. We show that ifand co-prime tothen one can run the minimal model program (MMP)f-equivariantly, after replacingfby a positive power, for a mildly singular threefoldXand reach a varietyYwith torsion canonical divisor (and also withYbeing a quasi-étale quotient of an abelian variety when). Along the way, we show that a power offacts as a scalar multiplication on the Neron-Severi group ofX(modulo torsion) whenXis a smooth and rationally chain connected projective variety of dimension at most three.