构造高度对称的紧流形与代数簇
Constructing Highly Symmetric Compact Manifolds and Algebraic Varieties
AI总结：
本文为每个代数闭域和自然数构造了代数簇与紧流形，使其自同构群包含所有满足特定条件的有限幂零群，并肯定回答了Mundet i Riera的问题。
摘要：
对于每个代数闭域$k$和自然数$r$，我们构造了若干（在$k$上的）代数簇，其双有理自同构群包含所有阶与$k$的特征互素、类至多为$2$且秩至多为$r$的有限幂零群。在特征$0$中，此构造是精确的，即至多相差有界扩张，命题中的群集合不能被更大的集合替换。利用类似的主要思想（但技术细节不同），对于每个$r$，我们构造了若干紧流形，其微分同胚群包含所有类至多为$2$且秩至多为$r$的有限幂零群。此结果肯定地回答了Mundet i Riera的一个问题，并推测在至多相差有界扩张的意义下是精确的。
英文摘要：
For every algebraically closed field $k$ and natural number $r$, we construct several algebraic varieties (over $k$) whose birational automorphism group contains every finite nilpotent group of class at most $2$, rank at most $r$ whose order is coprime to the characteristic of $k$. This construction is sharp in characteristic $0$, i.e. up to bounded extension, the set of groups from the statement cannot be replaced by a larger one. Using similar main ideas (with different technical details), for every $r$, we construct several compact manifolds whose diffeomorphism groups contain every finite nilpotent group of class at most $2$, rank at most $r$. This result answers a question of Mundet~i~Riera affirmatively and is conjecturally sharp up to bounded extension.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
