What question did this study set out to answer?

The study aims to construct algebraic varieties and compact manifolds whose automorphism and diffeomorphism groups exhibit specified properties related to nilpotent groups.

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March 21, 2026

Constructing highly symmetric compact manifolds and algebraic varieties

Key Points

The study aims to construct algebraic varieties and compact manifolds whose automorphism and diffeomorphism groups exhibit specified properties related to nilpotent groups.
Analyzed algebraically closed fields and natural numbers for constructions.
Developed models for algebraic varieties with specified birational automorphism groups.
Constructed compact manifolds whose diffeomorphism groups match given criteria.
Established constructions for algebraic varieties with birational automorphism groups containing finite nilpotent groups.
Showed similar constructions exist for compact manifolds with specified diffeomorphism groups.
Confirmed the sharpness of the results in characteristic zero.

Abstract

For every algebraically closed field k k and natural number r r , we construct several algebraic varieties (over k k ) whose birational automorphism group contains every finite nilpotent group of class at most 2 2 , rank at most r r whose order is coprime to the characteristic of k k . This construction is sharp in characteristic 0 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 r , we construct several compact manifolds whose diffeomorphism groups contain every finite nilpotent group of class at most 2 2 , rank at most r r . This result answers a question of Mundet i Riera affirmatively and is conjecturally sharp up to bounded extension.

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