Methods to compute ring invariants and applications: a new class of exotic threefolds

TL;DR

This paper introduces new methods for computing invariants of certain algebraic structures, enabling the distinction of exotic threefolds that are topologically but not algebraically equivalent to standard complex three-space.

Contribution

It develops tools to compute and distinguish invariants of k-domains, leading to the construction and analysis of new exotic threefolds that are topologically but not algebraically isomorphic to C^3.

Findings

Introduced exponential chains to analyze modifications of k-domains.

Constructed exotic C3 threefolds that are diffeomorphic but not isomorphic to standard C3.

Proved that these exotic threefolds are not isomorphic to any Russell C-domain.

Abstract

We develop some methods to compute the Makar-Limanov and Derksen invariants, isomorphism classes and automorphism groups for k-domains B, which are constructed from certain Russell k-domains. We propose tools and techniques to distinguish between k-domains with the same Makar-Limanov and Derksen invariants. In particular, we introduce the exponential chain associated to certain modifications. We extract C-domains from the class B that have smooth contractible factorial Spec(B), which are diffeomorphic to R 6 but not isomorphic to C 3, that is, exotic C 3. We examine associated exponential chains to prove that exotic threefolds Spec(B) are not isomorphic to Spec(R), for any Russell C-domain R.