On the Topologies on ind-Varieties and related Irreducibility Questions

TL;DR

This paper compares two topologies on affine ind-varieties, showing they can differ in reducibility properties, and provides counter-examples to existing irreducibility criteria, proposing a new criterion.

Contribution

It identifies classes of ind-varieties where Shafarevich's and Kambayashi's topologies differ and introduces a new irreducibility criterion for affine ind-varieties.

Findings

Existence of affine ind-varieties reducible in Shafarevich's topology but irreducible in Kambayashi's topology

Counter-examples to existing irreducibility criteria

Proposal of a new irreducibility criterion

Abstract

In the literature there are two ways of endowing an affine ind-variety with a topology. One possibility is due to Shafarevich and the other to Kambayashi. In this paper we specify a large class of affine ind-varieties where these two topologies differ. We give an example of an affine ind-variety that is reducible with respect to Shafarevich's topology, but irreducible with respect to Kambayashi's topology. Moreover, we give a counter-example of a supposed irreducibility criterion of Shafarevich which is different from a counter-example given by Homma. We finish the paper with an irreducibility criterion similar to the one given by Shafarevich.