A note on subvarieties of powers of OT-manifolds

TL;DR

This paper investigates the structure of OT-manifolds, showing that their finite-to-finite holomorphic correspondences are discrete and that their powers lack interesting complex-analytic subvarieties under certain conditions, using methods involving covers.

Contribution

It establishes discreteness of holomorphic correspondences on OT-manifolds and links the absence of proper infinite subsets to the non-existence of complex-analytic families in their powers.

Findings

Finite-to-finite holomorphic correspondences are discrete.

OT-manifolds with no proper infinite subsets have powers with no interesting subvarieties.

Methods involve studying finite unramified covers.

Abstract

It is shown that the space of finite-to-finite holomorphic correspondences on an OT-manifold is discrete. When the OT-manifold has no proper infinite complex-analytic subsets, it then follows by known model-theoretic results that its cartesian powers have no interesting complex-analytic families of subvarieties. The methods of proof, which are similar to [Moosa, Moraru, and Toma ``An essentially saturated surface not of K\"ahler-type", {\em Bull. of the LMS}, 40(5):845--854, 2008], require studying finite unramified covers of OT-manifolds.