Bogomolov Decomposition and Compact K{\"a}hler Manifolds of Algebraic Dimension Zero

TL;DR

This paper investigates the structure of compact Kähler manifolds with algebraic dimension zero, showing they are closely related to products of Kummer and simple manifolds, with specific results in four dimensions.

Contribution

It establishes a conditional classification of such manifolds, proving that four-dimensional strictly simple cases are either quotients of tori or holomorphically symplectic.

Findings

Compact Kähler manifolds of algebraic dimension zero are isogenous to products of Kummer and simple manifolds.

Four-dimensional strictly simple manifolds are either étale quotients of tori or holomorphically symplectic.

Conditional results depend on conjectural properties of simple manifolds.

Abstract

We prove conditionally that compact K\''ahler manifolds of algebraic dimension zero are (essentially) isogeneous to products of Kummer and `simple' ones, the latter being conjecturally bimeromorphically symplectic. `Simple' means: its general point is not contained in a nontrivial subvariety. We also prove that four-dimensional `strictly simple' manifolds are either \'etale quotients of tori or holomorphically symplectic. `Strictly simple' means: its only subvarieties are points and itself.