Bogomolov Decomposition and Compact K{\"a}hler Manifolds of Algebraic Dimension Zero
Algebraic Geometry
2026-05-20 v1
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.
Keywords
Cite
@article{arxiv.2605.19713,
title = {Bogomolov Decomposition and Compact K{\"a}hler Manifolds of Algebraic Dimension Zero},
author = {Frederic Bruno Campana},
journal= {arXiv preprint arXiv:2605.19713},
year = {2026}
}