Non-algebraic deformations of flat Kähler manifolds

2019/11/02 by Vasily Rogov, Rogov, Vasily · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1911.00798

openalex publication_date 2019/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let X be a compact Kähler manifold with vanishing Riemann curvature. We prove that there exists a manifold X', deformation equivalent to X, which is not an analytification of any projective variety, if and only if H⁰(X, Ω²) ≠ 0. Using this, we recover a recent theorem of Catanese and Demleitner, which states that a rigid smooth quotient of a complex torus is always projective. We also produce many examples of non-algebraic flat Kähler manifolds with vanishing first Betti number.

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• Hermitian geometrically formal manifolds

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