Non-algebraic deformations of flat Kähler manifolds

Abstract

Let $X$ be a compact Kähler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X^{\prime }$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X,{\Omega }^2)\ne 0$. 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.

14 pages; minor changes compared to v.1 and v.2

Cited by in corpus (1)

• Rigid-analytic varieties with projective reduction violating Hodge symmetry