Non-algebraic deformations of flat K\"ahler manifolds

TL;DR

This paper investigates the deformation properties of flat K"ahler manifolds, showing conditions under which they cannot be algebraized, and provides examples of non-algebraic flat K"ahler manifolds with specific topological features.

Contribution

It establishes a criterion linking the existence of certain deformations to the presence of holomorphic 2-forms, and recovers known results about projectivity of quotients of complex tori.

Findings

Existence of non-algebraic deformations when H^0(X, Ω^2) ≠ 0.

Recovery of Catanese and Demleitner's theorem on projective quotients of complex tori.

Construction of examples of non-algebraic flat K"ahler manifolds with vanishing first Betti number.

Abstract

Let $X$ be a compact K\"ahler 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 $H^0(X, \Omega^2) \neq 0$. 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 K\"ahler manifolds with vanishing first Betti number.