Non-algebraic deformations of flat Kähler manifolds

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 $H^0 (X , \Omega^2_X) \neq 0$. 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 Kähler manifolds with vanishing first Betti number.

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This study has been partially funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project ’5-100.

Received 25 March 2020

Accepted 13 September 2020

Published 23 February 2023