Mathematical Research Letters

Volume 29 (2022)

Number 4

Non-algebraic deformations of flat Kähler manifolds

Pages: 1229 – 1250



Vasily Rogov (Faculty of Mathematics, NRU HSE, Moscow, Russia)


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.

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