Non-algebraic Deformations of Flat Kahler Manifolds

Student: Rogov Vasilii

Educational Programme: Mathematics (Master)

Final Grade: 10

Year of Graduation: 2020

Let $X$ be a compact Kahler 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 $H^0(X, \Omega^2_X) \neq 0$. 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. Using the quaternionic double construction due to Verbitsky and Soldatenkov we produce many examples of non-algebraic flat K\"ahler manifolds with vanishing first Betti number. Finally, we show that algebraic reduction of a flat K\"ahler manifold can be modelled by a holomorphic submersion to flat K\"ahler orbifold and study the possible algebraic dimensions of flat K\"ahler manifolds.

Full text (added May 29, 2020)

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