The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

Abstract

AbstractLetXbe a normal compact Kähler space with klt singularities and torsion canonical bundle. We show thatXadmits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: ifXhas toroidal singularities, ifXhas finite quotient singularities and if the cohomology group${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$vanishes.