Non-Kaehler Deformations of Irreducible Holomorphic Symplectic Manifolds

Let $X$ be a compact hyperk\"ahler manifold. Consider a connected component of the Teichm\"uller space $\teich(X)$ of all complex structures on an underlying differentiable manifold of $X$. Complex structures of hyperk\"ahler type form a subspace $\teichk(X) \subset \teich(X)$; the points of $\overline{\teichk(X)}$ are called deformation limits of compact hyperk\"ahler manifolds. The purpose of the present paper is to study these deformation limits $X = \Xs_0$ assuming the existence of a holomorphic symplectic structure and $h^2(X, \mathscr{O}_X) = 1$. We prove that a holomorphic symplectic limit $X$ admits a so called strongly Gauduchon metric, and if $\Xs_s$ are projective then $X$ is bimeromorphic to a projective holomorphic symplectic manifold. Finally, we examine nesessary and sufficient conditions for these deformation limits to admit K\"ahler metrics.