The spaces of Kähler and holomorphically tamed symplectic forms on closed 4-manifolds

This paper investigates the uniqueness, connectedness and openness properties of spaces of Kähler forms on closed $4$-manifolds, extending discussions in \cite{Li08,Sal13}. Motivated by the Streets–Tian conjecture concerning the existence of Kähler metrics on Hermitian-symplectic complex manifolds, we also study holomorphically tamed symplectic forms, which are symplectic forms tamed by some integrable complex structure. We formulate a parallel question and relate it to the corresponding questions for Kähler-type symplectic forms.