On the Classification of Vaisman Manifolds with Vanishing First Basic Chern Class and Large First Betti Number

We show that every Vaisman manifold with large first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold. Furthermore, its complex structure is left-invariant, the characteristic foliation is regular, and the associated fibration is given by the Albanese map. Under the additional assumption that the LCK rank is $1$, the Vaisman structure is also left-invariant. We further prove that if all basic harmonic $1$-forms have constant length, then the Vaisman manifold with large first Betti number is diffeomorphic to a Kodaira-Thurston manifold and its complex structure is the standard complex structure. Finally, we discuss the relationship of this condition with transverse geometric formality in this setting.