Mori–Reid–Tian conjecture on disjoint positive-genus curves on complex surfaces

Let $X$ be a smooth complex surface with $b_1(X)=0$ and $b_2(X)>1$. Suppose that $D_i$ are smooth, pairwise disjoint complex curves on $X$, each of positive genus $g_i>0$, and that their homology classes span $H_2(X,\mathbb{Q})$. Mori–Reid–Tian conjecture. No such smooth complex surface $X$ exists. This conjecture would rule out the complex-surface configuration needed to construct a semi-regular Sasakian structure on the associated Seifert bundle, and is relevant to the distinction between semi-regular K-contact and Sasakian structures. Its resolution is not established in the supplied text.