In a finite quantum state space ${\cal D}$ with a monotone metric, a
family  of torsion-free affine connections is introduced, in analogy
with the classical $\alpha$-connections defined by Amari
\cite{amari}. The dual connections with respect to the metric are
found and it is shown that these are, in general, not torsion-free.
In ${\cal D}$, curvature and some statistical interpretations are
treated. Finaly, as in \cite{naga}, geodesics are used to define a
divergence function.
