  Divergences, also known as contrast functions, are distance-like quantities
defined on manifolds of non-negative or probability measures. Using the duality
in optimal transport, we introduce and study the one-parameter family of
$L^{(\pm \alpha)}$-divergences. It includes the Bregman divergence
corresponding to the Euclidean quadratic cost, and the $L$-divergence
introduced by Pal and the author in connection with portfolio theory and a
logarithmic cost function. They admit natural generalizations of exponential
family that are closely related to the $\alpha$-family and $q$-exponential
family. In particular, the $L^{(\pm \alpha)}$-divergences of the corresponding
potential functions are R\'{e}nyi divergences. Using this unified framework we
prove that the induced geometries are dually projectively flat with constant
sectional curvatures, and a generalized Pythagorean theorem holds true.
Conversely, we show that if a statistical manifold is dually projectively flat
with constant curvature $\pm \alpha$ with $\alpha > 0$, then it is locally
induced by an $L^{(\mp \alpha)}$-divergence. We define in this context a
canonical divergence which extends the one for dually flat manifolds.
