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 $L^{(\pm α)}$-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 $α$-family and $q$-exponential family. In particular, the $L^{(\pm α)}$-divergences of the corresponding potential functions are Rényi 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 $\pm α$ with $α> 0$, then it is locally induced by an $L^{(\mp α)}$-divergence. We define in this context a canonical divergence which extends the one for dually flat manifolds.