The compact four-manifolds that admit a Kahler metric with positive Ricci curvature have been classified in the 19th century: they come in 10 families. In analogy with conical Riemann surfaces (e.g., football, teardrop) and hyperbolic 3-folds with a cone singularity along a link appearing in Thurston's program, one may consider 4-folds with a Kahler metric having "edge singularities", namely admitting a 2-dimensional cone singularity transverse to an immersed minimal surface, a `complex edge'. What are all the pairs (4-fold, immersed surface) that admit a Kahler metric with positive Ricci curvature away from the edge? In joint work with I. Cheltsov we classify all such pairs under some assumptions. These now come in infinitely-many families and we then pose the "Calabi problem" for these pairs: when do they admit Kahler-Einstein edge metrics? This problem is far from being solved, even in this low dimension, but we report on some initial progress: some understanding of the non-existence part of the conjecture, as well as several existence results.