It's actually the shape style of the edge:
style = graph.getStylesheet().getDefaultEdgeStyle();
style[mxConstants.STYLE_CURVED] = '1';
Sets it for as the default for all edges.
Edge styles are really the positioning of the control points of the edge between the source and target. The curve isn't routing (this points positioning), it's just a styling through those points, thus it's not an edge style.