Attempting to synchronize local edits containing true curves from ArcMap to ArcGIS Online displays the following error message:
"The local edits were not synchronized to the server. Cannot convert polylines or polygons with curves to JSON format."
This is a known limitation. The error occurs if the edits contain true curves; a true curve is a curve that only has two vertices as endpoints. Currently, ArcGIS Online does not support true curves created in the geometry of lines and polygons.