Are copulas unimodal?

Three types of unimodality (central convex, block, and star) are considered and the corresponding sets of unimodal copulas determined. Examples of star unimodal copulas, absolutely continuous, with a nonnull singular part, and even singular, are given. Necessary and sufficient conditions for a diagonal to be the diagonal section of a star unimodal copula are also indicated. Attention is also paid to the Archimedean case.