Minkowski valuations in a 2-dimensional complex vector space

The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such valuation is given by the sum of a valuation of degree of homogeneity 1 and 3. In dimensions $m\ge 3$ such a classification was previously established and only valuations of a degree of homogeneity 2m-1 appear.