Multiplications of Maximal Rank in the Cohomology of P¹times P ¹

We show that the linear map defined by multiplication with a general bi-homogeneous form between two bi-graduated pieces of the first cohomology of a nonsingular quadric in the projective space is of maximal rank. This is the first non trivial case of a more general open problem on natural linear maps between vector spaces of tensors defined in terms of multiplications and contractions. An interpretation in terms of bi-homogeneous linear differential operators with polynomial coefficients is also given.