  Guided by a spinning particle model with U(N)-extended supergravity on the
worldline we derive higher spin equations on complex manifolds. Their minimal
formulation is in term of gauge fields which satisfy suitable constraints. The
latter can be relaxed by introducing compensator fields. There is an
obstruction to define these systems on arbitrarily curved spaces, just as in
the usual theory of higher spin fields, but we show how to couple them to
Kaehler manifolds of constant holomorphic curvature. Quite interestingly, the
first class gauge algebra defining the U(N) particles on these manifolds is
quadratic and realizes the zero mode sector of certain nonlinear U(N)
superconformal algebras introduced sometimes ago by Bershadsky and Knizhnik in
2D.
