Biharmonic holomorphic maps and conformally Kahler geometry
  We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions are shown to greatly simplify on l.c.K. manifolds and construction methods and examples are given in all dimensions. 
M. Benyounes, E. Loubeau, R. Slobodeanu
math.DG