Complex Gradient Systems

Let M be a complex manifold of complex dimension n+k. We say that the functions u1, . . . ,uk and the vector fields x1, . . .,xk on M form a complex gradient system if x1, . . . ,xk,Jx1, . . . ,Jxk are linearly independent at each point p ∈ eM and generate an integrable distribution of TM of dimension 2k and dua (xb ) = 0, dcua (xb ) = dab for a,b = 1, . . . ,k. We prove a Cauchy theorem for such complex gradient systems with initial data along a CR−submanifold of type (n,k). We also give a complete local characterization for the complex gradient systems which are holomorphic and abelian, which means that the vector fields xca = xa −iJxa , a = 1, . . . ,k are holomorphic and satisfy xca ,xcb = 0 for each a,b = 1, . . . ,k.