∂̄-free Maps Satisfy the Homotopy Principle

In this short note we show that the space of all ∂̄-free maps of any complex manifold Vc into Cn is always homotopically equivalent to the space of all sections of the corresponding bundle in the space of jets of smooth maps Vc → Cn. In particular, the space of all linear ordinary differential equations with complex-valued coefficients on an elliptic curve with the identical monodromy (defined as the conjugacy class in the corresponding loop group) is weakly homotopically equivalent to the space of all based maps of the curve in GLn(C). The proof is based on Gromov’s theory of convex integration, see e.g. [Gr,McD]. §0. Preliminaries and results A smooth map f : V → R is called free of order k if at any point x of a manifold V the vectors ∂f ∂ui (v); ∂ f ∂ui1∂ui2 (v); . . . ; ∂ f ∂ui1 ...∂uik (v) are linearly independent in R, where ui are some local coordinates in a neighborhood of x and v = f(x), see [Gr]. Obviously, q ≥ ( n+k k )