Levi geometry

The relationship between the Levi geometry of a submanifold of C” and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of C”, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in C". This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of C". In fact, we show that if S is a real analytic, generic, submanifold of C" (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then S is not extendible to any open set in C".