  In this paper we prove the following theorem.
  Main Theorem. Let n >= 3 and m >= 3n/2 +7. Then there exists no C^m Levi-flat
real hypersurface M in P_n.
  The condition that M is Levi-flat means that when M is locally defined by the
vanishing of a C^m real-valued function f, at every point of M the restriction
of d d-bar f to the complex tangent space of M is identically zero.
  The case of the nonexistence of C^\infty Levi-flat real hypersurface in P_2
is motivated by problems in dynamical systems in P_2.
