Microlocalization and Nonexistence of C 2 Levi-flat Hypersurfaces in Cp 2

Theorem. There exist no C Levi-flat real hypersurfaces in CP . This improves an earlier result of Siu [Si2] where C smoothness is required. For the nonexistence of Levi-flat hypersurfaces in CP with n ≥ 3, Lins-Neto [LN] first proved the nonexistence of real-analytic hypersurfaces in CP. Nonexistence of C Levi-flat hypersurfaces in CP was proved for n ≥ 3 by Siu [Si1]. It is proved in a recent paper [CS] that there exist no Lipschitz Levi-flat hypersurfaces in CP with n ≥ 3. In an earlier paper by Cao-Shaw-Wang [CSW], it is stated that there exist no C Levi-flat real hypersurfaces in CP with n ≥ 2, but the proof only works for n ≥ 3. While we are still unable to justify the estimate (4.18) which is necessary in the approach of [CSW], we have been able to combine the method of [CS] with microlocalization to prove the above theorem. The proof follows the same outlines as in the earlier paper of [Si1]. Suppose that there exists a C Levi-flat real hypersurface M in CP . Then M is foliated by complex leaves locally. It is known that the curvature form Θ̃ of complex normal bundle of M is a positive definite (1, 1)-form when restricted to the complex leaves on M . If we can find a continuous real-valued solution h of M to the equation