The Lefschetz theorem for CR submanifolds and the nonexistence of real analytic Levi flat submanifolds

In this paper we study the topology of a CR submanifold with degenerate Levi form embedded in a compact Hermitian symmetric space. In the case that the ambient manifold is ℙ^v we prove a Lefschetz hyperplane theorem for such CR submanifolds. In addition we relate the homotopy groups of the ambient manifold with the homotopy groups of the CR submanifold within a range depending on the nullity of the Levi form. The optimal results are obtained for Levi flat submanifolds. In particular we show that a compact Levi flat submanifold in a compact Hermitian symmetric space of complex dimension v is simply connected if the real dimension of the Levi flat manifold is greater than v +1. We then apply this result to give a proof of the nonexistence of real analytic Levi flat submanifolds of real dimension greater than v +1. The proof of the Lefschetz theorem is based on the Morse theory of paths between a CR submanifold and a complex submanifold. The proof of the other restrictions on the topology of a CR submanifold is based on the Morse theory of paths between two CR submanifolds. The crucial ingredient in both proofs is a second variation calculation in Schoen-Wolfson [S-W], that goes back to Frankel [F].

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Published 1 January 2003