Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

Abstract

Let H ⊂ ℙn be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian G(n + 1, n). Assuming H has a global defining function, we prove H is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension 2n - 2 or dimension 2n - 4. If the singular set is of dimension 2n - 4, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ℙn with a meromorphic (rational of degree 1) first integral. In this case, H is in some sense simply a complex cone over an algebraic curve in ℙ1. Similarly if H has a degenerate singularity, then H is also algebraic. If the dimension of the singular set is 2n - 2 and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ℙ2 of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ℙ2. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.