On higher dimensional complex Plateau problem

Let X be a compact connected strongly pseudoconvex CR manifold of real dimension 2n-1 in C^N. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For n≥3 and N = n+1, Yau found a necessary and sufficient condition for the interior regularity of the Harvey–Lawson solution to the complex Plateau problem by means of Kohn–Rossi cohomology groups on X in 1981. For n=2 and N ≥ n+1, the first and third authors introduced a new CR invariant g^(1,1)(X) of X. The vanishing of this invariant will give the interior regularity of the Harvey–Lawson solution up to normalization. For n≥3 and N>n+1, the problem still remains open. In this paper, we generalize the invariant g^(1,1)(X) to higher dimension as g^(Λn1)(X) and show that if g^(Λn1)(X)=0, then the interior has at most finite number of rational singularities. In particular, if X is Calabi–Yau of real dimension 5, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization.