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 problem has been open for over 30 years. In this paper we introduce 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. In the case n = 2 and N = 3, the vanishing of this invariant is enough to give the interior regularity.Dedicated to Professor Blaine Lawson on the occasion of his 68 th Birthday.

In this paper, we introduce some new invariants for complex manifolds. These invariants measure in some sense how far the complex manifolds are away from having global complex coordinates. For applications, we introduce two new invariants f (1,1) and g (1,1) for isolated surface singularities. We show that f (1,1) = g (1,1) = 1 for rational double points and cyclic quotient singularities.Dedicated to Professor Michael Artin on the occasion of his 78th Birthday.

Abstract. 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 (Λ n 1) (X) and show that if g (Λ n 1) (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.

In this paper, we classified the surfaces whose canonical maps are abelian covers over P 2 . Moveover, we give defining equations for Perssson's surface and Tan's surfaces with odd canonical degrees explicitly.

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