It seems self-evident that there is a significant gap between secondary school and university mathematics, even though the gap may take different forms, which vary with different education systems in different places and at different times. This paper attempts to capture some common forms of this gap, and in particular, it discusses some core factors that are directly related to the nature of mathematics. Further, it also attempts to find ways to bridge the gap more easily and surely for students.
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.
The purpose of this paper is to give a counterexample of Theorem 10.4 in [Ha-La]. In the Harvey-Lawson paper, a global result is claimed, but only a local result is proven. This theorem has had a big impact on CR geometry for almost a quarter of a century because one can use the theory of isolated singularities to study the theory of CR manifolds and vice versa.Example. Consider the following holomorphic map:Clearly for any c, F restricted on the line {v = c} is an embedding outside the two points (0, c) and (1, c). F sends (0, t) and (1, t) to (0, t, 0) for all t. Now take S, which is the boundary of a ball B = (u, v) ∈ C 2 : (u, v) ≤ 2 . It is easy to see that the mapping F restricted on S is still an embedding. The image of S under F is a strongly pseudoconvex CR manifold in C 3 . The variety that F (S) bounds is F (B). Observe that F (B) has curve singularities along the line (0, t, 0). We remark that F (C 2 ) is a hypersurface (x, y, z) ∈Theorem 10.4 of [Ha-La] was so powerful that it has been used by many researchers. Fortunately, we can replace it by the following theorem, the proof of which will appear elsewhere [Lu-Ya].Theorem. Let X be a strongly pseudoconvex CR manifold of dimension 2n − 1, n ≥ 2. If X is contained in the boundary of a bounded strictly pseudoconvex domain D in C N , then there exists a complex analytic subvariety V of dimension n in D − X such that the boundary of V is X. Moreover, V has boundary regularity at every point of X, and V has only isolated singularities in V |X.
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