Hing Sun Luk

Counterexample to boundary regularity of a strongly pseudoconvex CR submanifold: An addendum to the paper of Harvey-Lawson

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) ∈ C2 : ∥∥∥(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 C3. 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 (C2) is a hypersurface { (x, y, z) ∈ C3 : z2 − zx− x3 = 0 } in C3.

Explicit construction of graph invariant for strongly pseudoconvex compact 3-dimensional rational CR manifolds

Let X be a strongly pseudoconvex compact 3-dimensional CR manifolds which bounds a complex variety with isolated singularities in some CN. The weighted dual graph of the exceptional set of the minimal good resolution of V is a CR invariant of X; in case X has a tranversal holomorphic S1 action, we show that it is a complete topological invariant of except for two special cases. When X is a rational CR manifolds, we give explicit algebraic algorithms to compute the graph invariant in terms of the ring structure of ⊕k=0∞ mk/mk+1, where m is the maximal ideal of each singularity. An example is computed explicitly to demonstrate how the algorithms work.