On the middle dimension cohomology of $A_l$ singularity

Abstract

Let $(V, o)$ be a normal isolated singularity in a complex Euclidean space $(C^N, o)$. Let $M$ be the intersection of this singularity and the real hypersphere $S_{\epsilon}^{2N-1} (o)$, centered at the origin $o$ with an $\epsilon$ radius. Then, naturally, this link $M$ admits a CR structure, induced from $V$, and the deformation theory of this CR structures has been studied in [1], [2],[3]. Especially in [3], a particular subspace of the infinitesimal deformation space is found, and we propose to study the relation between this subspace and simultaneous deformation. We note that: if the canonical line bundle of the CR structure is trivial, then the infinitesimal space of the deformation of CR structures is a part of the middle dimension cohomology. And in this line, we conjecture that $Z^1$, introduced in [3], might be related to the simultaneous deformation of isolated singularity $(V, o)$ (see also [2]). We discuss this problem for $A_l$ singularities.