## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Complex Analysis in Several Variables — Memorial Conference of Kiyoshi Oka's Centennial Birthday, Kyoto/Nara 2001, K. Miyajima, M. Furushima, H. Kazama, A. Kodama, J. Noguchi, T. Ohsawa, H. Tsuji and T. Ueda, eds. (Tokyo: Mathematical Society of Japan, 2004), 37 - 44

### 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.

#### Article information

**Dates**

Received: 1 April 2002

First available in Project Euclid:
3 January 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1546542836

**Digital Object Identifier**

doi:10.2969/aspm/04210037

**Mathematical Reviews number (MathSciNet)**

MR2087037

**Zentralblatt MATH identifier**

1076.32027

#### Citation

Akahori, Takao. On the middle dimension cohomology of $A_l$ singularity. Complex Analysis in Several Variables — Memorial Conference of Kiyoshi Oka's Centennial Birthday, Kyoto/Nara 2001, 37--44, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/04210037. https://projecteuclid.org/euclid.aspm/1546542836