Nagoya Mathematical Journal

On holomorphic maps with only fold singularities

Yoshifumi Ando

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Abstract

Let $f:N\to P$ be a holomorphic map between $n$-dimensional complex manifolds which has only fold singularities. Such a map is called a holomorphic fold map. In the complex 2-jet space $J^2(n, n; \mathbf{C})$, let $\Omega^{10}$ denote the space consisting of all 2-jets of regular map germs and fold map germs. In this paper we prove that $\Omega^{10}$ is homotopy equivalent to SU$(n+1)$. By using this result we prove that if the tangent bundles $TN$ and $TP$ are equipped with SU$(n)$-structures in addition, then a holomorphic fold map $f$ canonically determines the homotopy class of an SU$(n+1)$-bundle map of $TN\oplus\theta_N$ to $TP\oplus\theta_P$, where $\theta_N$ and $\theta_P$ are the trivial line bundles.

Article information

Source
Nagoya Math. J., Volume 164 (2001), 147-184.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631659

Mathematical Reviews number (MathSciNet)
MR1869099

Zentralblatt MATH identifier
1037.58028

Subjects
Primary: 58K15: Topological properties of mappings
Secondary: 32S70: Other operations on singularities 57R45: Singularities of differentiable mappings

Citation

Ando, Yoshifumi. On holomorphic maps with only fold singularities. Nagoya Math. J. 164 (2001), 147--184. https://projecteuclid.org/euclid.nmj/1114631659


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