Calculate Brieskorn Manifold?

Question

I need show that Brieskorn Manifold is submanifold with dimension $2n-1$ and calculate specifically for $d=2$ and $n=1$

$W(d)=\lbrace (z_{0},z_{1},...,z_{n})\in \mathbb{C}^{n+1}\vert$ $ z_{0}^{d}+z_{1}^{2}+...+z_{n}^{2}=0$ and $\vert z_{0}\vert^{2}+\vert z_{1}\vert^{2}+...+\vert z_{n}\vert^{2}=2\rbrace$

The second cuestion I don't know what it means.

score 0 · Answer 1 · edited Apr 13 '17 at 12:21

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Hint: write $W(d)$ as inverse image of the regular value of a smooth function. See Inverse of regular value is a submanifold, Milnor's proof.

The second part is simply say what concrete set is

$$W(d)=\lbrace(z_{0},z_{1})\in\mathbb{C}^2\,\vert\, z_{0}^{2}+z_{1}^2=0\hbox{ and }\vert z_{0}\vert^{2}+\vert z_{1}\vert^{2}=2\rbrace.$$

edited Apr 13 '17 at 12:21

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answered Sep 29 '14 at 07:10

Martín-Blas Pérez Pinilla

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I don't know if I tell more about $W(d=2)$, it must be 1-manifold on $C^{2}\cong IR^{4}$ – Donyarley Sep 29 '14 at 07:43
What 1-manifold? $\Bbb R$? $S^1$? Another? – Martín-Blas Pérez Pinilla Sep 29 '14 at 07:48
A teacher told me that is a union of two $S^{1}$ – Donyarley Oct 01 '14 at 16:54
http://math.stackexchange.com/questions/1101325/proof-a-2n-1-compact-manifold – jimbo Jan 12 '15 at 21:17

Calculate Brieskorn Manifold?

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