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 Post subject: algebraic varieties
PostPosted: Mon, 12 Dec 2011 08:07:52 UTC 
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Hi all,

Need a help to find 'when certain geomtries can be viewed as algebraic varieties and determine when their automorphism groups are algebraic groups'


Thanks


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 Post subject: Re: algebraic varieties
PostPosted: Mon, 12 Dec 2011 09:50:36 UTC 
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Rayan_Rajapakse wrote:
Hi all,

Need a help to find 'when certain geomtries can be viewed as algebraic varieties and determine when their automorphism groups are algebraic groups'


Thanks


First define what exactly you mean by "certain geomtries".

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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 Post subject: Re: algebraic varieties
PostPosted: Wed, 14 Dec 2011 04:56:48 UTC 
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Yea I want to know about what type of geometries can be viewed as algebraic varieties .


Thanks.
Kanchana


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 Post subject: Re: algebraic varieties
PostPosted: Wed, 14 Dec 2011 05:11:46 UTC 
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Rayan_Rajapakse wrote:
Yea I want to know about what type of geometries can be viewed as algebraic varieties .


Thanks.
Kanchana


All real manifolds are algebraic varieties if that's what you mean.

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 Post subject: Re: algebraic varieties
PostPosted: Wed, 14 Dec 2011 06:28:08 UTC 
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Joined: Tue, 1 Nov 2011 05:23:23 UTC
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Thanks for your reply.

May I have brief introduction or any source of this (what type of geomtries can be viewed as algebraic varieties and determine when their automorphism groups are algebraic groups).


Kanchana


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 Post subject: Re: algebraic varieties
PostPosted: Wed, 14 Dec 2011 06:31:59 UTC 
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Rayan_Rajapakse wrote:
Thanks for your reply.

May I have brief introduction or any source of this (what type of geomtries can be viewed as algebraic varieties and determine when their automorphism groups are algebraic groups).


Kanchana


I believe it was Nash that originally proved this in his paper "Real Algebraic Manifolds" which should be available on JSTOR for example.

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