Tuesday, July 13, 2010

Projectivizations and orthogonal groups

Few days ago, I went to a summer school in algebraic geometry in Peru, and ithere was one course in projectivization of some special types of curves and properties of them.

One of the examples was doing the projectivization of a general conic over $\mathbb{C}^n$ and finding some nice properties that the resulting curves have.

The idea of the projectivization of a curve, is to grab the domain of definition and then do its compactification, for instance, in the case of a conic in $\mathbb{C}^2$, the result is to have a conic in a sphere (Riemann sphere). Algebraically, the idea is to have a curve defined as the zero set of a polynomial, for example, $p(x,y)=ax^2+bxy+cy^2=0$ with $(a,b,c)\in\mathbb{C}^3$, and then to extend the domain where the parameters are defined, i.e. $[a:b:c]\in\mathbb{P}(\mathbb{C}^2)$.

The speaker was talking about the cases in dimension 2 and 3, where the proyective spaces are $\mathbb{P}(\mathbb{C}^2)$ and $\mathbb{P}(\mathbb{C}^5)$ , and she defined a nice realization of the later one as the set of $3\times 3$ complex symmetric matrices, i.e.


$\mapsto \begin{pmatrix} a_1 & a_2 & a_3\\ a_2 & a_4 & a_5 \\ a_3 & a_4 & a_6 \end{pmatrix}$

This reminded me of the good old $O(3)$ with some differences. First of all, we have complex entries in our matrix, and second, the entries must satisfy and extra condition, we identify set of parameters with a common factor ($A\sim B$ iff $A=\lambda B$ for some $\lambda\in\mathbb{C}^x$), that is, the matrix defines the same curve up to a scalar multiple, which means that we need to scale it in some sense.

This two things can be fixed by considering $SO(3)$ instead of the whole $O(3)$ and then, by doing its complexification, $SO(3)\times_{\mathbb{R}}\mathbb{C}$.

So, my natural conjecture was that

$\mathbb{P}(S^n_2)\sim SO(n)\otimes_{\mathbb{R}}\mathbb{C}$

where $S^n_2$ is the space of homogeneous polynomials with degree 2 in $n$ variables and the isomorphism is as complex manifolds.

After discussing with a couple of people, I found that there were reasons to believe that this is true, since some topological properties of both spaces matched, but so far, I haven't found yet a formal proof of this fact.

In the case of this being true, it would turn out to be a really interesting property, since the object on the right is a Lie group, being isomorphic will imply that one can define a group structure in the set of conics, which is rather an interesting fact.

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