Subject: Re: Some volume calculations [for toplist] Date: Wed, 7 Mar 2001 21:17:17 -0800 From: Greg Kuperberg The following question from Doug Ravenel last year has been turning over in my mind lately: > >From Kuperberg's formula, we see that the volume of $B^{2n}$, the > $2n$-dimensional unit ball, is $\pi^n/n!$. It turns out that > $CP^n$ (complex projective $n$-space) has the same volume. This > is interesting for two reason: > > 1. $CP^n$ is a topological quotient of $B^{2n}$ obtained > by collapsing the boundary of the latter to $CP^{n-1}$. Thus the > collapsing map $B^{2n} \to CP^n$ preserves volume. WHY IS THAT? At the time I couldn't think of a decent volume-preserving map from B^{2n} to CP^n, but now I think I have one. If we think of B^{2n} as a subset of C^n, then it is a symplectic manifold and it also has an action of (S^1)^n given by coordinatewise multiplication by phases: (z_1,...,z_n) -> (e^{i t_1} z_1,e^{i t_2} z_2, ... , e^{i t_n} z_n) Since this is a symplectic group action, it means that there is a moment map mu from B^{2n} to R^n interpreted as the Lie algebra of (S^1)^n. As it happens the image of mu is the simplex x_1,...,x_n >= 0, x_1 + ... + x_n <= 1/2 Now moment maps in general preserve volume up to a factor of the volume of the Lie group acting. In this case the volume of the Lie group is (2*pi)^n and the volume of the simplex is 1/2^n/n!. (The latter is an application of the rule in R^n that the volume of a pyramid is 1/n times base times height.) So the moment map gives us a nice geometric interpretation of the formula pi^n/n! for the volume of B^{2n}. CP^n also admits a very similar (S^1)^n group action coming from its structure as a toric variety. I think that once again the image of the moment map is a simplex. Depending on conventions you can make it the same simplex as the one for B^{2n}. Finally I think that it is possible to make a collapsing map B^{2n} -> CP^n that is equivariant with respect to the group action and that makes a commutative triangle with the moment maps. If all of this is actually true, it would provide a symplectic explanation of Doug Ravenel's question. My inspiration for this construction, or proposal for a construction, is the classical fact in R^3 that if you project uniform measure on the unit 2-sphere S^2 onto the z axis, you get uniform measure on the interval [-1,1]. This projection is the moment map of the action of rotation of S^2 around the z axis. We can identify unit S^2 with Fubini-Study CP^1 at the expense of changing distances by a factor of 2 and area by a factor of 4. The counterpart for B^2 is also easy to describe if more routine. The moment map of rotation of B^2 is the function r^2/2; obviously it takes uniform measure on B^2 to uniform measure on [0,1/2]. -- /\ Greg Kuperberg (UC Davis) / \ \ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ \/ * All the math that's fit to e-print *