Subject: Re: volume, revisited Date: Mon, 12 Mar 2001 01:45:28 +0200 (IST) From: Yael Karshon Greg's explanation is totally correct: the volumes of B^{2n} and CP^n are equal to the Euclidean volumes of their moment map images, times (2 pi)^n. (This is because the torus acting is of half the dimension of the manifold). Both of these images are the tetrahedron X1 + ... + Xn <= 1/2 , X1 >= 0 , ... , Xn >= 0. Concretely, the symplectic embedding of B^{2n} in CP^n as an open dense subset is obtained by sending z to [w_0,...,w_n] , where w_0 = sqrt(1 - |z_1|^2 - ... - |z_n|^2) , w_1 = z_1 , ... , w_n = z_n. (This embedding, without moment maps, can be found in [McD, 2.7.1].) Densely embedding B^{2n} in CP^n is a special case of the "symplectic packing problem". ("How many symplectic balls of given radius can be disjointly embedded into a given symplectic manifold M?"). Whereas there are lots of deep "hard" results, concrete constructions typically use "soft" results. A simple "soft" result for M=CP^n can be found in my Appendix to [McD,Pol]. Since then, Lisa Traynor has obtained tons of more general beautiful constructions. The procedure of obtaining CP^n from C^n is a special case of Eugene Lerman's "symplectic cutting" [Ler]. Namely, let S^1 act on a symplectic manifold M with moment map Phi:M --> R , and let alpha be a regular value. Phi is S^1-invariant. The set { Phi <= alpha } is a manifold with boundary. Consider the space obtained from it by dividing the boundary by the S^1 action. Theorem: this space is naturally a symplectic orbifold. Special case: CP^n is obtained by cutting M=C^n with respect to the diagonal S^1-action with moment map Phi(z) = \half (|z_1|^2 + ... + |z_n|^2) at the value alpha=1. The volumes of Grassmannians, Flag manifolds, etc., should be computable by similar means (although I don't know a reference). For each of these, the moment map pushes Liouville measure to a measure on R^n (the "Duistermaat Heckman measure") given by a function f times Lebesgue measure. The total volume is obtained by integrating f. The function f is in principal computable. Alternatively, for each of these manifolds, the Duistermaat-Heckman formula [DH] gives concrete meromorphic functions on C^n whose sum is a holomorphic function whose value at z=0 is equal to the volume of the manifold. The individual summands have poles at z=0. References: [DH]: Duistermaat and Heckman, "On the variation in the cohomology of the symplectic form of the reduced phase space", Invent. Math. 69 (1982), 259-269. [McD]: Dusa McDuff, "Remarks on the uniqueness of symplectic blowing up", in: Symplectic Geometry (ed. D. Salamon), p.157-168, London Math. Soc. Lecture Note Series, 192, Cambridge Univ. Press.] [McD,Pol]: McDuff+Polterovich, "Symplectic packings and algebraic geometry", Invent. Math. 115, 431-434 (1994). [Ler]: Eugene Lerman, "Symplectic Cuts", Math.Res.Lett. 2 (1995), no.3, 247-258. Yael. On Sun, 11 Mar 2001, Assaf Libman wrote: > > > Hi Yael, > > You might be interested in this. > > Asi > > ------------- Begin Forwarded Message ------------- > > Date: Thu, 08 Mar 2001 07:41:27 -0500 > From: Don Davis > X-Accept-Language: en > MIME-Version: 1.0 > To: dmd1@lehigh.edu > Subject: volume, revisited > Content-Transfer-Encoding: 7bit > > 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 tomake 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 isthe 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 * > > > ------------- End Forwarded Message ------------- > >