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 *
>
>
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>
>