Polygon spaces and grassmannians

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(1)Article. Polygon spaces and grassmannians. HAUSMANN, Jean-Claude, KNUTSON, Allen. Reference HAUSMANN, Jean-Claude, KNUTSON, Allen. Polygon spaces and grassmannians. Enseignement mathématique, 1997, vol. 43, no. 1/2, p. 173-198. Available at: http://archive-ouverte.unige.ch/unige:12820 Disclaimer: layout of this document may differ from the published version..

(2) L'Enseignement Mathématique. Hausmann, Jean-Claude / Knutson, Allen. POLYGON SPACES AND GRASSMANNIANS. Persistenter Link: http://dx.doi.org/10.5169/seals-63276 L'Enseignement Mathématique, Vol.43 (1997). PDF erstellt am: 30 nov. 2010. Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung des Konsortiums der Schweizer Hochschulbibliotheken möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag.. SEALS Ein Dienst des Konsortiums der Schweizer Hochschulbibliotheken c/o ETH-Bibliothek, Rämistrasse 101, 8092 Zürich, Schweiz [email protected] http://retro.seals.ch.

(3) POLYGON SPACES AND GRASSMANNIANS by Jean-Claude. Hausmann. Allen Knutson. and. study the moduli spaces of polygons in R^ and R\ identify ingthem with subquotients of 2-Grassmannians using a symplectic version of the Gel'fand-MacPherson correspondance. We show that the bending flows defined by Kapovich-Millson arise as a réduction of the Gel'fand-Cetlin System on the Grassman. Abstract.. We. détermine the pentagon and hexagon spaces up to equivariant symplectomorphism. Other than invocation of Delzant's theorem, our proofs are purely. nian,and with. thèse. in nature.. polygon-theoretic. Introduction. 1 .. Let. m. V. homotheties structures a. :. k. be. the. (précise an action. function £:. m. V. k. of. space. ra-gons in. définitions of. —. O(k) R. >. ;. ". pis. (once the perimeter of. ,. in an. §2).. action. R. k. up. This of. S. m. translation and positive space cornes with several to. permuting the edges, and. polygon pto the lengths of its edges k fixed). The quotients of m V by SO^ (or O^) k m V ). Fixing a reflection in o{k) (respectively,. taking. a. moduli spaces V\ k k m provides an involution on m V and p + whose fixed point sets are in/ p - and twpk-i goal of this paper is to understand the topology of thèse various spaces and the géométrie structures that they naturally carry when k — 2 or 3 They are closely related to more familiar objects (Grassmannians, projective k l m V (a) := (a) of polygons with spaces, Hopf bundles, etc.) The spaces m. are the. k. {. .. £~l(a)~. given side-lengths. aG. R. ;. ". are of. particular interest.. k=3,. 3 The great miracle occurs when because R3R is isomorphic to the 3 is Kâhler. space /H of pure imaginary quaternions, and the 2-sphere in R3R The tools of symplectic geometry can then be used. Most prominent is a. ). Both authors thank the Fonds National Suisse de la Recherche Scientifique for its support..

(4) symplectic version of the Gel'fand-MacPherson correspondance identifying the spaces m V (a) as symplectic quotients of the Grassmannian of 2-planes in C m Earlier occurrences of symplectic geometry in the study of polygon spaces can be found in [Kl] and [KM2]. 3. .. While this paper illustrâtes many phenomena in symplectic geometry, the proofs are entirely polygon-theoretic and involve only classical differential topology. Nonetheless, many of the examples are new, interesting in their own right and instructive for both fields. Among our results 1.. :. The identification of the polygon. space. m. V. 3. with. G 2 (C. m. )/(U(l). m. ). intertwines complex conjugation on the complex Grassmannian (with fixed point set the real Grassmannian) and spatial reflection on the polygon moduli space (with fixed point set planar polygons). The fact that 3-dimensional and planar polygons hâve the same allowed values of £ is then an illustration of a theorem of Duistermaat ([Du]). (As is always true, and yet always mysterious,. helpful for studying the real case — hère planar polygons — the complex case — hère polygons in .). it is to. R3.)R. 2.. Identification of. the densely defined. to. extend. 3. "bending flows" ([Kl] and [KM2]). polygon spaces with the réduction of the Gel'fand-Cetlin System [GSl] the Grassmannian.. on the on. globally defined, and by Delzant's reconstruction theory the spaces are equivariantly symplectomorphic to toric varieties (for instance when m < 6, as noted in [KM2]). We give a précise description of the moment polytope and so explicitly identify the toric varieties. 3.. In some cases, the bending flows are. Contrary to the usual custom in be more natural to take symplectic manifold by the group, and to then Poisson space — the intermediate. symplectic réduction, it turned out hère to quotients by first quotienting the original pick out a symplectic leaf of the resulting quotient spaces ail hâve natural polygon. theoreticinterprétations. However, they are never complex; readers wishing more geometric-invariant-theoretic construction of thèse spaces should look [KM2]. This. paper. is. structured. as. follows.. Section. 2. gives. the. a. at. définitions and. elementary properties of polygon spaces. Sections 3 and 4 relate them to Grassmannians, and prove some facts about the moment map for the torus action on the Grassmannian by polygon-theoretic means. In section 4 is also. calculated the exact relation between the Kâhler structures in this paper and the ones in [KM2]. Section 5 relates the "bending flows" of [Kl] and [KM2] with the Gel'fand-Cetlin System on the Grassmannian. Section 6 uses this to.

(5) calculate the quadrilatéral, pentagon and hexagon spaces. Section. lists some. 7. open problems. The study of the polygon spaces will be pursued in a forthcoming paper [HK] in which we shall compute the cohomology ring of thèse spaces.. author was incited by Sylvain Cappell to introduce symplectic geometry in his study of polygon spaces. He is also grateful to Lisa Jeffrey and Michèle Audin for useful conversations. The two authors started this work The first. workshop in symplectic geometry organized in Cambridge by the Isaac Newton Institute (Fall 1994). The second author would like to thank Richard. at the. Montgomery for teaching him about dual pairs, and Michael Thaddeus for pointing out the link to moduli spaces of flat connections also the University of Geneva for its hospitality while this paper was being written. ;. The polygon spaces. 2.. Let. (2.1) m. jF(V). the. be. be. V. a. real. vector. space and m of ail maps p:. real. vector space that EJLiPO') = 0. An élément polygonal path in V. of m. steps,. polygon of. y;. and. a. the space. The m. on a m. V. k. .=.. m. as. a. m. Jr. (V). sides. We shall call an élément. J r (R. ~. G. k. be. V. regarded. to. (up. m. T(V). integer. V. —>. as. a. Let such. closed. translation) of. a. -polygon (in V/. We use the notation m T k for p G. an. m. ).. R+. {o}.. 0. ...,m}. {1,2,. will. configuration in. proper polygon when p(j) m. group. f{V). alternately,. or,. p. positive. a. of positive. The quotient. homotheties m. of m. V. acts. freely. and. properly then inherits. V(V) := T(V) ~ {0})/R + of smooth manifold diffeomorphic to a sphère. For instance, (mjrk = _ {0})/R + is diffeomorphic to the sphère S^'-D-i (. .. Suppose now that Vis oriented and is a Euclidean space, namely endowed with a scalar product. The group O(V) of isometries of V acts T m and m V(V) ;we deflne the moduli spaces. (2.2) V. on. is k. of m-polygons in V, up to similitude (where SO(V) is the identity component of O(V)). Observe that any orientation preserving isometry h: V Rk. produces identifications.

(6) We shall use the fact that thèse m. and thus. h. V(V)+. identified with. and. m. V\. identifications do not dépend on the choice of V(V), for any Euclidean space V, are canonically V. k .. The "degree of improperness" of polygons provides. (2.3). where. m. and. m. .. "open stratum" Ej m V(V) Viy) of dimension (j - \)k. The m m. stratification. a. V(V) -. EE. m. _i. m. V(V), open. -. EHE. ]li. V{V). if. 1. k. dense. and. =. in. -. polygons.. stratification. H{. is. submanifold. smooth. a. dimV. The m V(V), is the. of. open stratum space of proper. top. stratifications k {Ej m V+} and {Ej m V } of the moduli spaces (using the canonical identi ficationsof (2.2)). We shall see in §3 that the above stratifications describe the singular loci of smooth orbifold structures on the spaces m V(V), m p^_ this. As. and. m. V. is. O(V). it. projects. onto. k .. The map. (2.4) its total. perimeter. of radius. 2. S^FiV)). is. norm on. a. m. Jr. m. a. V(V). has. a. topological embedding i:. m. to. polygon. a. Si^TiV)). (V). We dénote by. for this norm. Each class in. which gives. which associâtes. := Y2jL\ \pU)\. \p\. i— ». p. p. the sphère. unique représentative in V(V) — m F(V) whose >. m m m image is S( F(V)) The image by i of E m^ V(V) is the subset of S( f(V)) where 5( m J r (V)) fails to be a smooth submanifold of m T(V). However, the .. m n V(V) is a smooth embedding. E/ V(V) - Ej^ The map £: m f(V) -> R m defined by £(p) :=J|p(l)|, |p(m)|) asso m ciâtesto a polygon its side-lengths. We define l: V(V) — R m by £:=loi.. restriction of. %. to. each. {. .. .. .. ,. >. We shall also use the notation £((p) for the. \p(ï)\. .. <9(V) -action and thus define maps (always. Thèse maps are invariant under. called £). which are smooth on each open stratum. (2.5). hyperplane m. Let 11. r: in. V. V.. —. V. One has. V(V) whose fixed-point. one has. >. be. the. orthogonal the involution p. space is naturally. mF{YI)m. F{Yl). reflection. m. some. \F{V) and m V(U). If h G SO(V),. p:= and. through. Top. on.

(7) does. to an. p. in. '"P*. Polygons. in. the. Observe that. =. p. V+. >. w. space of. t. The fixed point. not dépend on the choice of. k. m. involution (still denoted p i— p) on If t' is an orthogonal reflection with respect to another hyperplane IT the formula Top' = (t'ot)otop shows that the induced involution on Hence the involution descends. then. ,. m. m. is. .. k. V. V\ ~. x .. .. Examples (2.6) the. to. sphère. SS. m. ~2.~. 2. The. Une.. antipodal map and thus m V is a For example, V ~SI and V ~ RP x. {. 3. of antipodal points and thus side of length. one. E2E. X. V. 2. m. =. V\. is. 1 .. The stratum set of. a. EE. 3 2. 1. ). is. [. V. consists of. S^J 71. is. ). £:. Actually, the map. triangle.. a. diffeomorphic. is. the. "~. 7"7. 2 .. pairs. 3. points, the three triangles with. 3. This corresponds to the fact that. 0.. Oi\S( 3^. and. hexagon. l. 3VI3. 1. identification, the O\ -action becomes smooth manifold diffeomorphic to RP. Under this. .. "T. space. 3. V. regular. a X. —. >. R3R. 3. produces homeomorphisms. (2.7) is. a. the. in. Polygons. Identifying complex vector space isomorphic to C "" plane.. R2R. 7. corresponds. to. the. diagonal. U\. -action.. As. 2. 1. with and. in. the. (2.6). the. C,. (free). one. m. space. T. 2. -action. SO 2. establishes. the. diffeomorphisms. The above diffeomorphisms conjugate the involutions v with the complex 2 " m and CP conjugations of C" Also, the involution von V\ coincides 1. 7. 77. '~2.~. .. with its. the residual. sphère I. 3. O. action and therefore. x. m. V. 2. is. the. quotient of CP. mm. ~. 2. by. complex conjugation. For example,. 5I5. O. 2. V\. ,S. 3 .. 3V2,3. V. 2 ,. the. space of planar triangles,. -orbits (therefore, any is identified with CP. is. diffeomorphic to the ( V ) is a link of three circles which are two of themjîonstitute a Hopf link). The quotient. The singular stratum 1. and. E2E. 3. l. 2. E2. QV\). is. a. set. of three points. in. CP. 1. ..

(8) 3V23. Finally,. V. 2. l. ~ CP. /{z. ~z}is. homeomorphic, via the length-side map. £, to. the solid triangle. with boundary. 3VI.3. V. l .. Quaternions, Grassmannians and structures on the full polygon spaces 3.. H= C©Cj. skew-field of quaternions; the space /H of pure imaginary quaternions is equipped with the orthonormal basis /, j and which turns the pure imaginary part k= ij, giving rise to an isometry with Let. (3.1). be the. R3R. 3. of the quaternionic 3. T is identifications space. m. multiplication pq into the usual cross product p xq. The thus identified with T(IYl) which gives rise to the canonical mT(IYl)m. on the the various. moduli spaces. (2.2)).. (see. Recall that the correspondent. H — A4(2x2)(C). This R-algebra homomorphism rj P G enables a to act on the right or on the left on H. It also of unit quaternions with SU 2 identifies the group. gives. an. injective matrix. >. :. U2U. S3S. 2. 3. .. (3.2). The Hopf map. o:H—>. /H defined by. sends the 3-sphere of radius \]~r in H onto the 2-sphere of radius. /H. (The. in. r. formulae given in the original paper by Hopf [Ho, §5] actually correspond to the map q I— qkq.) The equality <p(q) = cj)(q') occurs if and only if q = é° q. The map <fi satisfies the equivariance relation cj)(q-P) — P~ -(/)(q)-P. Writing q — u + vj with m, v G C, one has f. l. (3.3). Observe that if. q—s+tj. with. Rj of its images is the fixed point that will be used later. Its image under. R 0. (3.4) is. the. REMARK.. Lie. algebra. /H, with for. the. the Lie. group. s,. tGR,. set of the </>. is. bracket U\(JS). then <p(q). iq. —. involution. 2 .. This plane. + bj. a. h-». â. + bj. R/©Rfc. \p,q] ~. SU2. =pq-qp= 3. —. .. The. 2lm(pq), pairing.

(9) H~COC. (q^qi) ,_> -Re(^) = (g,q ) identifies /H with its dual. If \ç is the moment is endowed with the standard Kàhler form, then the map H (the factor \ can be checked map for the Hamiltonian action of £/i(H) on f. restricting the action. by. Let V 2 (C m. (3.5). that. such. \a\. =. =. space of (m x 2) -matrices. 2-frames in. of orthonormal. m. C. V 2 (C m. 0.. (a,b) =. and. 1. -action on C).. 1. S. be the. ). \b\. the. to. The group. .. Um. is. ). Stiefel manifold. the. transitively. acts. on. the left. (C") producing the diffeomorphism V (C") = U m /U m - 2 One has the conjugation on V (C") given by (a,b) h-> Çà,b) with fixed-point space of orthonormal 2-frames in R'\ the Stiefel manifold V (R m ) = O m /O m m ) (. a r + b r j, Finally, the embedding V (C ) CHm given by (a, b) intertwines the conjugation on V (C m ) with the involution of (2.5) on H". V2. on. 2. .. 2. 2. 2. 2. .. .. .. .. .. .. 2. V 2 (R m. One thus gets an embedding the. Using. O:. V 2 (C. m. —. ). m >. \a\. =. r+ —1,. \b\. with the projection. b. the. T. m. 3. =ois. j). r. équivalent. Ois. of. image. m. .. defines. of (3.2), one ç map 3 T(IR) ~ m \F by the formula. Hopf. The fact that As. (ROR/) m. C. ). S^J. exactly. 3. =0. (a. b). to. smooth. the. 73. and. = \b\. .. composing. By. ).. \a\. map. surjective smooth map One checks that &(a.b) = ®(a ,b') if and only if O: V 2 (C m ) — m V (a.b) and {a! ,b') are in the same orbit under the action of the maximal torus U™ of diagonal matrices in U m This action is free when none of the (c//./?/)'s vanishes, namely if and only if Q?(a.b) is a proper polygon. As. - {0} —. >. P. one. ,. gets. a. 3. f. >. .. .. O(â, b) = O(<z, by, the restriction. —. ;. 2 (R ") R map hâve thus proved ORO. :. V2(R;")V. THEOREM. 3.6.. a). >. m. of. O. V(RiQRk). The. smooth. m. ~. fixed points gives a smooth with analogous properties. We. the. to. V. 2. O:. map^. V 2 (C. m. ). —. >. m. V. 3. induces. a. homeomorphism 6: £/;"\V 2 (C ) -=-> V such thaï 6(â,fe) = O(a, b)\ The restriction of O above the space of proper polygons is a smooth principal /n. U'I. 1. 3. ). —. -bundle. b). $R. m. :. smooth map^ 2 m V O7\V (R W ) The. 2. planar polygons. -^. is. a. ORO. .. R. :. The. principal. V 2 (R m. >. m. restriction of O;". -covering.. V ORO. 2. R. induces. a. homeomorphism. above the space of proper.

(10) m. COROLLARY 3.7.. Let. (3.8). m. G 2 (C. m. 3. V. ~. map V 2 (C ) — G 2 (C ) a and b is the projection. O:. m. Y 2 (C. U2U. ). m. >. V. —. 2. .. m. bundle), )/U (a principal m C M mX 2(C). This projection is on V (C x Um projection U m /U m - — m /U2 V 2(V. >. (C. 2. U2U. 2. Um/U2U. >. 2. 3. 2. ). 2. 2. the. to. —. ). m. V 2 (C. -equivariant, équivalent The map. ~ O?\O m /O m -. .. for the natural right action of Um. 2. V. Grassmann manifold of 2-planes in C m The which associâtes to (a,b) the plane generated by. m. >. m. and. 2. the. be. ). U?\U m /U m -. 2. .. satisfies. The. conjugation by P being an élément of SO(IH), one thus gets a m m 7^. The space m V\ has a map (still called O) from G (C ) onto smooth structure on the open-dense subset of non-lined polygons (which is where the SO3 -action was free) and, above this open-dense subset, 2. the. new. O. map. restricts. and. so. the. Grassmannian. object. is. 2-planes in action of. Rm. U\. OR:O. map. R. :. G 2 (R. of. V. on. effective. replacing. a. m. 2. m. (C. descends. ). kernel. its. :. isomorphic. ,. to. (the. O\. by. ORO. intertwines. O. map. with the smooth map. U™. Um. by. to. The. m 2-planes in R Grassmannian G 2 (R m ) =. the. longer center of. no. smooth.. is. —. ). SO R. to. G. >. m. action. an. SO. x. m. >. 2. (R m ). is. intermediate. - 2 of ~ CP. m. V\. G 2 (C. on. G. an. case,. —. ). involutions. where. ,. /SO 2. (R m. 2. 2. V. this. In. .. m. the. m. ). oriented mm. ~. 2. The. .. which. is. the subgroup À of U'{ The same holds true in the real case, U\ diagonal subgroup of O™ is also denoted the. is. 1. diagonal. ,. .. A).. Using Theorem 3.6, the reader will easily prove the following THEOREM 3.9.. phism$: [/f\G. 2. (C. m. 6R6. R. :. smooth map. m <9^\G 2 (R ) -=->. V. 3. The. map m. 6R6. Or -=^. <9f\G 2 (R ) non-lined polygons R. :. is. m. ). —. >. m. V. 3. such thaï O(â,fc) =. m. Or. V\.. G 2 (R m. :. It. G. :. m. a. V. 2 .. 2. (R. m. The. principal. ). —. ). >. m. V\. induces. O(a,^) is. .. homeomor The restriction a*. smooth principal. a. induces. v. a. homeomorphism. smooth branched covering and, restricted. is a. above the space of proper polygons, c). G 2 (C. of proper non-lined polygons. above the space (U?/ A) -bundle. The. m. -=+. ). of O. b). O:. The map. a). a. —. principal >. m. V. 2. (O™ / A). induces. restriction of 6 above {O™ / A). -covering.. -covering. a. homeomorphism. the. space of proper.

(11) One has homeomorphisms between the. COROLLARY 3.10.. polygon spaces. and the double cosets a). m. b). m. c). "'P. V. - U?\U /(U. 3. m. V\. ~ S(O. 2. ~. ]. ï)\SO m /(SO. Of\O m /(O. O. x. 2. in. As. Example.. (3.11). Um. x. 2. -. 2. ). x 5O. 2. -. m. 2. -2).. w. ).. example of planar triangles. the. (2.7). k=2)is. interesting. The Stiefel manifold V2(R 2 in turn diffeomorphic the unit tangent bundle to S and. ,. Grassmannian. G. 2. (R. 3. identified with. can be. ). S2S. 2. 3. ). diffeomorphic. is. SO3.. to. =3. (m. to. oriented. The. by associating to an oriented. plane its unit normal vector. The smooth map. branched over the 3 points. This map can be visualized as 2 with equilateral triangles. Divide follows tesselate by the subgroup of isometries which préserve the tesselation and the orientation (it thus préserves of degree. is. 4,. R2R. :. 2. R2R. checkerboard coloring of the triangle tesselation). This quotient is a well 2 known orbifold structure on with three branched points. The projection 2 2 factors through an octahedron with a chess-board coloring of its —. a. S2S. R2R. S2S. >. 2. faces. The residual map from this octahedron to. Take the pullback by. of degree. ORO. R. S. is. 2S. 3. of the Hopf bundle. our map. —. S3S. >. S. 2 .. O. R. .. One gets. a. map. 3. from some lens space L onto S with branched locus the link formed by three S0 2 -orbits. The lens space will be doubly covered by SO3 4. ,. .. We thus get the map. of degree 8. Finally, one has G 2 (R. action. the. by. of. O\. 2-simplex and one. each. on. ~ RP 2 and. ). homogeneous 3V23. again that. sees. 3. 2. V. is. a. ORO. R. is. the. 2 quotient of RP. coordinate. This solid triangle.. is. quotient. a. (3.12) Orbifold structures. The maps 6R6 R and R provide, for the spaces 2 " 2 m " and V ~ a smooth orbifold structure. Each point V\ ~ CP" ORO. 2V22. _. 2. 1. S. 3. 1. 2S. ,. has. a. neighbourhood homeomorphic. 2 open set of the quotient of (R / 2 O\ acts on each R2R via the antipodal map. a "small cover" in the sensé of [DJ]. The. where subgroup of O\ Observe that the map Oç^ is branched loci are E m - m V 2 and by. a. 9. E. {. hâve. to. add. the. branched. neighbourhood modelled. on. locus the. to. m. V. an. m^ V\ n. x .. respectively. As for The generic points of m V. quotient of. CC m. ~2~. 2. by. m x. V. 2. hâve. we a. complex conjugation..

(12) the m. 3. V. to. ,. smooth. a. locally modelled. O: G 2 (C m ). map. —. m. >. 3. V. orbifold structure.. complex. on the. that. By. for. rise,. gives. the. mean. we. space. a. space We define the. s. quotient of C by a subgroup of U\ 3 m V to the reals as the subspace of space C°°( V ) of smooth maps from m C°°(G 2 (C )) which is invariant by the action of Uf m. .. 3. .. Riemannian and Poisson structures. Let TC(m) be the space of Hermitian (m x m) -matrices, identified with u^ via the pairing (3.13). identification turns. This. action for. P. Consider the. H(m). := (a,b). on. ¥(a,fc). t/ 2. e. m. (a,b)*. and. through diag(l,. co-adjoint action of. the. Q. 1, 0,. .. .. ,. One. t/ m. G. .. .. map. —. H(m). (a,b) •P). •. v. :=. C. 2(C). the. >. conjugation given. by. Q• ¥((«,&) •ô*. =. F(V (C m )). the. is. 2. £/m-orbit/. m. -orbit. diffeomorphism. a. -^C.. ). complex vector space Hermitian structure (A, B) := ,. thus. mX. into. 0). This proves that *F descends to. The. eu(. ¥(Q. has. and. M. :. Um. )= -Im(,. ). .. ¥. The map. .A4 mX. with. tr(As*),. with. endowed. is. 2(C). associated. O: M. above and the map. classical. its. form. symplectic. — Hq(2). mx2 (C). >. given by. are. moment maps for the Hamiltonian actions of. One has. V 2 (C m ) =. O-1(0)O. -1. (0). and thus. G. 2. (C. m. ). Um. occurs. as. and. f/ 2. respectively. symplectic réduction. of the Hermitian vector space A^ mX 2(C) and thereby inherits a U m -invariant Kàhler structure, using, for instance [Ki], §1.7. (Strictly speaking, one deals. [Ki] with compact Kâhler manifolds; to fulfill this condition, one can first divide .M mX 2(C) — {0} by the diagonal action of C* to put oneself into a m C C H(m) is a complex projective space.) The residual map *F G 2 (C ) m moment map for the action of U m on G (C ). m Being thus a Kàhler manifold, G (C ) is a Riemannian Poisson manifold. the algebra C°°(T ) This structure descends to the complex orbifold m V in. :. 2. 2. 3. 3. :. admits. a. unique Lie bracket. that the. so. projection. G 2 (C. m. ). —. >. 3. m. V. m. 'PV\. is. a. Poisson map. It is possible to endow with. (3.14). a. Poisson structure the space. 3. of. configurations of ail m-gons in R without fixing the perimeter to 2. It suffices in the above construction, to replace the U 2 -réduction G 2 (C m ) = &~ (0)/U 2 m by the S U -réduction G (C ) := <J>~ (0)/U 2 The latter is a non-compact m space, the total space of the déterminant bundle over G 2 (C ) with the zéro ,. <. l. 2. 2. .. {.

(13) M. section collapsed. The trace fonction on to. £:. now use the map m. that £(p), for pG. k. V. a=. (a. u. .. k. V (a). The space. .. .. am. ,. ). k. ™V. m. m. V%. V. k. a. représentative. 2. G. R. ;. £. with. o. £"ii. a. t. =2,we. invariant under the action of. is. defined in (2.4). Recall. Rm. -*. length of the successive sides of. is the. ,. of r with total perimeter. For. .. POLYGONS WITH GIVEN SIDES - KÄHLER STRUCTURES. 4.. We. m yn(C) descends to G 2 (C ) and. VV\. m. Casimir fonction "perimeter" on. the. m. O. k. .. define. We derme the moduli. spaces. and. l. The. of. consists. (a) space empty. We call a. generic if. THEOREM 4.1.. The map. for. Vl(a)V. the. action of As. Proo/ the. Um. U. l. on. [. l Vl(a)V. \i. (a). is. generically. =0.. := 100. :. G 2 (C /M. —. ). >. R/MR. /M. wa moment. map. H. G 2 (C' ).. (3.13), the moment map G (C /M ) is induced from *F. seen. -action on. ]. finite number of points and. a. ¥:. in. 2. :. M. G. 2. (C. mx2 (C). /H. ). for. — + Uinî). — Him). given. >. (a.b)* A moment map /i for the action of U" is obtained by composing *F with the projection 7Y(m) — R'" associating to a with matrix its diagonal entries. So, if n G G (C W ) is generated by a and := (a./?). by. 1. •. .. >. Z?. 2. (ai)çV. (C" ), one has !. 2. classic theorem of Atiyah and Guillemin-Sternberg [Au, §111.4.2] asserts that the image of a moment map for a torus action is a convex polytope (the moment polytope). The restriction of the moment map to the fixed point A now. set. of an. one gets. anti-symplectic involution thèse facts directly :. has. the. same. image [Du]. In our case,.

(14) COROLLARY. /i(G. 2. (C. m. )). Proof.. 4.2.. The. moment. /i(G 2 (R m )) =. =. One. has. E. m. where. ,. Image (fi). m fi\ G 2 (C ). map. the. is. EmisE. m. >. Rm. satisfies. hypersimplex. Further. Image (£).. —. —. it. manifest. is. that. actually provided in [KMl], Lemma 1, or [Ha]. We give hère however another argument, for the pleasure 2 of constructing a continuous section a:Em — m V of £. If m=3,we hâve. Image (£) c. E. m. proof that Image (f) =Em. A. .. is. >. already mentioned in (2.7) that V Let aG E m Deflne fi := Ylj=\ a j 3V23. .. The numbers. /? r. ,. a. r. ,. 2—. /3. r+. a. to. form. i. :>2. unique triangle r(a) G 7 deflne the élément a(a) G 37:>2,3. ,. m. 2. is. homeomorphic. to. S 3via. triple of S 3and are then the lengths of which can be subdivided in the obvious way a. 2. V (a). Figure. (see. 1. :. 1).. r(a). The continuity of a cornes from the fact that if the map at some a, the triangle r(a) is then lined.. 1). Corollary 4.2. is. also. a. r. is. discontinuous. conséquence of our stronger re. introduced in [GM]. Observe that H obtained by taking the convex hull of the middle point of each edge of 2). The word. standard (m—. "hypersimplex". l)-. We also obtain the. £.. an<^. Figure. Remarks. suit(5.4).. the map. is. critical values of. jjl. (compare [Ha]). :. is a.

(15) G2(R"?)G. 2. (R". ?. ). critical values of. The set of. PROPOSITION 4.3.. consists of those points (x. -> S m. following conditions. .. .* m ). .. G. S. Em. or. satisfying one of the. w/. :. a). one. Xj. vanishes;. b). one. Xi. is. c). there exist. equal to e. u. G?(C m ) -». on. \x. 1. ;. =±1. {. such that. J2"Li. cie. i. x. =°> wiîh. >. at ieast hvo. '. Si. °f. s. each sign.. satisfying a) and b) constitute the boundary of E m Points satisfying c) are "inner walls". Points satisfying a) correspond to non properpolygons. Those satisfying b) or c) are non-generic a 's (Condition b). Remark.. Points. implies that there exist. £. =±1. (. such that. X^/li. £. i. xi. ~®wâ^. ut one. £. .. '. of the same sign.). Proof for which. The critical points of the moment map /i, are the points of G 2 (C ") the U™ -action has a stabilizer of dimension bigger than They are 7. 1. .. the images of those. actionhas. (2 x m). \. -matrices in. non-discrete stabilizer. There. a. contained in U". 1. {1}. x. ;. 2. (C'. are. n. ). for which the (U". 1. x. U{. Ui). such points whose stabilizer is. they are the matrix with one row vanishing and their. values under^// are the points of E m satisfying a). The other points give rise to 3 m m ) V = U'? so that the action of £/ 2 /{center of U 2 } ~ SO 3 points in 2 (C. /\. non discrète Their values in. stabilizer.. has. satisfying. b). E. m. points are the lined configurations m V are the non generic a 's, which are the points in E lu Those. x. or c).. We hâve proven most of the main resuit of this section for generic 3 /7/ proper a, the space V (a) is a Kâhler sub-quotient of G?(C ). :. THEOREM. isomorphic. to. For. 4.4. the. morphicand V (a). 3.9 that. in. We. R. ,. E. m. O(ô, b) = ®(a, b)\. shall. now. generic, 2. U'(. \/.r. compare to. the. £~l{a)~. l. V\(a). (a). The. real part of. can be seen as the 3. ([KM2], §3). Using 3. int. By 4.1, one has V (a) =. Grassmannian. the. aG. Kâhler réduction. 2. Proof. .. l. {a) =. and. Kâhler manifold involution v is antiholo is. a. V\(a).. U^/n-â). and we hâve seen. D. Kâhler. structure. obtained. on. V\(a). from. that introduced by the. Klyachko [Kl] or Kapovich-Millson standard cross product x and scalar product (.. .} on. thèse authors put on the sphère. S;,. of radius. r. the complex. structure. 1.

(16) defined by. Kàhler metric. and the. with associated symplectic form lj(u, v) := 3 The map (5 :Wa — defined by (3(z\, >. R3R. map for the diagonal action of SO3 as. the symplectic réduction. PROPOSITION 4.5.. compare with those. (û. J. SO 3 \f3~. on l. .. .. Wa. ,z m. .. .. v). x ). Let W(a) := Yl™=\ $%. .. := Ya=\. The space. z. sme moment. î. V\(a). (0).. The. complex structure. and. h. J. and Kàhler metric. of Kapovich-Millson in the. h. of 4.4. following way. Starting from the Hermitian vector space .M = A / mX 2(C) that V (a) is obtained by two successive symplectic réductions. Proof sees. thus occurs. f. :. one. 3. (we use the notation of §3). One can perform the réductions in the reverse order. We first get. where CPj. is. the quotient of the. 3-dimensional sphère. M — H{2) gives a diagonal action of £/i The moment map O a moment map (still called O) from the product of projective spaces into Ho(2). One has a commutative diagram. by the. where. :. i/j. :. Ho(2). —>. R3R. 3. c±. RxC. sends the matrix. f_. >. jto. (u,z)..

(17) that for ail prove Proposition 4.5, it is enough to establish T a CP\ — T^S? satisfles the tangent map T a (p To. U2U. 2. space. 1. ,. -equivariance, we can restrict ourselves to a = [y/r.O]. The tangent T a CP). is identifled with {0} x C and one can take v = (0.1) and. Jv = (0, 0- One has. (p(a) = (r. 0, 0),. =4, while uo(v,Jv). uj(T ô(v),T ô(Jv)). and. CP,. G. >. :. By. a. a. a. =. 1.. Remarks. a m. symplectic leaves of the Poisson structure. are the. V\,. V\(a). The results of this section show that the spaces. (4.6) m. or. VV\. on. for generic. regular part of. the. given in (3.13) and (3.14).. If one works in the pure quaternions /H, the complex structure on Si becomes. J. co-adjoint orbit of £/i(H) and the Hermitian form the Kirillov-Kostant form (see [Gu, Theorem 1.1]).. is. (4.7). The sphère. is. Sj.. w. a. The isomorphism between the symplectic réductions of the Grass mannianG?(C m ) and the product of CP 's that underlies our results 3.9,. (4.8). 1. symplectic version of the Gel'fand-MacPherson correspondence ([GM] and [GGMS]). The fact that this isomorphism cornes from two réductions of M is the philosophy of "dual pairs" (see [Mo] and. 4.4 and the proof of 4.5. is. a. the références therein).. The. 5.. On. m. distances d(p). =. Tk. hâve. we. between. (|p(l)|. ;. Gel'fand-Cetlin action. far. deflned the length functions successive veitices. We now introduce so. p(2)|,...,|^/ liP(0|), ?. |p(l) +. Connecting the veitices. to the. origin. (Only m as d(p)\ = £(p)i, û?(p) _i = £(p) m and s(p) m £j,dj and the p is to be understood.) IM. ,. the 3. lengths. i. measuring the d: m T k — R" >. ,. of the. diagonals. of thèse functions are new,. =0.. Hereafter we write only.

(18) with. As. don. functions. smooth. function. the. £,. m. vanishes, that. is. to say the. We call. such. a. polygon. The. of prodigal. set. codimension For a. torus. k. ~3~. V. k. m ,. V\. and. continuous but only generically k m It is smooth where no d V. t. .. polygon does not return. prodigal is. polygons. and call. dense. open. to the. origin prematurely.. (£{P),d(P)) in. m. p^_. a. with. prodigal value. complément of. k.. [KM2] (see also [Kl], §2.1) introduced an action of prodigal polygons; the /th circle acts by rotating the section. = 3, there. m TT. P. descends to. d. 3. on. is. in. of the polygon formed by the first. i. edges about the zth diagonal. (When that. diagonal is length zéro, there is no well-defined axis about which to rotate, and indeed the action cannot be extended continuously over this subset.) This action plainly préserves the level sets of the functions. the. THEOREM 5.1. (KM2).. function. a. d. is. but more is true:. of prodigal polygons of. On the subspace. moment map for thèse. d,. V\(a),. "bending flows".. One important conséquence of this is that the torus action also préserves the symplectic structure. It does not, seemingly, préserve the Riemannian metric. nor the complex see. structure (the codimension of the singular set. not even;. also §6).. Thèse functions. prétations.For. (a M —\. x. (a, b). \ai. £, d G. lifted. V2(C m. bÂ \. t. \, the first. i. V2(C m ) hâve simple matrix-theoretic inter ), /= 1. m, introduce the truncated matrices to. .. .. .. ,. rows of (a,b). Then the. bi). has the eigenvalues. Thèse are calculable from. and. is. £. and. d, since. 2x2. matrix.

(19) X)'=i. So. i s. £j. me sum of the two eigenvalues of. différence. (Note that This. H. x. =. as. 2)-matrix MfM. (2 x. t. promised; M*Mi. d\. is. the. lesser eigenvalue. is. 0.). MfMj, 's. has the same nonzero. whereas. eigenvalues. the. as. i. x. i. matrix MjM* The latter matrix is more relevant in that it is the upper left x / submatrix of the m x m matrix (a, b) (a, b)* introduced in section (3.11). .. /. This family of Hamiltonians — the eigenvalues of the upper left submatrices — has been studied already in [Th] and is called the classical Gel'fand-Cetlin System (our main référence is [GSl]). The linear relations established above. between them and d, THEOREM 5.2.. £. are. summed up in the following. The bending. flows. from the Gel' fand-Cetlin System on The Gel fand-Cetlin 7. mysterious (at least. to. on. the. m. are the residual îorus action. V\(a). Grassmannian G2(C m ).. action on the flag manifold has us); it. is. always been rather pleasant that in this case it has a natural. géométrie interprétation.. Gel'fand-Cetlin fonctions. The. left. (the. {£/_/}_/</. jth eigenvalue of. upper some linear inequalities that can be established. submatrix) satisfy using the minimax description of eigenvalues [Fr,. For /. x. /. = 0,. But. /. the .. polygon .. .. ,. n —. thèse. inequalities. (1). the. are !. 1. space the. functions. /, d. most. nontrivial inequalities. transparent. in. our. of. p.. 149]. thèse. :. say. 0. <. 0;. for each. are. situation,. as. they. are. just. the. triangle.

(20) (The first one,. starting from satisfied. X^l=i^>. as. an. can. e. P. rove. d. inductively from. others. the. = 0.). do. [GSl]. In. <. dt. it is left. exercise to show that (1) are the only inequalities. equivalently, that every point in the convex polytope T m CRm xRm defined by them (and d 0 — d m = 0 and J2iî = 2) is realized by some Hermitian matrix. We show this directly ;. :. THEOREM 5.3. Tm. polytope. image of. m. k. V. >. 2. under the map. (£,d). whole. the. is. .. We. Proof.. The. construct the polygons m. directly, vertex by vertex — really. k. V (a,ô) is nonempty (and so its quotient by establishing that each space SO(k) is as well). We must place each new vertex on the intersection of two k ~ 's, one of radius d i+ from the origin, the other of radius £ i+ from the Y. SS. \. \. previous vertex. The inequalities £[+\ < d -f di+\ and 4"+i < A"+i +d{ rule k out one containing the other; the third inequality d < 4fi +d i+ rules 2 k a point or the out their being separated balls. So they intersect in an t. SS. l. ~l~. \. t. SS. ~2,~. ,. whole. SS. k. ~1,~. Remarks While the map. 1). R. on. m. [i. (n. the. anywhere on which we may place the new vertex.. ,. (5.4). S. l. m. —. ,. the. 01. ,. d. map. and. hypersimplex. the E. m. £. equivariant with respect to the usual action of can only be made equivariant under the involution is. polytope. Tm. is. correspondingly. less. symmetric than. .. That the image of (£, d) is the same when restricted to planar polygons has the flavor of a more gênerai theorem of Duistermaat [D] on restricting moment maps to the fixed-point sets of antisymplectic involutions. In fact 2). Duistermaat's theorem does not apply directly, because the subset where d is smooth (and a moment map) is noncompact; in any case we preferred to give a. polygon-theoretic proof. When. 3). k. =. 3. Theorem. 5.1. guarantees. that. the. bending torus. acts. interior point of T m making this flber a torus U(l) ~ (or 0(l) ~ when k— 2). Over a prodigal boundary point of T m the flber is still a product of 0- or -sphères, but fewer of them. simply transitively mm. on the fiber over an. 3. mm. ,. 3. 1. ,. Bending around other diagonals than the ones above can be done in the m same way, the moment map lifted to V2(C ) being the différence of the two m eigenvalues of M* M for a corresponding submatrix Mof (a,b) G V 2 (C ). 4). For instance, we take.

(21) for the diagonal. d 2A. := p(2) + p(3) + p(4). The bending flows around two commute if and only if the pairs {p. q} and {p'.q'} q. diagonals d p q and dd p intersect or are unlinked in R/mZ. \\. .. '. Toric manifold structures on. 6.. $m. section, we study examples of V]_{a). In this. for. P_+^3(\alpha)(a)$. m. C. V. 3. m. = 4,5,6. such that the. m-3. R never vanish. The whole space diagonal fonctions w _ 2 :£+ (a) —^ V\{a) consists of prodigal polygons and, by §5, the bending flows give an action of a big (i.e. half-dimensional) torus on Vl(a) By Delzant's theorem dd. d2,...,d. 2. ,. .. .. .. ,. .. [De], or [Gu, §1]), we can construct from the moment polytope A a alone a toric manifold which is equivariantly symplectomorphic to the space. (see. V\{a).. This can be achieved also by [DJ,§ 1.5], though only up to equivariant diffeomorphism. The latter also gives the real part, the planar polygon space V. 2. (a),. as. a. 22. W. ~3~. 3. -sheeted branched cover of. A. Q. We. .. sum. up. below some. results of thèse constructions without writing ail the détails.. Without explicit mention of the contrary, a is supposed to be generic. Contrary to the previous sections, we do not require that the perimeter of our polygons is 2. It was necessary to fix the perimeter in order to derme the map t and the value 2 is the natural choice to deal with the map k But m T (a) makes sensé for any aG R'^ o and so do O: V (C ) — m V k the various moduli spaces m V (a), etc. When Yl a = A is a -> me polytope slice through the Gel'fand-Cetlin moment polytope T m of §5: for gênerai a a it is a homothetic copy of this section. k. /M. r. 2. .. i. (6.1) Qi. 7^. q2q 2. m=4: or. The condition which guarantees that : The space of quadrilaterals V i(a) a4. d2d. never vanishes. 2. 4V4. q3q. 3. .. toric manifold of dimension 2, therefore diffeomorphic has image the interval A a := /j Dh where map. to. is. CP. then 1 .. is. compact The moment a. d2d. 2. The space V 2 (a) is RP The quadrilatéral spaces been classified (see for instance [Ha]). One has 4V4. 1. .. 4V4. V. 2. (a)+ hâve long since.

(22) Observe also that a and. I\. I2I. 2. is. generic if and only if the boundaries of the intervais. do not meet.. By the Duistermaat-Heckman Theorem [Gu, §2], the symplectic volume of 4V4. 3. V (a). is. equal to the length of. Aa. .. We would then obtain the same length. if we had used the other diagonal |p(2) + p(3)| This produces a statement of elementary Euclidean geometry the variation intervais of the two diagonals of a quadrilatéral with given sides in are the same length. .. :. R3R. m=s:. 3. never vanish are s for instance a\ and a 5 The space of pentagons 2 4 V\(a) is then 4. The moment polytope A a GR2 for {d 2^d^) a toric manifold of dimension. (6.2). Conditions for which both. a2a. is. a4a. d2d. 2. and. d3d. 3. .. the intersection of the rectangle. I. a. with the non-compact rectangular région. FIGURE. 2. :. The moment polytope A. a. Figure 2). One sees that A a has at most 7 sides. The generic exactly those for which the boundary of £l a contains no corner of 2 s T\(a) is then obtained by symplectic blowings up from CP or. a. (see. S2S. s. Ia 2. are. and x S. 2. The space of planar polygons V\(a) is a closed surface obtained by gluing 4 copies of A and its Euler characteristic is given by the formula a. ..

(23) orientable if and only if I a C a One has of course x( 5^+(«)) = 2x( P («)) and V\(a) is an orientable surface m ( V+(a) is always orientable). The possible cases, depending on the number (see. [DJ], Example 1.20) and. is. 5. of sides of À a. ,. va.u. 2. are summed up in the. Figure. Some. (6.3). 3:. A(l5])1;1)l)A. following table.. (l5])1;1)l). embeddings of the regular are not prodigal. However none are lined Vo := V (a) is diffeomorphic for small e to 5V5. .. s. pentagon. a. and. the. thus. moduli. 3. V£. where. V£. (1,1,1,11). =. :=. 5V5. space V (a £ ) and 3.

(24) a. £. thus. :=(l+£,l,l,l,l+£). ~. The moment polytope for a £ has then. diffeomorphic The "limit moment polytope" Vo. V£. is. to. (S. 2. 2. S. x. À^îî). 3CF (if. #. ). A:. =2,. shown in Figure. is. sides and. 7. 2. 5P5. P (a). ~ Z4. +. ).. 3.. The pre-image in V £ of the segments {x = e} Pi A^ and {y —e)P A' a are 2-spheres of symplectic volume proportional to e, by the Duistermaat-. Heckman Theorem. Passing. limit. to the. Vo. thèse sphères become Lagrangian,. ,. and so cannot be complex. This shows that the action of the bending torus is. not complex — thèse polygon spaces are only equivariantly symplectomorphic, not equivariantly isometric, to toric varieties. k unique représentative in pG P (a) with p(5) = (-«5,0,0) and 7(7-) := p(l) + p(2) in the half-plane H — {z = o,y > o} This provides a map 7 V (a) — H whose image A a is the intersection Ri DR2 DH where R\ and i? 2 are the rings. (6.4). 5V5. Any class rG. V. k. ~. 2. 3. '. 5P5. (a) has. a. 3. 5V5. >. :. .. Figure. The idea of reconstructing. 5V5. V. 2. 4. :. A a. (a) by gluing copies of. early works of W. Thurston on planar linkages (see relationship with our theory is the following the domain A a. the. goes back to [TW, p. 100]). The. :. into. PL-polytope A a in just the moment polytope for and &(p) = |p(3) + p(4)|. R2R. a. m=6:. 2. by the map the bending. The conditions. v. (\v\,. \v. -. (0,. Aa. is as)|)a. straightened. up. and. is. 5. )|). A. a. Hamiltonians d\(p) = |p(l) + p(2)|. a^ imply that d^ and ai 7^ «2 and as 0. But we <^4 never vanish. However, one cannot guarantee generically d^ can replace the d = (d\,d2,d?) by 6 := (81,82,83) where. (6.5).

(25) non-vanishing of the. and guarantee. Observe that <9,00 given (on (a. 6). V 2 (C m. :. ô. (. 's. — R(z=. ). a /condition aa /-i the fonctions on V 2 (C"). by the generic. 1,2 3) are. >. :. 2. 2. (2x2). V 2 (C W )) by the différence of the eigenvalues of the. G. matricesM* Mi, where. The moment polytope in. R3R. 3. intersection of the rectangular parallelepiped. the. is. with the région. The domain. The polytope A a has then at R+ •E3 on the hypersimplex E facets. The length-system a is generic when the boundary of £1 does. or the cône. most. 9. 3. not contain corners of 6V6. V. l. (l.. 1). contains. For. generalization. m. jF. la.I. a. 10. k. —. >. n. k. T. m. V. k. -^. n. k. V. Above. .. the. even, the regular hexagon is not generic. occuring in (6.4) and. d. 2/7. 1. —. e(p)(ï) :=. by. We. .. the. e. call. 2n,. or. induced maps. pG. m. F. of. k. even. admit the following. derme. we. -1)+. p(2z. 6. V. space. k. -^ V\. generic the. >. :. even-step. map. taking^ e(p) (n) := p(m). p(2ï) m. the. m. n. if. e(p). map proper polygons, locally trivial bundle whose flber is a product of (k — 1) d= (d v ....d n ) m T k — R" by d:= £oc. The map d. polygon.. :. éléments.. =. m. if mis odd. We also call and. 6is. As. .. The bending flows. (6.6). e:. of the three half-lines. can be described as the convex hull. Q.. e. -^. V\ is. is. gives. a a. n. V\. proper smooth. Define the. side. always continuous and smooth when e(p) is a proper polygon. As the map e is a submersion on even-generic polygons, the critical values of d are the same as those of £, the walls of 4.3. k As for the map £, the map d can be deflned on each m V {a). Call a G R m. lengths of the new polygon. <?v£/7. q. is. generic if. m. k. V (a). even-generic if. aa 2. e{p). .. It is. only consists of even-generic polygons. For instance, «2/ for ail /. When k = 3, d is a moment map /-i. for the corresponding bending action of T n deflned on even-generic polygons. Restrict to m V (a)+ for an even-generic a. Define the right-angled 3. polytope.

(26) and consider the convex polytope. Proposition polytope. =. 3,. 1). The. a. cRn. image of. d:. m. k. —. V (a)+. >. Rn. is. the. regular value of d, the even-step map e induces, for l symplectomorphism from the symplectic réduction T n \d~ (x) onto. a. is. a. Remarks and open problems. 7.. Is. (7.1). there an octonionic version of Section 3? Alternately, are there. bendings in dimension 5 (like the U\ (C) bending flows in dimension U\ (R) flippings in dimension 2) ?. U\ (H). and. Observe that the inclusion. (7.2) k. whole. Àq,.. If xGAa. 2) m. 6.7.. A. > m. —. m<pm-\. (7.3). m. V. k. C. m. V k+l. becomes. (triangles are always planar, etc.). In what ways more natural than the unstable ones ? 1. The. m-polygons whose first diagonal. sphère bundle over. l)-. is. of. a. a. 3. bijection when. are thèse. spaces. given length forms. (For k = 3 this is just space of (m — symplectic réduction by the first bending circle.) This gives an inductive way. a. a. l)-. construct the space of m-polygons by gluing together (sphère bundles over) it would require identification of thèse sphère the spaces of (m— bundles, which in k — 3 might be done using the Duistermaat-Heckman theorem (where the circle bundle is determined by its Euler class). to. Alternately one might work out Unfortunately in dimensions above the planar polygons). (7.4). for. In. [KMl]. identifying the combination of [Du]. the fibers of the whole map d of section 5. 3. thèse are always singular (at, in particular,. [Wa] there are presented "wall-crossing arguments" 2 m V (a). It would be nice to relate thèse to a spaces and. and the paper [GS2], which présents its own. arguments for any symplectic réduction by. a. torus.. wall-crossing.

(27) (7.5). A space of great interest nowadays. is. moduli space of flat 517(2). the. punctured Riemann sphère — in the language of this paper, 3 (rather than R ). The spaces hère can be seen as géodésie polygons in limiting versions where the radius of goes to infinity. We do not know how connections. on. a. 3. S. 3S. 3. S3S. Gel'fand-MacPherson correspondence to this case; one definite complication is that it is no longer the symmetric group but the braid group which permutes the edges, and that action is not complex.. to. adapt the. averaging the Riemannian metric with respect to the bending torus, one can deform the complex structure on a space of prodigal polygons to that of the corresponding toric variety. Is the original complex structure that of a toric variety (not just in the same déformation class) ? (7.6). By. REFERENCES [Au]. M.. AUDIN,. The. topology. Birkhàuser, [DJ]. Davis, M. and. of. Delzant,. [Du]. DuiSTERMAAT,. [Fr]. FRANKLIN,. [GGMS]. J.. 62. Hamiltoniens périodiques. T.. actions. on. symplectic. manifolds.. Januszkiewicz. Convex polytopes, Coxeter orbifolds. T.. torus actions. Duke Math.. [De]. torus. 1991. and. (1991), 417-451.. image convexe de l'application moment. Bull. Soc. Math. France 116 (1988), 315-339. et. Convexity and tightness for restrictions of Hamiltonian functions to fixed-point sets of an antisymplectic involution. Trans. AMS 275 (1983), 417-429. J.. J.. Matrix theory. Prentice-Hall, 1968. Gel'fand, L, M. Goresky, R. MacPherson and V. Serganova. Combi natorialgeometries, convex polyedra and Schubert cells. Adv. Math 63 J.. (1987), 301-316.. [GM] [Gu]. Gel'fand,. MacPherson. Geometry in Grassmannians and a generalization of the dilogarithm. Adv. Math. 44 (1982), 279-312. GUILLEMIN, V. Moment maps and combinatorial invariants of Hamiltonian T -spaces. Birkhàuser, 1994. and. I.. R.. n. [GSl]. Guillemin. V. zationof. Sternberg.. and. S.. the. complex flag manifolds.. The Gelfand-Cetlin System and J.. Funct. Anal.. quanti. 5?(1) (1983). 106-128.. [GS2]. Guillemin,. [Ha]. category. Inventiones Math. 97 (1989), 485-522. Hausmann, J.-C. Sur la topologie des bras articulés. In "Algebraic Topology, Poznan". Springer Lectures Notes 1474 (1989), 146-159.. [HK]. Hausmann, J.-C.. V.. spaces.. [Ho]. Hopf,. H.. and. S.. Sternberg. Birational équivalence. in the. symplectic. Knutson. The cohomology ring of polygon Preprint (1997). http://www.unige.ch/matWbiblio. Ùber. and. die. A.. Abbildungen der dreidimensionalen Sphàre auf die Kugelflàche. Math. Annalen 104 (1931), 637-665..

(28) [KMl]. KAPOVICH, M.. [KM2]. KAPOVICH, M. and. MILLSON. The symplectic geometry of polygons in the Euclidean plane. J. Diff. Geometry 42 (1995), 430-464. and. J.. MILLSON. The symplectic geometry of polygons in Euclidean space. /. Diff. Geometry 44 (1996), 479-513. KIRWAN, F. Cohomology of quotients in symplectic and algebraic geometry.. [Ki]. J.. Princeton University Press, 1984. KLYACHKO, A. Spatial polygons and stable configurations of points in the projective line, in Algebraic geometry and its applications (Yaroslavl, 1992), Aspects Math., Vieweg, Braunschweig (1994), 67-84.. [Kl]. :. [Mo]. MONTGOMERY, R. Heisenberg and isoholonomic inequalities "How to use dual pairs to ignore the rest of the universe". Symplectic geometry and mathematical physics. Souriau Volume. Birkhàuser, 1991.. [Th]. THIMM, A. Integrable géodésie flows on homogeneous spaces. Ergodic theory and dynamical Systems 1 (1980), 495-517. THURSTON, W. and J. WEEKS. The mathematics of three-dimensional manifolds. Scientific American 251 (1984), 94-107. Walker, K. Configuration spaces of linkages. Undergr. Thesis, Princeton (1985).. :. [TW] [Wa]. le. (Reçu. Jean-Claude Hausmann. Mathématiques-Université C.P.. 240. CH-1211 Genève 24 Switzerland [email protected]. Allen Knutson MSRI 1000 Centennial Drive. Berkeley, Ca 94720 U.S.A. allenk @ alumni caltech edu .. .. 20 octobre. 1996; version révisée,. le. 10. février 1997).

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