Higher codimension CR structures, Levi-Kähler reduction and toric geometry

Transcription

1 Higher codimension CR structures, Levi-Kähler reduction and toric geometry Vestislav Apostolov (Université du Québec à Montréal) May, 2015 joint work in progress with David Calderbank (University of Bath); Paul Gauduchon (Ecole polytechnique, Palaiseau); Eveline Legendre (Université de Toulouse).

2 Goal. Overview Construct (non-trivial) explicit extremal Kähler metrics on certain complex (toric) manifolds/orbifolds

3 Goal. Overview Construct (non-trivial) explicit extremal Kähler metrics on certain complex (toric) manifolds/orbifolds Construct (relative) K-stable toric varieties.

4 Goal. Overview Construct (non-trivial) explicit extremal Kähler metrics on certain complex (toric) manifolds/orbifolds Construct (relative) K-stable toric varieties. Idea. A CR structure on a manifold N is a co-rank l distribution D TN equipped with a complex structure J : D D satisfying [D 1,0, D 1,0 ] D 1,0. Under suitable positivity conditions, N is equipped with a family of horizontal Kähler structures on D. If a Lie group G acts freely and transversely to D, preserving J, then the quotient N/G acquires a family of Kähler metrics.

5 Goal. Overview Construct (non-trivial) explicit extremal Kähler metrics on certain complex (toric) manifolds/orbifolds Construct (relative) K-stable toric varieties. Idea. A CR structure on a manifold N is a co-rank l distribution D TN equipped with a complex structure J : D D satisfying [D 1,0, D 1,0 ] D 1,0. Under suitable positivity conditions, N is equipped with a family of horizontal Kähler structures on D. If a Lie group G acts freely and transversely to D, preserving J, then the quotient N/G acquires a family of Kähler metrics. Main Example. N = S 2m1+1 S 2ml+1 C d with d = l + l i=1 2m i and G T d is an l-dimensional subtorus.

6 Plan. 1. Toy Example: Weighted projective spaces as CR reductions of the sphere. 2. CR structures and Levi Kähler reduction in arbitrary co-dimension. 3. Applications to extremal toric varities.

7 Weighted projective spaces and CR spheres N = S 2m+1 = {z = (z 0,..., z m ) C m+1 : z 2 = 1} C m+1 CR structure: D = ker(d c σ) N TN with σ = 1 2 m i=0 z i 2 is J-invariant. T m+1 acts on C m+1 preserving (S 2m+1, D, J)

8 Weighted projective spaces and CR spheres N = S 2m+1 = {z = (z 0,..., z m ) C m+1 : z 2 = 1} C m+1 CR structure: D = ker(d c σ) N TN with σ = 1 2 m i=0 z i 2 is J-invariant. T m+1 acts on C m+1 preserving (S 2m+1, D, J) For a = (a 0,..., a m ) N m+1 with gcd(a 0,..., a m ) = 1: G a = {exp 2iπta, t R} T m+1 is a circle subgroup with g a = Lie(G a ) transversal to D i.e. x N : T x N = D x span{k v (x) v g a }.

9 Weighted projective spaces and CR spheres N = S 2m+1 = {z = (z 0,..., z m ) C m+1 : z 2 = 1} C m+1 CR structure: D = ker(d c σ) N TN with σ = 1 2 m i=0 z i 2 is J-invariant. T m+1 acts on C m+1 preserving (S 2m+1, D, J) For a = (a 0,..., a m ) N m+1 with gcd(a 0,..., a m ) = 1: G a = {exp 2iπta, t R} T m+1 is a circle subgroup with g a = Lie(G a ) transversal to D i.e. x N : T x N = D x span{k v (x) v g a }. S 2m+1 /G a = CPa m is the m-dimensional complex weighted projective space, where the complex structure comes from (D, J). It is a toric complex orbifold with respect to T m = T m+1 /G a.

10 Kähler metrics via the Levi form D TN a constant rank distribution. The Levi form L D C (N, 2 D TN/D) is L D (X, Y ) = [X, Y ] mod D.

11 Kähler metrics via the Levi form D TN a constant rank distribution. The Levi form L D C (N, 2 D TN/D) is L D (X, Y ) = [X, Y ] mod D. D is called Levi non-degenerate or contact if x N, α (TN/D) x with α L D non-degenerate on D x ;

12 Kähler metrics via the Levi form D TN a constant rank distribution. The Levi form L D C (N, 2 D TN/D) is L D (X, Y ) = [X, Y ] mod D. D is called Levi non-degenerate or contact if x N, α (TN/D) x with α L D non-degenerate on D x ; If D admits complex structure J with [D 1,0, D 1,0 ] D 1,0, then L D is of type (1, 1) on D: (D, J) is Levi definite if there exists a section s C (N, (TN/D) ) with s L D > 0 (Levi definite Levi non-degenerate).

13 Kähler metrics via the Levi form D TN a constant rank distribution. The Levi form L D C (N, 2 D TN/D) is L D (X, Y ) = [X, Y ] mod D. D is called Levi non-degenerate or contact if x N, α (TN/D) x with α L D non-degenerate on D x ; If D admits complex structure J with [D 1,0, D 1,0 ] D 1,0, then L D is of type (1, 1) on D: (D, J) is Levi definite if there exists a section s C (N, (TN/D) ) with s L D > 0 (Levi definite Levi non-degenerate). S 2m+1 C m+1 is Levi non-degenerate (i.e. a strictly pseudo convex CR manifold).

14 Kähler metrics via the Levi form Let K a C (TS 2m+1 ) be a generator for the circle action G a on S 2m+1. By the transversality condition TS 2m+1 = D R K a, TS 2m+1 /D = R K a and η a L D = dη a D, where η a is the unique 1-form with η a D = 0, η a (K a ) = 1. Furthermore, dη a > 0 on (D, J).

15 Kähler metrics via the Levi form Let K a C (TS 2m+1 ) be a generator for the circle action G a on S 2m+1. By the transversality condition TS 2m+1 = D R K a, TS 2m+1 /D = R K a and η a L D = dη a D, where η a is the unique 1-form with η a D = 0, η a (K a ) = 1. Furthermore, dη a > 0 on (D, J). Theorem (S. Webster-77, R. Bryant-01, David Gauduchon-06) Any weighted projective space CPa m admits a (unique up to scale) T m -invariant Kähler metric ǧ a obtained as the projection of the Levi form dη a to S 2m+1 /G a.

16 Kähler metrics via the Levi form Let K a C (TS 2m+1 ) be a generator for the circle action G a on S 2m+1. By the transversality condition TS 2m+1 = D R K a, TS 2m+1 /D = R K a and η a L D = dη a D, where η a is the unique 1-form with η a D = 0, η a (K a ) = 1. Furthermore, dη a > 0 on (D, J). Theorem (S. Webster-77, R. Bryant-01, David Gauduchon-06) Any weighted projective space CPa m admits a (unique up to scale) T m -invariant Kähler metric ǧ a obtained as the projection of the Levi form dη a to S 2m+1 /G a. ǧ a is Bochner-flat the Chern-Moser tensor of (S 2m+1, D, J) is zero;

17 Kähler metrics via the Levi form Let K a C (TS 2m+1 ) be a generator for the circle action G a on S 2m+1. By the transversality condition TS 2m+1 = D R K a, TS 2m+1 /D = R K a and η a L D = dη a D, where η a is the unique 1-form with η a D = 0, η a (K a ) = 1. Furthermore, dη a > 0 on (D, J). Theorem (S. Webster-77, R. Bryant-01, David Gauduchon-06) Any weighted projective space CPa m admits a (unique up to scale) T m -invariant Kähler metric ǧ a obtained as the projection of the Levi form dη a to S 2m+1 /G a. ǧ a is Bochner-flat the Chern-Moser tensor of (S 2m+1, D, J) is zero; ǧ a is extremal i.e. Jgrad(Scalǧa ) is Killing for ǧ a the Cartan 4-th order tensor of (S 2m+1, D, J) is zero.

18 Kähler metrics via the Delzant construction (after V. Guillemin, M. Abreu and S. Donaldson) Theorem (Delzant, Lerman Tolman) There is a bijective correspondence between simply-connected compact toric symplectic manifolds/orbifolds (M 2m, ω, T m ); compact convex simple labelled polytopes (, L) in t = Lie(T m ), where = {y t L j (y) 0, j = 1,..., d} with dl j t spanning a lattice of t.

19 Kähler metrics via the Delzant construction (after V. Guillemin, M. Abreu and S. Donaldson) Theorem (Delzant, Lerman Tolman) There is a bijective correspondence between simply-connected compact toric symplectic manifolds/orbifolds (M 2m, ω, T m ); compact convex simple labelled polytopes (, L) in t = Lie(T m ), where = {y t L j (y) 0, j = 1,..., d} with dl j t spanning a lattice of t. Given (, L), M = N/G (,L) with N = µ 1 G (,L) (c) C d where: G (,L) T d with Lie algebra g (,L) = ker{(x 1,..., x d ) j x jdl j }; µ G(,L) : C d g G (,L) is the momentum map for G (,L). ω is the projection of the standard symplectic form of C d (not the Levi form of N!) and T m = T d /G (,L) acts on M.

20 Kähler metrics via the Delzant construction Theorem (Abreu, Guillemin) The standard Kähler metric on C d defines a T m -invariant Kähler metric (g G, ω) on M = N/G (,L) of the form with u G = 1 2 d j=1 L j log L j. g 0 = (u G ),ij dy i dy j + (u G ),ij dt i dt j,

21 Kähler metrics via the Delzant construction Theorem (Abreu, Guillemin) The standard Kähler metric on C d defines a T m -invariant Kähler metric (g G, ω) on M = N/G (,L) of the form with u G = 1 2 d j=1 L j log L j. g 0 = (u G ),ij dy i dy j + (u G ),ij dt i dt j, Any T m -invariant ω-compatible Kähler metric g on (M, ω) is given by g = u,ij dy i dy j + u,ij dt i dt j, with u u G smooth and det(u,ij )/det(u G,ij ) smooth and positive on ;

22 Kähler metrics via the Delzant construction Theorem (Abreu, Guillemin) The standard Kähler metric on C d defines a T m -invariant Kähler metric (g G, ω) on M = N/G (,L) of the form with u G = 1 2 d j=1 L j log L j. g 0 = (u G ),ij dy i dy j + (u G ),ij dt i dt j, Any T m -invariant ω-compatible Kähler metric g on (M, ω) is given by g = u,ij dy i dy j + u,ij dt i dt j, with u u G smooth and det(u,ij )/det(u G,ij ) smooth and positive on ; g is extremal iff u satisfies the Abreu equation m i,j=1 (u,ij ),ij + ( m j=1 p jy j ) + p 0 = 0.

23 Kähler metrics via the Delzant construction Theorem (R. Bryant-01, M. Abreu-01) The Bochner-flat metric on the weighted projective space corresponds to the function u B = 1 ( m m m ) L j log L j ( L j ) log( L j ), 2 j=0 defined on a labelled simplex (, L). In particular, the (unique) solution of the Abreu equation on a labelled simplex is explicit and induced by the Levi form of the (standard) CR structure S 2m+1. It is different in general from the Guillemin metric j=0 u G = 1 ( m L j log L j ). 2 j=0 j=0

24 CR structures and Levi Kähler reduction in arbitrary codimension. Recall: D TN a rank 2m, codimension l, distribution with Levi form L D C (N, 2 D TN/D) L D (X, Y ) = [X, Y ] mod D. If D admits complex structure J with [D 1,0, D 1,0 ] D 1,0, then L D is of type (1, 1) on D: (D, J) is Levi definite if there exists a section s C (N, (TN/D) ) with s L D > 0. (N, D, J) is called a codimension l Levi-definite CR manifold.

25 CR structures and Levi Kähler reduction in arbitrary codimension. Transversal CR action of a torus: G an l-dimensional torus which acts smoothly and semi-freely on N, by preserving (D, J) and satisfying the transversality condition TN = D {K v v g = Lie(G)}.

26 CR structures and Levi Kähler reduction in arbitrary codimension. Transversal CR action of a torus: G an l-dimensional torus which acts smoothly and semi-freely on N, by preserving (D, J) and satisfying the transversality condition TN = D {K v v g = Lie(G)}. N/G is a complex m-dimensional orbifold.

27 CR structures and Levi Kähler reduction in arbitrary codimension. Transversal CR action of a torus: G an l-dimensional torus which acts smoothly and semi-freely on N, by preserving (D, J) and satisfying the transversality condition TN = D {K v v g = Lie(G)}. N/G is a complex m-dimensional orbifold.! η Ω 1 (N, g) s.t. D = Ann(η) and η(k v ) = v v g

28 CR structures and Levi Kähler reduction in arbitrary codimension. Transversal CR action of a torus: G an l-dimensional torus which acts smoothly and semi-freely on N, by preserving (D, J) and satisfying the transversality condition TN = D {K v v g = Lie(G)}. N/G is a complex m-dimensional orbifold.! η Ω 1 (N, g) s.t. D = Ann(η) and η(k v ) = v v g TN/D = g N and η L D = dη D.

29 CR structures and Levi Kähler reduction in arbitrary codimension. Transversal CR action of a torus: G an l-dimensional torus which acts smoothly and semi-freely on N, by preserving (D, J) and satisfying the transversality condition TN = D {K v v g = Lie(G)}. N/G is a complex m-dimensional orbifold.! η Ω 1 (N, g) s.t. D = Ann(η) and η(k v ) = v v g TN/D = g N and η L D = dη D. If for c g, L D,c = c, η L D = c, dη D > 0 on (D, J) then the Kähler metric on N/G induced by L D,c is called Levi Kähler quotient of (N, D, J) by (G, c).

30 Product of spheres Model (flat) example: N = S 2m 1+1 S 2m l+1 C d with d = l + l i=1 2m i.

31 Product of spheres Model (flat) example: N = S 2m1+1 S 2ml+1 C d with d = l + l i=1 2m i. T d acts on N preserving the CR structure a Levi Kähler quotient via a transversal l-dimensional subtorus G T d is toric undert m = T d /G.

32 Product of spheres Model (flat) example: N = S 2m1+1 S 2ml+1 C d with d = l + l i=1 2m i. T d acts on N preserving the CR structure a Levi Kähler quotient via a transversal l-dimensional subtorus G T d is toric undert m = T d /G. M = CP m 1 CP m l endowed with a product of Fubini Study metrics is a (codimension l) Levi Kähler reduction of the flat CR model N = S 2m1+1 S 2ml+1.

33 Product of spheres Model (flat) example: N = S 2m1+1 S 2ml+1 C d with d = l + l i=1 2m i. T d acts on N preserving the CR structure a Levi Kähler quotient via a transversal l-dimensional subtorus G T d is toric undert m = T d /G. M = CP m 1 CP m l endowed with a product of Fubini Study metrics is a (codimension l) Levi Kähler reduction of the flat CR model N = S 2m1+1 S 2ml+1. As a (symplectic) toric manifold, M = CP m 1 CP m l corresponds to a product of standard (Delzant) simplices Σ = Σ 1 Σ l.

34 Product of spheres Model (flat) example: N = S 2m1+1 S 2ml+1 C d with d = l + l i=1 2m i. T d acts on N preserving the CR structure a Levi Kähler quotient via a transversal l-dimensional subtorus G T d is toric undert m = T d /G. M = CP m 1 CP m l endowed with a product of Fubini Study metrics is a (codimension l) Levi Kähler reduction of the flat CR model N = S 2m1+1 S 2ml+1. As a (symplectic) toric manifold, M = CP m 1 CP m l corresponds to a product of standard (Delzant) simplices Σ = Σ 1 Σ l. For any polytope, the set of its faces is partially ordered (by the inclusion of faces); has the combinatorial of Σ if there is a bijection between the corresponding faces which preserves the partial ordering.

35 Product of spheres Model (flat) example: N = S 2m1+1 S 2ml+1 C d with d = l + l i=1 2m i. T d acts on N preserving the CR structure a Levi Kähler quotient via a transversal l-dimensional subtorus G T d is toric undert m = T d /G. M = CP m 1 CP m l endowed with a product of Fubini Study metrics is a (codimension l) Levi Kähler reduction of the flat CR model N = S 2m1+1 S 2ml+1. As a (symplectic) toric manifold, M = CP m 1 CP m l corresponds to a product of standard (Delzant) simplices Σ = Σ 1 Σ l. For any polytope, the set of its faces is partially ordered (by the inclusion of faces); has the combinatorial of Σ if there is a bijection between the corresponding faces which preserves the partial ordering. E.g. any quadrilateral in R 2 has the combinatorial type of the square (product of simplices in R).

36 Main Results Model (flat) example: N = S 2m1+1 S 2ml+1 C d with d = l + l i=1 2m i. Any Levi Kähler quotient via a transversal l-dimensional subtorus G T d is toric undert m = T d /G. Theorem 1 (ACGL) A simply-connected compact toric symplectic orbifold M is the Levi Kähler quotient of the flat example w.r.t. a transversal subtorus G T d and c Lie(G) iff its (labelled) Delzant polytope (, L) has the same combinatorial type as Σ = Σ 1 Σ l. The ω-compatible Levi Kähler quotient metric g LK corresponds to the potential u LK = 1 2 l ( m i L ir log L ir ( m i ) ( m i ) ) L ir log L ir. i=1 r=0 r=0 r=0

37 Main Results Model (flat) example: N = S 2m1+1 S 2ml+1 C d with d = l + l i=1 2m i. Any Levi Kähler quotient via a transversal l-dimensional subtorus G T d is toric undert m = T d /G. Theorem 1 (ACGL) A simply-connected compact toric symplectic orbifold M is the Levi Kähler quotient of the flat example w.r.t. a transversal subtorus G T d and c Lie(G) iff its (labelled) Delzant polytope (, L) has the same combinatorial type as Σ = Σ 1 Σ l. The ω-compatible Levi Kähler quotient metric g LK corresponds to the potential u LK = 1 2 l ( m i L ir log L ir ( m i ) ( m i ) ) L ir log L ir. i=1 r=0 r=0 r=0 l = 1: the results by Abreu, Bryant, David Gauduchon

38 Application to extremal Kähler metrics Question. Given a labelled polytope (, L) with the combinatorics of Σ = Σ 1 Σ l, is the Levi Kähler quotient metric g LK extremal?

39 Application to extremal Kähler metrics Question. Given a labelled polytope (, L) with the combinatorics of Σ = Σ 1 Σ l, is the Levi Kähler quotient metric g LK extremal? Equivalently, is the function u LK = 1 2 l ( m i L ir log L ir ( m i ) ( m i ) ) L ir log L ir i=1 r=0 r=0 r=0 verifies Abreu equation m m LK ),ij + ( p j y j ) + p 0 = 0? (u,ij i,j=1 j=1

40 Computing the scalar curvature Assumption: Suppose (, L) has the combinatorics of the cube C m = [0, 1] [0, 1] in R m and that is projectively equivalent to C m (i.e. opposed facets of intersect on an affine hyperplane).

41 Computing the scalar curvature Assumption: Suppose (, L) has the combinatorics of the cube C m = [0, 1] [0, 1] in R m and that is projectively equivalent to C m (i.e. opposed facets of intersect on an affine hyperplane). By the Main Theorem, (, L) is the Delzant polytope of a simply-connected orbifold obtained as the quotient of S 3 S 3 and carries a Levi Kähler quotent metric (g LK, ω LK ).

42 If is generic (no parallel opposite facets), then g LK is given by the following ansatz (on int( ) T m ): ξ 1 = 1 m j=2 y j y 1, ξ j = y j + 1 y 1, m j=2 ξ jdt j θ 1 = dt 1 + m j=2 ξ jdt j, θ r = dt 1 + dt r, ξ ξ m ξ ξ m m m ( dξ 2 ) g LK = i ξ ξ m A i (ξ i ) + A i(ξ i )θ i, i=1 m m ω LK = dξ i θ i, ξ ξ m i=1 (1) where ξ i (α 1i, α 2i ) and (t 1,, t m ) are coordinates on T m and A i (x) = a i (x α 0i )(x α 1i )(x α 2i ), a i 0.

43 One thus computes [ s LK = (ξ ξ m ) (m + 1)(m + 2) ξ ξ m ( m i=1 ( m i=1 ) A i (ξ i ) + 2(m + 1) )] A i (ξ i ). ( m i=1 ) A i(ξ i )

44 One thus computes [ s LK = (ξ ξ m ) (m + 1)(m + 2) ξ ξ m ( m i=1 ( m i=1 ) A i (ξ i ) + 2(m + 1) )] A i (ξ i ). For m = 2 the extremal condition corresponds to A 1 (x) + A 2 ( x) = 0. ( m i=1 ) A i(ξ i )

45 One thus computes [ s LK = (ξ ξ m ) (m + 1)(m + 2) ξ ξ m ( m i=1 ( m i=1 ) A i (ξ i ) + 2(m + 1) )] A i (ξ i ). For m = 2 the extremal condition corresponds to A 1 (x) + A 2 ( x) = 0. This leads to the following Theorem 2 (ACGL) ( m i=1 ) A i(ξ i ) There exist a four parameter family of simply connected compact topic orbifolds M 4 p,q,r,s whose Delzant polytopes are (all possible) compact convex quadrilaterals and for which the Levi Kähler metric g LK is extremal. These extremal metrics are explicitly given by two cubics depending on 6 real coefficients.

46 Theorem (ACG-13) Any extremal toric metric (g, J, ω) on a simply connected compact toric symplectic 4-orbifold (M 4, ω) whose Delzant polytope is a quadrilateral is ambitoric in the sense that g admits an invariant complex structure I which induces the opposite orientation, and which is Kähler with respect to g = e f g.

47 Theorem (ACG-13) Any extremal toric metric (g, J, ω) on a simply connected compact toric symplectic 4-orbifold (M 4, ω) whose Delzant polytope is a quadrilateral is ambitoric in the sense that g admits an invariant complex structure I which induces the opposite orientation, and which is Kähler with respect to g = e f g. Theorem 3 (ACGL) The Levi Kähler metric g LK is the unique compatible toric Kähler metric on (M 4, ω) which is ambitoric, and Scal g is a Killing potential for g LK.

48 Theorem (ACG-13) Any extremal toric metric (g, J, ω) on a simply connected compact toric symplectic 4-orbifold (M 4, ω) whose Delzant polytope is a quadrilateral is ambitoric in the sense that g admits an invariant complex structure I which induces the opposite orientation, and which is Kähler with respect to g = e f g. Theorem 3 (ACGL) The Levi Kähler metric g LK is the unique compatible toric Kähler metric on (M 4, ω) which is ambitoric, and Scal g is a Killing potential for g LK. In the non-product case, g LK is extremal iff g is CSC.

49 Idea of proof of the Theorem 1 ((, L) (G, c)) N = S 2m1+1 S 2ml+1 C m1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 m i z iq 2 = } D = l i=1ker(d c σ i ) q=0

50 Idea of proof of the Theorem 1 ((, L) (G, c)) N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 1. (, L) = {y t : L iq (y) 0, i = 1,..., l; q = 0,..., m i } a labelled polytope; G = G (,L) T d, g = Lie(G (,L) ) R d as in the Delzant construction: g = ker{(x 1,..., x d ) d i=1 x jdl j t}. Can show g satisfies TN = D {K v v g} when is compact and convex. m i q=0

51 Idea of proof of the Theorem 1 ((, L) (G, c)) N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 1. (, L) = {y t : L iq (y) 0, i = 1,..., l; q = 0,..., m i } a labelled polytope; G = G (,L) T d, g = Lie(G (,L) ) R d as in the Delzant construction: g = ker{(x 1,..., x d ) d i=1 x jdl j t}. Can show g satisfies TN = D {K v v g} when is compact and convex. N/G (,L) is a compact complex orbifold and η Ω 1 (N, g) with dη D = η L D. m i q=0

52 Idea of proof of Theorem 1 N = S 2m1+1 S 2ml+1 C m1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 m i z iq 2 = } q=0 D = l i=1ker(d c σ i ) Step 2. Put c := ı (L iq (0)) g where ı : g R d ; then η c = η, c = l χ i d c σ i, (dη c ) D = i=1 l = χ i (dd c σ i ) D i=1 determines a Levi Kähler metric on N/G (,L) iff χ i > 0, with momentum map η c (K eiq ) = L iq µ c (z) = χ i (z) z iq 2 (µ c : N/G (,L) t always exists).

53 Idea of proof of Theorem 1 N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 3. Show that χ i > 0 on N: m i q=0

54 Idea of proof of Theorem 1 N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 3. Show that χ i > 0 on N: L iq µ c (z) = χ i (z) z iq 2 any point z 0 N with χ i (z 0 ) = 0 is mapped by µ c to m i q=0 {L iq = 0} m i q=0

55 Idea of proof of Theorem 1 N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 3. Show that χ i > 0 on N: L iq µ c (z) = χ i (z) z iq 2 any point z 0 N with χ i (z 0 ) = 0 is mapped by µ c to m i q=0 {L iq = 0} j i and p r s.t. L jp (µ c ([z 0 ]) = χ j (z 0 ) (z 0 ) jp 2 > 0 > L jr (µ c ([z 0 ])) = χ j (z 0 ) (z 0 ) jr 2, a contradiction. m i q=0

56 Idea of proof of Theorem 1 N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 3. Show that χ i > 0 on N: L iq µ c (z) = χ i (z) z iq 2 any point z 0 N with χ i (z 0 ) = 0 is mapped by µ c to m i q=0 {L iq = 0} j i and p r s.t. L jp (µ c ([z 0 ]) = χ j (z 0 ) (z 0 ) jp 2 > 0 > L jr (µ c ([z 0 ])) = χ j (z 0 ) (z 0 ) jr 2, a contradiction. χ i > 0 or χ i < 0 on N dη c is symplectic on N/G (,L) m i q=0

57 Idea of proof of Theorem 1 N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 3. Show that χ i > 0 on N: L iq µ c (z) = χ i (z) z iq 2 any point z 0 N with χ i (z 0 ) = 0 is mapped by µ c to m i q=0 {L iq = 0} j i and p r s.t. L jp (µ c ([z 0 ]) = χ j (z 0 ) (z 0 ) jp 2 > 0 > L jr (µ c ([z 0 ])) = χ j (z 0 ) (z 0 ) jr 2, a contradiction. χ i > 0 or χ i < 0 on N dη c is symplectic on N/G (,L) im µ c is the convex hull of the images of the fixed points of the T d /G (,L) action m i q=0

58 Idea of proof of Theorem 1 N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 3. Show that χ i > 0 on N: L iq µ c (z) = χ i (z) z iq 2 any point z 0 N with χ i (z 0 ) = 0 is mapped by µ c to m i q=0 {L iq = 0} j i and p r s.t. L jp (µ c ([z 0 ]) = χ j (z 0 ) (z 0 ) jp 2 > 0 > L jr (µ c ([z 0 ])) = χ j (z 0 ) (z 0 ) jr 2, a contradiction. χ i > 0 or χ i < 0 on N dη c is symplectic on N/G (,L) im µ c is the convex hull of the images of the fixed points of the T d /G (,L) action im µ c χ i > 0 m i q=0

59 Idea of proof of Theorem 1 N = S 2m 1+1 S 2m l+1 C m 1+1 C m l+1 = {z = (z iq ), i = 1,..., l, q = 0,... m i σ i (z) = 1 z iq 2 = } D = l i=1ker(d c σ i ) Step 3. Show that χ i > 0 on N: L iq µ c (z) = χ i (z) z iq 2 any point z 0 N with χ i (z 0 ) = 0 is mapped by µ c to m i q=0 {L iq = 0} j i and p r s.t. L jp (µ c ([z 0 ]) = χ j (z 0 ) (z 0 ) jp 2 > 0 > L jr (µ c ([z 0 ])) = χ j (z 0 ) (z 0 ) jr 2, a contradiction. χ i > 0 or χ i < 0 on N dη c is symplectic on N/G (,L) im µ c is the convex hull of the images of the fixed points of the T d /G (,L) action im µ c χ i > 0 = im µ c. m i q=0