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Geometric structures on moment-angle manifolds
Taras Panov Moscow State University

based on joint works with Victor Buchstab er, Andrey Mironov, and Yuri Ustinovsky

80th Transp ennine Top ology Triangle meeting Manchester, 14 July 2011


Intersections of quadrics. Given a set of m vectors =
{

k = (1,k , . . . , m-n,k )t Rm-n,

k = 1, . . . , m ,

}

and a vector c = (c1, . . . , cm-n)t Rm-n, we consider the following intersections of m - n real quadrics Rm and Cm:
{
m k=1

R = u = (u1, . . . , um) Rm : Z = z = (z1, . . . , zm) Cm :
{

j k u2 = cj , k j k |zk |2 = cj ,

}

for 1

j

m-n ,
}

m k=1

for 1

j

m-n .

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Prop 1. Intersections of quadrics R and Z are nonempty and nondegenerate if and only if the following two conditions are satisfied: (a) c 1, . . . , m; (b) if c i1 , . . . ik , then k m - n.

Under these conditions, R and Z are smo oth submanifolds in Rm and Cm of dimension n and m + n resp ectively, and the vectors 1, . . . , m span Rm-n. From now on we assume that the conditions of Prop osition 1 are satisfied. Moreover, we assume that (c) the vectors 1, . . . , m generate a lattice L in Rm-n.
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Let L = { Rm-n : , Z for all L} b e the dual lattice. The torus Tm = (e2i1 , . . . acts on Z co ordinatewise. (corresp onding to (1, . . . ,
{ }

, e2im ) Similarly, 1m m) 2 Z )

Cm , where (1, . . . , m) Rm, the `real torus' (Z/2)m Tm acts on R.

The vectors i define an (m - n)-dimensional torus subgroup in Tm whose lattice of characters is L: T =
{(

e2 i1,, . . . , e2im,) Tm} = Tm-n,
(Z/2)m-n. =

where Rm-n. We also define 1 D = L /L 2 Note that D emb eds canonically

as a subgroup in T = Rm-n/L.
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Given a subset I [m] = {1, . . . , m}, define the sublattice LI = Zi : i I L. / Given u = (u1, . . . , um) Rm, define its zero set as Iu = {i : ui = 0} [m], and define Iz similarly for z = (z1, . . . , zm) Cm. A G-action is almost free if all isotropy subgroups are finite. Prop 2. The torus T acts on Z almost freely. The isotropy subgroup of z Z is given by Lz /L, where LIz = Zk : k Iz L. / I Pro of. An element (e2 i1,, . . . , e2 Z whenever e2 ik , = 1 for every equivalent to k , Z, that is, 1 T, the isotropy subgroup of z is
im , ) T fixes the given z k Iz . The latter condition is / Lz . Since L maps to I indeed Lz /L. I
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Lagrangian immersions. Let (M , ) b e an n-manifold then i(N ) is a is Hamiltonian a symplectic 2n-manifold. An immersion i : N M of N is Lagrangian if i( ) = 0. If i is an emb edding, Lagrangian submanifold of M . A vector field on M if the 1-form ( · , ) is exact.

Assume that a compatible Riemannian metric is chosen on M . A Lagrangian immersion i : N M is Hamiltonian minimal (H -minimal ) if the variations of the volume of i(N ) along all Hamiltonian vector fields with compact supp ort are zero, that is, d vol(it(N )) = 0, t=0 dt where i0(N ) = i(N ), it(N ) is a deformation of i(N ) along a Hamiltonian vector field, and vol(it(N )) is the volume of the deformed part of it(N ). An immersion is minimal if the variations of the volume of i(N ) along all vector fields are zero.
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Consider the map j : R â T - Cm, (u , ) u · = (u1e2 i1,, . . . , ume2 im,). Note that j (R â T) Z. The quotient N = R âD T is an m-dimensional manifold. Lemma 1. The map j : R â T Cm induces an immersion i : N Cm . Thm 1 (Mironov). The immersion i : N Cm is H-minimal La grangian. Moreover, if m k = 0, then i is a minimal Lagrangian k=1 immersion.
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Lagrangian emb eddings and moment-angle manifolds. Thm 2. The following conditions are equivalent: (1) i : N Cm is an emb edding of an H-minimal Lagrangian submanifold; (2) LIu = L for every u R; (3) T acts on Z freely.

This result op ens a way to construct explicitly new families of Hminimal Lagrangian submanifolds, once we have an effective metho d to pro duce nondegenerate intersections of quadrics R satisfying conditions (2) or (3) of Theorem 2. Toric top ology provides such a metho d.
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The quotient of R by the action of (Z/2)m (or the quotient of Z by the action of Tm) is identified with the set of nonnegative solutions of the following system of m - n linear equations:
m k=1

k yk = c .

This set may b e describ ed as a convex p olyhedron obtained by intersecting m halfspaces in Rn: P=
{

x Rn : a i, x + bi

0

for i = 1, . . . , m ,

}

(1)

Note that P may b e unb ounded; in fact P is b ounded if and only if R is b ounded (compact). Bounded p olyhedra are known as p olytop es. We refer to (1) as a presentation of P by inequalities. A presentation is generic if P is n-dimensional, has at least one vertex, and the hyp erplanes defined by the equations a i, x + bi = 0 are in general p osition at every p oint of P . If P is a p olytop e, then the existence of a generic presentation implies that P is simple.
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Given a generic presentation of a p olyhedron P , we may reconstruct the intersections of quadrics R and Z as follows. Consider the affine map iP : Rn Rm, iP (x ) = a 1, x + b1, . . . , a m, x + bm .
( )

It is monomorphic onto a certain n-dimensional plane in Rm (b ecause P has a vertex), and iP (P ) is the intersection of this plane with Rm. We define the space ZP from the commutative diagram ZP - Cm
µ
iZ

P

- Rm

iP

where µ(z1, . . . , zm) = (|z1|2, . . . , |zm|2). Note that Tm acts on ZP with quotient P , and iZ is a Tm-equivariant emb edding.
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If the presentation of P is generic, then ZP is a smo oth manifold of dimension m + n, known as the (p olytopal) moment-angle manifold corresp onding to P . Now we can write the n-dimensional plane iP (Rn) by m - n linear equations in Rm. Replacing each yk by |zk |2 we obtain a presentation of the moment-angle manifold ZP as an intersection of quadrics. By replacing Cm by Rm we obtain the real moment-angle manifold RP .

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Clearly, 1, . . . , m generate a lattice L in Rm-n if and only if a 1, . . . , a m generate a lattice in Rn. The corresp onding P are called rational. If P is rational, then we have a map of lattices x (a 1, x , . . . , a m, x ). conjugate gives rise to a map of tori Rm/Zm Rn/, whose kernel denote by TP . It b ecomes T under the identification of ZP with . We also have DP = (Z/2)m-n and NP = RP âDP TP . AP : Zm,

Its we Z

The manifolds RP , ZP , NP represent the same geometric objects as R, Z, N, although a different initial data is used in their definition. P is Delzant if it is rational and for every vertex x P the vectors a j1 , . . . , a jn normal to the facets meeting at x constitute a basis of = Za 1, . . . , a m. Thm 3. The map NP = RP âDP TP Cm is an emb edding if and only if P is a Delzant p olyhedron.
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Top ology of Lagrangian submanifolds N . Toric top ology provides large families of explicitly constructed Delzant p olytop es: · simplices and cub es in all dimensions; · pro ducts and face cuts; · asso ciahedra (Stasheff p olytop es), p ermutahedra, and general nestohedra. Nevertheless, the top ology of ZP (and therefore of NP ) is very complicated in general. Cohomology rings of ZP are describ ed by [Buchstab er-P.], and explicit homotopy and diffeomorphism typ e for some particular families of P are given by [Bahri­Bendersky­Cohen­ Gitler], [Gitler­Lop ez de Medrano], [Grbi´ c­Theriault], and others.
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Prop 3. (a) The immersion of N in Cm factors as N

Z Cm;

(b) N is the total space of a bundle over a torus T m-n with fibre R; (c) if N Cm is an emb edding, then N is the total space of a principal T m-n-bundle over the n-dimensional manifold R/DP . Pro of. Statement (a) is clear. Since DP acts freely on TP , the projection N = R âDP TP TP /DP onto the second factor is a fibre bundle with fibre R. Then (b) follows from the fact that TP /DP = T m-n. If N Cm is an emb edding, then TP acts freely on Z . The action of DP on R is also free. Therefore, the projection N = RâDP TP R/DP onto the first factor is a principal TP -bundle, which proves (c).
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Ex 1 (one quadric). Let m - n = 1, that is, R is given by 1u2 + . . . + mu2 = c. m 1 If R is compact, then R = S m-1, and
m-1 â S 1 1 S S m -1 â N= Z/2 S = m K

(2)

if preserves the orientation of S m-1, if reverses the orientation of S m-1,

where Km is an m-dimensional Klein b ottle. Prop 4. We obtain an H-minimal Lagrangian emb edding of N = S n-1 âZ/2 S 1 in Cm if and only if 1 = . . . = m in (2). The top ological typ e of N = N (m) dep ends only on the parity of m, and is given by N (m) N (m) = S m-1 â S 1 = Km if m is even, if m is o dd.

The Klein b ottle Km with even m do es not admit Lagrangian emb eddings in Cm [Nemirovsky, Shevchishin].
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Ex 2 (two quadrics). Thm 4. Let m - n = 2.

(a) R is diffeormorphic to R(p, q ) = S p-1 â S q-1 given by u2 + . . . + u2 + u2+1 + . . . + u2 = 1, p 1 k k +u2+1 + . . . + u2 = 2, u2 + . . . + u2 m 1 p k where p + q = m, 0 < p < m and 0 k p. (b) If N Cm is an emb edding, then N is diffeomorphic to Nk (p, q ) = R(p, q ) âZ/2âZ/2 (S 1 â S 1), (4) where R(p, q ) is given by (3) and the two involutions act on it by 1 : (u1, . . . , um) (- 2 : (u1, . . . , um) (- u1, . . . , -uk , -uk+1, . . . , -up, up+1, . . . , um), (5) u1, . . . , -uk , uk+1, . . . , up, -up+1, . . . , -um). (3)

There is a fibration Nk (p, q ) S q-1 âZ/2 S 1 = N (q ) with fibre N (p) (the manifold from the previous Example), which is trivial for k = 0.
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Ex 3 (three quadrics). In the case m - n = 3 the top ology of compact manifolds R and Z was fully describ ed by [Lop ez de Medrano]. Each manifold is diffeomorphic to a pro duct of three spheres, or to a connected sum of pro ducts of spheres, with two spheres in each pro duct. The simplest p olytop e P with m - n = 3 is a p entagon. It has many Delzant realisations, for instance, P=
{

(x1, x2) R2 : x1

0, x2

0, -x1+2

0, -x2+2

0, -x1-x2+3

0.

}

In this case RP is an oriented surface of genus 5, and ZP is diffeomorphic to a connected sum of 5 copies of S 3 â S 4. We therefore obtain an H-minimal Lagrangian submanifold NP C5 which is the total space of a bundle over T 3 with fibre a surface of genus 5.
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Manifolds RP corresp onding to p olygons are describ ed as follows. Prop 5. Assume that n = 2 the 2-dimensional p olytop e P corresp onding to R is an m-gon. Then R is an orientable surface Sg of genus g = 1 + 2m-3(m - 4). The H-minimal Lagrangian submanifold N Cm corresp onding to R from Prop osition 5 is a total space of a bundle over T m-2 with fibre Sg . It is an aspherical manifold (for m 4) whose fundamental group enters the short exact sequence 1 - 1(Sg ) - 1(N ) - Zm-2 - 1. For n > 2 and m - n > 3 the top ology of R and Z is even more complicated.
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Other geometric structures on moment-angle manifolds ZP include · non-K¨ ahler complex-analytic structures [Bosio­Meersseman, P.Ustinovsky, Tamb our] · T m-invariant metrics of p ositive Ricci curvature [Bazaikin]

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[1] Victor Buchstab er and Taras Panov. Torus Actions and Their Applications in Top ology and Combinatorics. University Lecture Series, vol. 24, Amer. Math. So c., Providence, R.I., 2002. [2] Andrey Mironov and Taras Panov. Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian emb eddings. Preprint (2011); arXiv:1103.4970. [3] Taras Panov. Moment-angle manifolds and complexes. Trends in Mathematics - New Series. Information Center for Mathematical Sciences, KAIST. Vol. 12 (2010), no. 1, pp. 43­69; arXiv:1008.5047. [4] Taras Panov and Yuri Ustinovsky. Complex-analytic structures on moment-angle manifolds. Moscow Math. J. (2011), to app ear (2010); arXiv:1008.4764.
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