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Non-K¨ ahler complex structures on moment-angle manifolds and other toric spaces

Taras Panov joint with Yuri Ustinovsky Moscow State University

Several Complex Variables Steklov Institute, Moscow, 27 February­3 March 2012


1. Moment-angle complexes and manifolds. K an (abstract) simplicial complex on the set [m] = {1, . . . , m}. I = {i1, . . . , ik } K a simplex. Always assume K. Allow {i} K for some i (ghost vertices). / Consider the unit p olydisc in Cm, Dm = Given I [m], set BI :=
{

(z1, . . . , zm) Cm : |zi|
{

1,

i = 1, . . . , m .

}

} m : |z | = 1 for j I . (z1, . . . , zm) D / j
I K

Define the moment-angle complex ZK =
{

BI Dm
}

It is invariant under the co ordinatewise action of the standard torus Tm = on Cm.
2

(z1, . . . , zm) Cm : |zi| = 1,

i = 1, . . . , m


Constr 1 (p olyhedral pro duct). Given spaces W X and I [m], set (X, W )I =
{

(x1, . . . , xm) X m : xj W for j I /

}

X â W, =
iI i I /

and define the p olyhedral pro duct of (X, W ) as (X, W )K =

I K

(X, W )I X m.

Then ZK = (D, T)K , where T is the unit circle. Another example is the complement of a co ordinate subspace arrangement: U (K) = Cm \ namely, U (K) = (C, Câ)K = where Câ = C \ {0}. Clearly, ZK U (K). Moreover, ZK is a Tm-equivariant deformation retract of U (K) for every K [Buchstab er-P].
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{i1 ,...,ik }K /

{z Cm : zi1 = . . . = zik = 0},
(
I K iI

Câ


iI /

) â, C


Prop 1 ([Buchstab er-P]). Assume |K| = S n-1 (a sphere triangulation with m vertices). Then ZK is a closed manifold of dimension m + n. We refer to such ZK as moment-angle manifolds. If K = KP is the dual triangulation of a simple convex p olytop e P , then ZP = ZKP emb eds in Cm as a nondegenerate (transverse) intersection of m - n real quadratic hyp ersurfaces. Therefore, ZP can b e smo othed canonically. Now assume K is the underlying complex of a complete simplicial fan (a starshap ed sphere).

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A fan is a finite collection = {1, . . . , s} of strongly convex cones in Rn such that every face of a cone in b elongs to and the intersection of any two cones in is a face of each. A fan = {1, . . . , s} is complete if 1 . . . s = Rn. Let b e a simplicial fan in Rn with m one-dimensional cones generated by a1, . . . , am. Its underlying simplicial complex is K = I [m] : {ai : i I } spans a cone of Note: is complete iff |K| is a triangulation of S n-1.
{ }

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Given with 1-cones generated by a1, . . . , am, define a map R : Rm Rn, ei a i ,

where e1, . . . , em is the standard basis of Rm. Set Rm = {(y1, . . . , ym) Rm : yi > 0}, > and define R := exp(Ker R) =
{
m i=1

(y1, . . . , ym) Rm : >

} ai ,u n, yi = 1 for all u R

R Rm acts on U (K) Cm by co ordinatewise multiplications. > Thm 1. Let b e a complete simplicial fan in Rn with m one-dimensional cones, and let K = K b e its underlying simplicial complex. Then (a) the group R = Rm-n acts on U (K) freely and prop erly, so the quotient


U (K)/R is a smo oth (m + n)-dimensional manifold; (b) U (K)/R is Tm-equivariantly homeomorphic to ZK . Therefore, ZK can b e smo othed canonically.
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2. Complex-analytic structures. We shall show that the even-dimensional moment-angle manifold ZK corresp onding to a complete simplicial fan admits a structure of a complex manifold. The idea is to replace the action of Rm-n on U (K) (whose quotient > is ZK ) by a holomorphic action of C
m- n 2

on the same space.

Rem 1. Complex structures on p olytopal moment-angle manifolds ZP were describ ed by Bosio and Meersseman. Existence of complex structure on moment-angle manifolds corresp onding to complete simplicial fans has b een also recently and indep endently established by Tamb our. Assume m - n is even from now on. We can always achieve this by formally adding an `empty' one-dimensional cone to ; this corresp onds to adding a ghost vertex to K, or multiplying ZK by a circle.
- Set = m2 n .
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Constr 2. Cho ose a linear map : C Cm satisfying the two conditions: (a) Re : C Rm is a monomorphism. (b) R Re = 0. The comp osite map of the top line in the following diagram is zero:
C - R Re m m C - R - - - exp exp

Rn

exp

|·| exp R (Câ)m - Rm - - - Rn - - > > where | · | denotes the map (z1, . . . , zm) (|z1|, . . . , |zm|). Now set

C, = exp (C) = where w = (w1, . . . , w) C, = (ij ).

{(

e1,w, . . . , em,w

)

(Câ)m

}

i denotes the ith row of the m â -matrix

Then C, = C is a complex-analytic (but not algebraic) subgroup in (Câ)m. It acts on U (K) by holomorphic transformations.
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Ex 1. Let K b e empty on 2 elements (that is, K has two ghost vertices). We therefore have n = 0, m = 2, = 1, and R : R2 0 is a zero map. Let : C C2 b e given by z (z , z ) for some C, so that C = C, = (ez , ez )} (Câ)2. Condition (b) of Constr 2 is void, while (a) is equivalent to that R. Then / exp : C (Câ)2 is an emb edding, and the quotient (Câ)2/C with the natural 2 complex structure is a complex torus TC with parameter C:
2 (Câ)2/C = C/(Z Z) = TC ().

{

Similarly, if K is empty on 2 elements (so that n = 0, m = 2), we may obtain 2 any complex torus TC as the quotient (Câ)2/C,.

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Thm 2. Let b e a complete simplicial fan in Rn with m one-dimensional cones, and let K = K b e its underlying simplicial complex. Assume that m - n = 2. Then (a) the holomorphic action of the group C, = C on U (K) is free and prop er, so the quotient U (K)/C, is a compact complex (m - )-manifold; (b) there is a Tm-equivariant diffeomorphism U (K)/C = Z defining a
,

complex structure on ZK in which Tm acts holomorphically.

K

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Ex 2 (Hopf manifold). Let b e the complete fan in Rn whose cones are generated by all prop er subsets of n + 1 vectors e1, . . . , en, -e1 - . . . - en. To make m - n even we add one `empty' 1-cone. We have m = n + 2, = 1. Then R : Rn+2 Rn is given by the matrix (0 I -1), where I is the unit n â n matrix, and 0, 1 are the n-columns of zeros and units resp ectively. We have that K is the b oundary of an n-dim simplex with n + 1 vertices and 1 ghost vertex, ZK = S 1 â S 2n+1, and U (K) = Câ â (Cn+1 \ {0}). Take : C Cn+2, z (z , z , . . . , z ) for some C, R. Then / C = C , =
{ } z , ez , . . . , ez ) : z C (Câ)n+2 , (e

and ZK acquires a complex structure as the quotient U (K)/C :

( )/ ( )/ â â Cn+1 \ {0} z t, ez w)} Cn+1 \ {0} C {(t, w) (e {w e2 iw}, =

where t Câ, w Cn+1 \ {0}. The latter quotient of Cn+1 \ {0} is known as the Hopf manifold.
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3. Holomorphic bundles over toric varieties, and Ho dge numb ers. Manifolds ZK corresp onding to complete regular simplicial fans are total spaces of holomorphic principal bundles over toric varieties with fibre a complex torus. This allows us to calculate invariants of the complex structures on ZK . A toric variety is a normal algebraic variety X on which an algebraic torus (Câ)n acts with a dense (Zariski op en) orbit. Toric varieties are classified by rational fans. Under this corresp ondence, complete fans compact varieties normal fans of p olytop es projective varieties regular fans nonsingular varieties simplicial fans orbifolds

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complete, simplicial, rational; a1, . . . , am primitive integral generators of 1-cones. Constr 3 (`Cox construction'). Let C : Cm Cn, ei ai, exp C : (Câ)m (Câ)n, (z1, . . . , zm)
( m
m i=1

zi i1 , . . . ,

a

zi

ain

)

i=1

Set G = Ker exp C. This is an (m - n)-dimensional algebraic subgroup in (Câ)m. It acts almost freely (with finite isotropy subgroups) on U (K). If is regular, then G = (Câ)m-n and the action is free. X = U (K)/G the toric variety asso ciated to . The quotient torus (Câ)m/G = (Câ)n acts on X with a dense orbit.

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Observe that C = C, G = (Câ)m as a complex subgroup. Prop 2. (a) The toric variety X is homeomorphic to the quotient of ZK holomorphic action of G/C,.


by the

(b) If is regular, then there is a holomorphic principal bundle ZK X with fibre the compact complex torus G/C, of dimension . Rem 2. For singular varieties X the quotient projection ZK X is a holomorphic principal Seifert bundle for an appropriate orbifold structure on X.

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p,q hp,q (M ) = dim H (M ): the Ho dge numb ers of a complex manifold M . ¯

The Dolb eault cohomology of a complex torus is given by
( )( ) 1,0 0,1 2 2 2 where 1, . . . , H (TC ), 1, . . . , H (TC ). Hence, hp,q (TC ) = p q . ¯ ¯
, 2 H (TC ) = [1, . . . , , 1, . . . , ], ¯

The Dolb eault cohomology of a complete nonsingular toric variety X is given by [Danilov­Jurkiewicz]:
, H (X) = C[v1, . . . , vm]/(IK + J), ¯

where vi H (X), ¯ ) ( / IK = vi1 · · · vik : {i1, . . . , ik } K (the Stanley­Reisner ideal), m J = ( k=1 akj vk , 1 j n). We have hp,p(X) = hp, where (h0, h1, . . . , hn) is the h-vector of K, and hp,q (X) = 0 for p = q .
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1,1


By an application of the Borel sp ectral sequence to the holomorphic bundle ZK X we obtain the following description of the Dolb eault cohomology. Thm 3. Let b e a complete rational nonsingular fan. Then the Dolb eault p,q cohomology group H (ZK ) is isomorphic to the (p, q )-th cohomology group ¯ of the differential bigraded algebra
[
, [1, . . . , , 1, . . . , ] H (X), d ¯

]

whose differential d of bidegree (0, 1) is defined on the generators as dvi = dj = 0,
2 where c : H (TC ) H 2( ¯ the torus principal bundle 1,0

dj = c(j ),
1,1

1

i

m, 1

j

,

X, C) = H (X) is the first Chern class map of ¯ ZK X.

This result may b e compared to the analogous description of the ordinary cohomology of ZK from [Buchstab er-P]: Thm 4. H (ZK ) is isomorphic to the cohomology of the dga
[ ] (X ), d , [u1, . . . , um-n] H

with deg uj = 1, deg vi = 2, and differential d defined on the generators as dvi = 0, duj = j 1v1 + . . . + j mvm, 1 i m, 1 j m - n.
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( ) ( ) k - [k/2] p,0 (a) h for p 0; p p () 0,q = for q (b) h 0; q ( ) ( ) 1,q = ( - k ) 1,0 +1 for q (c) h 1; q -1 + h q (d) (3+1) - h2(K) - k + ( + 1)h2,0 h2,1 2

Thm 5. Let ZK b e as in ab ove, and let k b e the numb er of ghost vertices in K. Then the Ho dge numb ers hp,q = hp,q (ZK ) satisfy

(3+1) - k + ( + 1)h2,0. 2

Rem 3. At most one ghost vertex is required to make dim ZK = m + n even. Note that k 1 implies hp,0(ZK ) = 0, so that ZK do es not have holomorphic forms of any degree in this case. If ZK is a torus, then m = k = 2, and h1,0(ZK ) = h0,1(ZK ) = . Otherwise Thm 5 implies that h1,0(ZK ) < h0,1(ZK ), and therefore ZK is not K¨ ahler.

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Ex 3 (Calabi­Eckmann manifold). Let K = p â q with p p + q , m = n + 2 and = 1.

q , so n =

Then U (K) = (Cp+1 \ {0}) â (Cq+1 \ {0}). Cho ose = (1, . . . , 1, , . . . , )t where the numb er of units is p + 1 and R. Have exp : C (Câ)m. / This gives ZK = U (K)/C = S 2p+1 â S 2q+1 a structure of a complex manifold. It is the total space of a holomorphic principal bundle over X = CP p â CP q with fibre a complex torus C/(Z Z), a Calabi­Eckmann manifold CE (p, q ). By Thm 3, where dx = dy = d = 0 and d = x - y for an appropriate choice of x, y . We therefore obtain
( ) , H CE (p, q ) = [ , ] C[x]/(xp+1), ¯ ( ) ( ) [ ] , H [ , ] C[x, y ]/(xp+1, y q+1), d , H CE (p, q ) = ¯

q +1,q CE (p, q ) where H ¯

xq+1 -y q+1 is the cohomology class of the co cycle . x-y

This calculation is originally due to Borel.

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Ex 4. The pro duct S 3 â S 3 â S 5 â S 5 has has two complex structures as a pro duct of Calabi­Eckmann manifolds, namely, CE (1, 1) â CE (2, 2) and CE (1, 2) â CE (1, 2). In the first case h2,1 = 1, and h2,1 = 0 in the second.


[BM] Fr´ ´ ederic Bosio and Laurent Meersseman. Real quadrics in Cn, complex manifolds and convex p olytop es. Acta Math. 197 (2006), no. 1, 53­127.

[BP] 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.

[LV] Santiago L´ ez de Medrano and Alb erto Verjovsky. A new family of op complex, compact, non-symplectic manifolds. Bol. So c. Mat. Brasil. 28 (1997), 253­269.

[PU] Taras Panov and Yuri Ustinovsky. Complex-analytic structures on moment-angle manifolds. Moscow Math. J. 12 (2012), no. 1; arXiv:1008.4764

[Ta] J´ ome Tamb our. LVMB manifolds and simplicial spheres. er^ (2010); arXiv:1006.1797.

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