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Complex-analytic structures on moment-angle manifolds

Taras Panov joint with Yuri Ustinovsky Moscow State University

2010


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

Following [BP] define the moment-angle complex ZK =
{

BI Dm
}

It is invariant under the co ordinatewise action of the standard torus Tm = on Cm.
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(z1, . . . , zm) Cm : |zi| = 1,

i = 1, . . . , m


Constr 1 (K-p ower). Let X b e a space, and W a subspace of X . Given I [m], set { } I = (x , . . . , x ) X m : x A for j I (X, W ) / Xâ W, =
1 m j i I i I /

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


(X, W )I X m.

I K

Then ZK = (D, T)K , where T is the unit circle. Another imp ortant example is the complement of the co ordinate subspace arrangement corresp onding to K: 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 [BP, Th. 8.9].
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{i1 ,...,ik }K /

{z Cm : zi1 = . . . = zik = 0},


Prop 1 ([BP]). 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 [BM], [BP]. Therefore, ZP can b e smo othed canonically. Now we shall lo ok at a wider class of simplicial complexes K: starshap ed spheres, or underlying complexes of complete simplicial fans.

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A set of vectors a 1, . . . , a k Rn generates a convex p olyhedral cone = {µ1a 1 + . . . + µk a k : µi R, µi 0}.

A cone is rational if its generators can b e chosen from Zn Rn, and is strongly convex if it do es not contain a line. A cone is simplicial (resp ectively, regular) if it is generated by a part of basis of Rn (resp ectively, Zn). 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 is rational (resp ectively, simplicial, regular) if every cone in is rational (resp ectively, simplicial, regular). 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 a 1, . . . , a m. Its underlying simplicial complex is K = I [m] : {a i : i I } spans a cone of Note: is complete iff |K| is a triangulation of S n-1.
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{

}


Now consider the linear map R : Rm Rn, e i a i,

where e 1, . . . , e m 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 : >

} a i ,u n, yi = 1 for all u R

Note: R = Rm-n if is complete (or contains an n-dimensional cone). > Both Rm and its subgroup R act on the complement U (K) Cm by co or> dinatewise 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 acts on U (K) freely and prop erly, and the quotient U (K)/R has a canonical structure of 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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Rem 1. The construction of the smo oth structure on ZK from Thm 1 do es dep end on the geometry of the fan . However, we exp ect that the smo oth structures coming from fans and are equivalent whenever the underlying simplicial complexes K and K are the same. This question is equivalent to that the quotients ZK /Tm and ZK /Tm are diffeomorphic as manifolds with corners whenever K = K . It is true in the p olytopal case, and also for those fans which are shellable.

Question 1. Describ e the class of sphere triangulations K for which the moment-angle manifold ZK admits a smo oth structure.

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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.

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. The column of matrix R corresp onding to the `empty' 1-cone is set to b e zero.
- Set = m2 n .

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Constr 2. Cho ose a linear map : C Cm satisfying two conditions: (a) Re : C Rm is a monomorphism. (b) R Re = 0. This corresp onds to cho osing a complex structure and sp ecifying a complex basis in the real vector space Ker R = R2. We also obtain that 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, [Meersseman].

{

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, on U (K) is free and prop er, and the quotient U (K)/C, has a canonical structure of a compact complex manifold of complex dimension m - ; (b) there is a Tm-equivariant diffeomorphism U (K)/C = Z defining a
K , m acts holomorphically. complex structure on ZK in which T
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Rem 2. Unlike the smo oth structure, the complex structure on ZK dep ends on b oth the geometry of and the choice of . (The latter is already clear from the torus example (Ex 1). 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 e 1, . . . , e n, -e 1 - . . . - e n. 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, by a generalisation of the construction of Meersseman and Verjovsky. This allows us to calculate invariants of complex structures on ZK . A toric variety is a normal algebraic variety X on which an algebraic torus (Câ)n acts with a dense 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; a 1, . . . , a m primitive integral generators of 1-cones. Constr 3 (`Cox construction'). Let C : Cm Cn, e i a i, 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, G as a complex -dimensional 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 3. For singular varieties X the quotient projection ZK X is a holomorphic principal Seifert bundle for an appropriate orbifold structure on X (same as in the projective case of [Meersseman­Verjovsky]).

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Given a complex n-dimensional manifold M , there is a decomp osition (M ) = p,q (M ) of the space of differential C-forms on M into a diC n, and the Dolb eault rect sum of the subspaces of (p, q )-forms for 1 p, q ¯ differential : p,q (M ) p,q+1(M ). p,q hp,q (M ) = dim H (M ): the Ho dge numb ers of 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), J = ( m akj vk , 1 j n). k=1 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.

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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 4. Let ZK b e as in Thm 3, 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 4. 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 4 implies that h1,0(ZK ) < h0,1(ZK ), and therefore ZK is not K¨ ahler (this was observed by [Meersseman] in the p olytopal case).

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Ex 3 (Calabi­Eckmann manifold). Let X = CP pâCP q with p q , so n = p+q , m = n + 2 and = 1. The cohomology ring is C[x, y ]/(xp+1, y q+1). Cho ose = (1, . . . , 1, , . . . , )t where the numb er of units is p + 1 and R. / This provides ZK = S 2p+1 â S 2q+1 with a structure of a complex manifold. It is the total space of a holomorphic principal bundle over 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 ) = ¯

This calculation is originally due to Borel.

q +1,q CE (p, q ) where H ¯

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

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.
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[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. [PU] Taras Panov and Yuri Ustinovsky. Complex-analytic structures on moment-angle manifolds. Preprint (2010), no. 2; arXiv:1008.4764.

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