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Complex geometry of moment-angle manifolds

Taras Panov joint with Yuri Ustinovsky and Misha Verbitsky Moscow State University

Group Action Forum Tsinghua Sanya International Mathematics Forum, China, 15­19 Decemb er 2014


1. Moment-angle manifolds from simplicial fans. a complete simplicial fan in Rn (not necessarily rational!) a1, . . . , am Rn generators of 1-dimensional cones K = K = I [m] : {ai : i I } spans a cone of the underlying simplicial complex of . ZK = (D2, S 1)K =
(
I K iI

{

}

D2 â


i I /

S1

)



(D2)m

the moment-angle manifold corresp onding to K (or ). U (K) = Cm \

{i1,...,ik }K /

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

Câ


i I /

Câ

)

the complement of a co ordinate subspace arrangement corresp onding to K. Note: ZK is a deformation retract of U (K) for every K.
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Define a map A : 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 A) =
{
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. They identified ZP with a class of complex manifolds known as LVM-manifolds (named after L´ ez de Medrano, op Verjovsky and Meersseman). Top ology of p olytopal moment-angle manifolds ZP is interesting and compliop cated. L´ ez de Medrano and Gitler identified their diffeomorphism typ es for many imp ortant series of p olytop es. 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 1. Cho ose a linear map : C Cm satisfying the two conditions: (a) Re : C Rm is a monomorphism. (b) A Re = 0. The comp osite map of the top line in the following diagram is zero:
C -

Cm

exp

- Rm -
|·|

Re

exp

-
exp A

A

Rn

exp

- Rm - - Rn > -- > where | · | denotes the map (z1, . . . , zm) (|z1|, . . . , |zm|). Now set C = exp (C) = where w = (w1, . . . , w) C, = (ij ).
{(

(Câ)m

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 A : R2 0 is a zero map. Let : C C2 b e given by z (z , z ) for some C, so that C = (ez , ez )} (Câ)2. Condition (b) of Constr 1 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 = ZK defining a complex structure on ZK in which Tm acts holomorphically.

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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 A : 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=
{ } 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. Manifolds ZK corresp onding to complete regular (in particular, rational ) 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 , such as Ho dge numb ers and Dolb eault cohomology. 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; ai = (ai1, . . . , ain) Zn. Constr 2 (`Cox construction'). Let AC : Cm Cn, ei ai, exp AC : (Câ)m (Câ)n, (z1, . . . , zm)
( m
m i=1

zi i1 , . . . ,

a

zi

ain

)

i=1

Set G = Ker exp AC. 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. V = U (K)/G the toric variety asso ciated to . The quotient torus (Câ)m/G = (Câ)n acts on V with a dense orbit.

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Observe that C = C G = (Câ)m-n as a complex subgroup. Prop 1. (a) The toric variety V is homeomorphic to the quotient of ZK morphic action of G/C .


by the holo-

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

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4. Submanifolds and analytic subsets. The complex structure on ZK is determined by two pieces of data: ­ the complete simplicial fan with generators a1, . . . , am; ­ the -dimensional holomorphic subgroup C (Câ)m. If this data is generic (in particular, the fan is not rational), then there is no holomorphic principal torus fibration ZK V over a toric variety V. However, there still exists a holomorphic -dimensional foliation F with a transverse K¨ ahler form F . This form can b e used to describ e submanifolds and analytic subsets in ZK .

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Consider the complexified map AC : Cm Cn, ei ai. and the following complex (m - n)-dimensional subgroup in (Câ)m: G = exp(Ker AC) = Note C G. The group G acts on U (K), and its orbits define a holomorphic foliation on U (K). Since G (Câ)m, this action is free on op en subset (Câ)m U (K), so that the generic leaf of the foliation has complex dimension m - n = 2. The -dimensional closed subgroup C G acts on U (K) freely and prop erly by Theorem 2, so that U (K)/C carries a holomorphic action of the quotient group D = G/C . F : the holomorphic foliation on U (K)/C = ZK by the orbits of D.
{(

ez1 , . . . , ezm

)

} â )m : (z , . . . , z ) Ker A . (C m 1 C

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The subgroup G (Câ)m is closed if and only if it is isomorphic to (Câ)2; in this case the subspace Ker A Rm is rational. Then is a rational fan and V is the quotient U (K)/G. The foliation F gives rise to a holomorphic principal Seifert fibration : ZK V with fibres compact complex tori G/C . For a generic configuration of nonzero vectors a1, . . . , am, G is biholomorphic to C2 and D = G/C is biholomorphic to C.

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A (1, 1)-form F on the complex manifold ZK is called transverse K¨ ahler with resp ect to the foliation F if (a) F is closed, i.e. dF = 0; (b) F is nonnegative and the zero space of F is the tangent space of F . A complete simplicial fan in Rn is called weakly normal if there exists a (not necessarily simple) n-dimensional p olytop e P such that is a simplicial sub division of the normal fan P . Thm 3. Assume that is a weakly normal fan. Then there exists an exact (1, 1)-form F on ZK = U (K)/C which is transverse K¨ ahler for the foliation F on the dense op en subset (Câ)m/C U (K)/C .

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For each J [m], define the corresp onding co ordinate submanifold in ZK by ZKJ = {(z1, . . . , zm) ZK : zi = 0 Obviously, ZKJ is identified with the quotient of U (KJ ) = {(z1, . . . , zm) U (K) : zi = 0 for i J } / by C = C. In particular, U (KJ )/C is a complex submanifold in ZK = U (K)/C . Observe that the closure of any (Câ)m-orbit of U (K) has the form U (KJ ) for some J [m] (in particular, the dense orbit corresp onds to J = [m]). Similarly, the closure of any (Câ)m/C -orbit of ZK = U (K)/C has the form ZKJ . for i J }. /

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Thm 4. Assume that the data defining a complex structure on ZK = U (K)/C is generic. Then any divisor of ZK is a union of co ordinate divisors. Furthermore, if is a weakly normal fan, then any compact irreducible analytic subset Y ZK of p ositive dimension is a co ordinate submanifold.

Cor 1. Under generic assumptions, there are no non-constant meromorphic functions on ZK .

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[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.

[GL] Samuel Gitler and Santiago L´ ez de Medrano. Intersections of quadrics, op moment-angle manifolds and connected sums. Preprint (2009); arXiv:0901.2580.

[MR] Laurent Meersseman and Alb erto Verjovsky. Holomorphic principal bundles over projective toric varieties. J. Reine Angew. Math. 572 (2004), 57­96.

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

PUV] Taras Panov, Yuri Ustinovsky and Misha Verbitsky. Complex geometry of moment-angle manifolds. Preprint (2013); arXiv:1308.2818.
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