Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/people/taras/talks/2013tianjin-talk.pdf Äàòà èçìåíåíèÿ: Wed Jul 10 20:43:04 2013 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:07:44 2016 Êîäèðîâêà:

Complex Geometry and Toric Top ology

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

International Conference on Algebraic and Geometric Top ology Chern Institute, Nankai University, Tianjin, 8­12 July 2013


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].
3


{i1 ,...,ik }K /

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

Câ


iI /

) â, C


Prop 1. 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).

4


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.
{ }

5


Given with 1-cones generated by a1, . . . , am, 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.
6


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 op complex manifolds known as LVM-manifolds (named after L´ ez de Medrano, Verjovsky and Meersseman). 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 .
7


Constr 2. 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.
8


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 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 .

9


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.

10


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.
11


3. Holomorphic bundles over toric varieties. 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

12


complete, simplicial, rational; a1, . . . , am primitive integral generators of 1-cones; ai = (ai1, . . . , ain) Zn. Constr 3 (`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.

13


Observe that C = C G = (Câ)m as a complex subgroup. Prop 2. (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.

14


4. Submanifolds and analytic subsets. The complex structure on ZK is determined by two pieces of data: ­ the complete simplicial fan with primitive 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 .

15


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

16


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.

17


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 .

18


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 }. /

19


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 .

20


[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. Moscow Math. J. 12 (2012), no. 1; arXiv:1008.4764

PUV] Taras Panov, Yuri Ustinovsky and Misha Verbitsky. SubmanifoldsComplexanalytic structures on moment-angle manifolds. Preprint (2013).

21