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Analogous p olytop es, toric manifolds and complex cob ordisms
Victor M. Buchstab er Steklov Mathematical Institute, Russian Academy of Sciences, Moscow
< buchstab@mendeleev o.ru >

Scho ol of Mathematics, University of Manchester, Manchester
< V ictor.B uchstaber@manchester.ac.uk >

The results of joint pap ers with Taras Panov and Nige Ray
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Our aim is to bring geometric and combinatorial metho ds to b ear on the study omnioriented toric manifolds M , in the context of stably complex manifolds with compatible torus action. We interpret M in terms of combinatorial data (P, ), where P is the combinatorial typ e of an oriented simple p olytop e, and is an integral matrix whose prop erties are controlled by P . By way of application, we study conditions on (P, ) such that the corresp onding toric manifold admits sp ecial unitary and level-N structures, and develop combinatorial formulae for the evaluation of genera in terms of sub circles of the torus action. We provide a discussion of the complex cob ordism ring in the term of omnioriented toric manifolds.
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Prelude to toric manifolds Consider a smo oth 2n-dimensional manifold M 2n, endowed with a smo oth action of n-dimensional torus T n. The orbit space M 2n/T n is a manifold with corners. The most basic manifolds with corners are the simple convex p olytop es, i.e. n-dimensional convex p olytop es with exactly n facets meeting at each vertex. A simple convex p olytop e is generic; its b ounding hyp erplanes are in general p osition. Each face of a simple p olytop e is again a simple p olytop e.
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Complex projective space

CP n = S

2n+1 /S 1

where S 2n+1 Cn+1 consists of vectors
n+1

z = (z1, . . . , zn+1)

with

|z |2 =
k=1

|z k | 2 = 1

and t1 S 1 acts by t1 · z = (t1z1, . . . , t1zn+1). The standard action of T n+1 on Cn+1 is t · z = (t1z1, . . . , tn+1zn+1), and induces an action of T n on CP n by t · [z ] = [t1z1, . . . , tnzn, zn+1]. The orbit space is the n-simplex n, given by
n+1

{(x1, . . . , xn+1) R

n+1 ,

xi

0,
k=1

xi = 1 } .

The projection : CP n - n acts by (z1 : z2 : · · · : zn+1) - (|z1|2, |z2|2, . . . , |zn+1|2).
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The b ounded flag manifolds Bn A p oint U = (U1 U2 · · · Un+1) Bn is a complete flag in Cn+1, such that each Uk , contains the subspace Ck-1, spanned by the first (k - 1) standard basis vectors in Cn+1 for 2 k n. So U is equivalent to a sequence of lines {Lk Ck Lk+1, 1 k n},

where Ck is k-th co ordinate line and L1 = U1, Ln+1 = Cn+1. The standard action of T n+1 on Cn+1 induces an action of T n on Bn, whose orbit space is the cub e I n = I â · · · â I . The projection : Bn - I n satisfies (U ) = ( (L1), . . . , (Ln)), where (Lk ) = |lk |2 and lk is the projection of a unit vector from Lk Ck Lk+1 into Ck .
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The manifold Bi,j Bi,j is a 2n-dim smo oth CP j -1-bundle over Bi, where n = (i + j - 1) and 0 i Each U B a line p oint of Bi,j is a pair (U, W ), where i+1 and W is i is a b ounded flag in C in U1 Cj -i. The projection : Bi,j - Bi satisfies (U, W ) = U . So T n = T i â T j -1 acts on Bi,j by (t1z1, . . . , tizi, zi+1, ti+1w1, . . . , tnwj -1, wj ). The orbit space is I i â j -1, and the quotient map Bi,j - I i â j -1 is given by (U, W ) = ( (U ), (W )).
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j.


Moment-angle manifolds We deal only with simple p olytop es, and reserve the notation m = m(P ) for the numb er facets of P . Every face of co dimension k may b e written uniquely as FI = Fi1 · · · Fik for some subset I = {i1, . . . , ik } of [m]. We denote the i-th co ordinate sub circle of the standard m­torus T m by Ti for every 1 i m. Given any subset I [m], we define the subgroup by TI =
iI

(1)

Ti T m,

in particular, T is the trivial subgroup {1}.

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Every p oint p of P lies in the interior of a unique face FIp , where Ip = {i : p Fi}, and it is convenient to abbreviate FIp to F (p) and TIp to T (p). If p is a vertex, then T (p) has dimension n (the maximum p ossible), and if p is an interior p oint of P , then T (p) is trivial. We now construct the identification space

ZP = T

m â P /

(2)

where (t1, p) (t2, p) if and only if t-1t2 T (p). 1 So ZP is an (m + n)-dim manifold with a canonical left T m-action, whose isotropy subgroups are precisely the subgroups T (p).
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Many imp ortant examples of manifolds in top ology and geometry arise as factor spaces of ZP by an action of an appropriate subtorus T k T m. Define s = s(P ) to b e the maximal dimension for which there exists a subgroup H = T s in T m acting freely on ZP . The numb er s(P ) is obviously a combinatorial invariant of P . We have 1 s (P ) m - n.

In the case s(P ) = m - n the quotient space M 2n = ZP /T m-n is smo oth and called toric manifold. Any nonsingular compact toric variety is a toric manifold.
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Dicharacteristic In order to construct toric manifolds over P , we need one further set of data. This consists of a homomorphism : T m T n, whose prop erties are controlled by P (cf. Davis and Januszkiewicz), namely FI a face of co dim k = monic on TI . (3)

Any such is called a dicharacteristic; condition (3) ensures that the kernel K ( ) of is isomorphic to an (m - n)­dim subtorus of T m. Wherever p ossible we abbreviate K ( ) to K . We write the sub circle (Ti) T n as T (Fi) for any 1 i m, and the subgroup (TI ) as T (FI ) for any face FI . For each p oint p in P let S (p) denote the subgroup T (F (p)); it is, of course, (T (p)). For example, S (w) = T n for any vertex w, and S (p) = {1} for any p oint p in the interior of P .
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Refined form of dicharacteristic matrix Applied to the initial vertex v = F1 · · · Fn, (3) ensures that the restriction of to T1 â · · · â Tn is an isomorphism. So we may use the circles T (F1), . . . , T (Fn) to define a basis for the Lie algebra of T n, and represent the homomorphism induced by by an n â m integral matrix 1 0 . . . 0 1,n+1 0 1 . . . 0 2,n+1 = . . .. . . . . . .. . . . . 0 0 . . . 1 n,n+1


. . . 1,m . . . 2,m . . ... . . . . . n,m



(4)

Given any other vertex v = Fj1 · · · Fjn , (3) implies that the corresp onding columns j1 , . . . , jn form a basis for Zn, and have determinant ±1. We refer to (4) as the refined form of the dicharacteristic matrix.
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A mo del for toric manifolds Since K = K ( ) acts freely on ZP there is a principal K -bundle : ZP M , whose base space is a 2n­dim manifold. By construction, M = T n â P / where (s1, p) (s2, p) if and only if s-1s2 S (p). 1 Furthermore, M admits a canonical T n-action , which is lo cally isomorphic to the standard action on Cn, and has quotient map : M P. Note that · is the natural projection P : ZP P .
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(5)


The fixed p oints of project to the vertices of P , and we refer to -1(v ) as the initial fixed p oint x . Then (5) identifies a neighb ourho o d of x with Cn, on which is standard; its representation at other fixed p oints -1(v ) may b e read off from the corresp onding columns of . The quadruple (M , , , P ) is an example of Davis and Januszkiewicz's toric manifolds. Any manifold with a similarly well-b ehaved torus action over P is equivariantly diffeomorphic to one of the form (5).

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Facial submanifold structure The facial submanifolds Mi of M are defined as the inverse images of the facet Fi under , for 1 i m. Every Mi has co dim 2, with isotropy subgroup T (Fi) T n. The quotient map

ZP â

K

Ci - M

(6)

defines a canonical complex line-bundle i, whose restriction to Mi is isomorphic to the normal bundle i of its emb edding in M . The submanifolds Mi are mutually transverse, and we write any non-empty intersection as MI = Mi1 · · · Mik . (7) So MI is the inverse image under of the co dim k face FI . MI has co dim 2k, and its isotropy subgroup is T (FI ). The restriction of I = i1 · · · ik to MI is isomorphic to the normal bundle I of its emb edding in M , for any face FI .
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The cohomology ring of a toric manifold The bundles i are imp ortant in understanding the integral cohomology ring of M . Let ui b e the first Chern class c1( then H (M ) is generated by the u1, . . . , um, mo dulo two sets of i) in H 2(M ); elements relations.

The first are linear, and arise from the refined form (4) of the dicharacteristic; the second are monomial, and arise from the Stanley-Reisner ideal of P . The linear relations take the form ui = -i,n+1un+1 - . . . - i,mum (8)

for 1 i n. So, un+1, . . . , um suffice to generate H (M ) multiplicatively.
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Fixed p oints of sub circle actions Given the action of T n on M , it is natural to study its restriction to an arbitrary sub circle T T n. We may decomp ose the fixed p oint set of T into the union of its comp onents as Fix(T ) = MI (1) · · · MI (s) · · · MI (d), (9) Following (4), we represent T by a primitive column vector l = l(T ) in Zn.

Prop osition. The comp onents of Fix(T ) are sp ecified by those I (s) for which none of the co efficients i(s)j is zero in any expansion of the form l(T ) = i(s)1 i(s)1 + · · · + i(s)k i(s)k , for 1 s d.
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(10)


Stably complex, sp ecial unitary, and level-L structures On a smo oth manifold N of dimention d, a stably complex structure is an equivalence class of real 2k­plane bundle isomorphisms (N ) R2k-d ,

where is a fixed GL(k, C)-bundle over N and k is suitably large. Two such isomorphisms are equivalent when they agree up to stabilisation. If the first Chern class c1( ) is zero, then the stably complex structure is sp ecial unitary, (or S U ); and if it is divisible by a p ositive integer L, then it is level-L. We identify the geometric data required to induce such structures on a toric manifold. Note that CP n is level-L for any divisor L of (n + 1), where n 1.
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Combinatorial data underlying an omnioriented toric manifold An omniorientation of a toric manifold M consists of a choice of orientation for M and for every normal bundle i. An interior p oint of the quotient p olytop e P admits an op en neighb orho o d U , whose inverse image under the projection is canonically diffeomorphic to T n â U as a subspace of M . Since T n is oriented by the standard choice of basis, orientations of M corresp ond bijectively to orientations of P . Moreover, the dicharacteristic determines a complex structure on every i, so it enco des an orientation for every i. Every pair (P, ) therefore determines a 2n­dim omnioriented toric manifold, where P is the combinatorial typ e of an oriented finely ordered n­dim simple p olytop e, and is a matrix of the form (4).
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Stably complex structures Theorem. Any omnioriented toric manifold admits a canonical stably complex structure, which is invariant under the T n-action. Pro of. Using the theory of analogous p olytop es we obtained an emb edding iP : P - Rm which resp ects facial co dimensions and gives a pullback diagram

ZP -Z Cm
P

i





(11)

P

i -P Rm

of identification spaces. Here (z1, . . . , zm) is given by (|z1|2, . . . , |zm|2), the vertical maps are projections onto the quotients by the T m-actions, and iZ is a T m-equivariant emb edding.
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So, there is a K -equivariant decomp osition (ZP ) (iZ )

Z P â Cm ,

obtained by restricting the tangent bundle (Cm) to ZP . Factoring out K yields (M ) ( /K ) ( (iZ )/K )

ZP â

K

Cm, (12)

where denotes the (m - n)­plane bundle of tangents along the fibres of : ZP - M . The right-hand side of (12) is isomorphic
m

to
i=1

i as GL(m, C)-bundles.

This is all an example of Szczarba's Theorem. The emb edding iZ : ZP - C m R2m is T m-equivariantly framed, so (iZ)/K is trivial. The bundle /K canonically isomorphic to the adjoint bundle of the principal K -bundle, which is trivial, b ecause K is an ab elian group.
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So, (12) reduces to an isomorphism (M ) R2(m-n) 1 . . . m ,

although different choices of trivialisations may lead to different isomorphisms. Since M is connected and GL(2(m - n), R) has two connected comp onents, such isomorphisms are equivalent when and only when the induced orientations agree on R2(m-n). We cho ose the orientation which is compatible with those on (M ) and 1 . . . m, as given by the omniorientation. The induced structure is invariant under the action of T n, b ecause iZ is T m-equivariant.

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S U , and level-L structures Corollary. The omniorientation induces an S U -structure on M precisely when the refined matrix of (4) has every column-sum equal to 1; it induces a level-L structure when every column-sum is congruent to 1 mo d L. Pro of. The stably complex structure induced by the omniorientation has first Chern class m u . It is zero in H 2 (M ) if and only if i=1 i
n n

1-
i=1

i,n+1 un+1+· · ·+ 1-
i=1

i,m um = 0.

The same argument shows that it is divisible by L if and only if every column-sum is congruent to 1 mo d L.
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Complex Cob ordism Complex cob ordism functor U (X ) is a generalized cohomology theory with dual elements (b ordism classes) u Uk (X ) represented by maps f : M k X of closed U -manifolds. The U -structure on M k is given by the complex structure in the stable normal bundle of some emb edding M k R2N +k . The op eration of intersection of b ordism classes is dual to the pro duct of cob ordism classes. U (X ) is a commutative and asso ciative Z-graded ring with a unit 1 U (pt) = . U By the Milnor and Novikov theorem = Z[a2, a4, a6, . . .], U deg ak = -2k.

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The problem of cho osing the appropriate generators for the ring is very imp ortant U in the cob ordism theory and its applications. The standard set of multiplicative generators for is constructed using the projective U spaces CP i, i 0, and Milnor hyp ersurfaces Hi,j CP i â CP j , 1 i j . The hyp ersurface Hi,j is defined by Hi,j = z CP i, w CP j |
i

zq w q = 0 .
q =0

Note, that Hi,j is not a toric manifold if i > 1. Theorem. An alternative set of multiplicative generators for is toric manifolds {Bi,j }. U

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Application to complex cob ordism Theorem. Every complex cob ordism class is represented by a disjoint union of omnioriented toric manifolds, which are suitably oriented pro ducts of the Bi,j . Our mo dification of connected sum of toric manifolds gives: Theorem. In dimensions > 2, every complex cob ordism class contains a toric manifold, necessarily connected, whose stably complex structure is induced by an omniorientation, and is therefore compatible with the action of the torus. Similar metho ds combine with Diagram(11) to give: Theorem. Every complex cob ordism class may b e represented by the quotient of a free torus action on a real quadratic complete intersection.
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Davis and Januszkiewicz's space B P Let B P b e the Borel construction E T nâT n M 2n. We have the fibration p : B P - B T n and the following results: U (B T n) = U [[v1, . . . , vn]]; U (B P ) = U [[u1, . . . , um]]/I , where I is Stanley-Reisner ideal of P ; U (B P ) is generators and pvi = a free U (B T n)-mo dule un+1, . . . , um, where uk ui + i,n+1un+1 + · · · + with = c U (k ) 1 i,mum.
M 2n

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The universal index for a toric manifold
2n) = [p : B P - B T n] U -2n(B T n).

I (M

We have

I (M
where

2n) = p (1), !

p! : U (B P ) - U (B T n) is the Gysin homomorphism and I(M 2n) = I(M 2n)(0) = [M 2n] -2n, U where : U (B T n) induced by U : (pt) B T n. The expression for Gysin homomorphism in terms of isolated fixed p oints of T n-action on M 2n gives a universal formula for I(M 2n) (esp ecially for[M 2n]) in terms of combinatorial data (P, ).
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Krichever's results Let Fs b e a connected comp onent of the set {Fs} of fixed p oints under the action of S 1 on a stably complex S 1-manifold M . Supp ose that the representation of S 1 in the normal bundle to Fs is given by Then if c1(M ) = 0, all the sums rs =
i

js,i .
i js,i

are equal. The resulting integer is called the typ e of the circle action on the S U -manifold M . Theorem. If the action of S 1 on any S U -manifold M has nonzero typ e, then Tell ([M ]) = 0, where Tell is Krichever's generalized elliptic genus.
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The generalized elliptic genus of a toric manifold Let M 2n b e a toric S U -manifold and T T n b e an arbitrary sub circle. So l(T ) = (z1, . . . , zn) Zn. Prop osition. The action of T on M 2n
n

has typ e
i=1

zi .

Corollary. The generalized elliptic genus Tell of any toric S U -manifold M 2n is zero.



Corollary. If M 2n is any toric S U -manifold, where n < 5, then [M 2n] = 0.

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References 1. Victor M Buchstab er, Taras E Panov and Nigel Ray. Analogous Polytop es, Circle Actions, and Toric Manifolds. in preparation. 2. Victor M Buchstab er and Taras E Panov. Torus Actions and Their Applications in Top ology and Combinatorics. Volume 24 of University Lecture Series, Amer Math So c, Providence, RI, 2002. 3. Victor M Buchstab er and Nigel Ray. Tangential structures on toric manifolds, and connected sums of p olytop es. Internat Math Res Notices, 4:193­219, 2001. 4. Victor M Buchstab er and Nigel Ray. Toric Manifolds and Complex Cob ordism (Russian). Usp ekhi Mat Nauk 53(2):139­140, 1998. English translation in Russian Math Surveys 53(2):371­373, 1998.
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