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From real quadrics to p olytop es via manifolds
Taras Panov Moscow State University

CAT'09, 6­11 July 2009, Warszawa


1. From p olytop es to quadrics. Rn: Euclidean vector space. Consider a convex p olyhedron P = {x Rn : (a i, x ) + bi 0 for 1 i m} , a i Rn, bi R.

Assume dim P = n, no redundant inequalities, P is b ounded, and b ounding hyp erplanes Hi = {(a i, x ) + bi = 0}, 1 i m, intersect in general p osition at every vertex. Then P is an n-dim convex simple p olytop e with m facets Fi = {x P : (a i, x ) + bi = 0} = P Hi and normal vectors a i, for 1 of facets. i m. At every vertex meets an n-tuple

Two p olytop es are said to b e combinatorially equivalent if their face p osets are isomorphic.
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We may sp ecify P by a matrix inequality P = {x : AP x + b P 0}, where AP = (aij ) is the m â n matrix of row vectors a i, and b P is the column vector of scalars bi. The affine injection iP : Rn - Rm, emb eds P into Rm = {y Rm : yi x AP x + b P 0}.

Now define the space ZP by a pullback diagram ZP -Z Cm

i

(z1, . . . , zm)


P Here iZ is a T m-equivariant emb edding.

- Rm

iP

(|z1|2, . . . , |zm|2)
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Prop 1. ZP is a smo oth T m-manifold with canonically trivialised normal bundle of iZ : ZP Cm. Idea of pro of. 1) Write the image iP (Rn) Rm as the set of common solutions of m m - n linear equations k=1 cj k (yk - bk ) = 0, 1 j m - n; 2) replace every yk by |zk |2 to get a representation of ZP as an intersection of m - n real quadratic hyp ersurfaces:
m k=1

cj k |zk |2 - bk

(

)

= 0, for 1

j

m - n.

3) check that 2) is a non-degenerate intersection, i.e. the gradient vectors are linearly indep endent at each p oint of ZP . ZP is called the moment-angle manifold corresp onding to P . In fact, the top ological typ e of ZP dep ends only on the combinatorial typ e of P (the original construction of [Davis­Januszkiewicz]).
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m Write the system k=1 cj k (|zk |2 - bk ) = 0, 1

j

m - n, as

ZP = {z Cm : C |z |2 = C b P } with C = (cj k ) (m - n) â m-matrix, |z |2 column of |zk |2. Rows of C constitute a basis in the space of linear relations b etween the a i's. That is, C AP = 0 and rank C = m - n (note rank AP = n). Given P , how to cho ose C ?

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1st metho d: Assume the first n facets F1, . . . , Fn meet at a vertex, and take their normals as the basis for Rn (after linear transformation). Then
(

AP =

E A P

)

with E unit n â n-matrix and A an (m - n) â n-matrix. Then we may P take C = -A P (Rememb er C AP = 0!) This is convenient for applications in cob ordism (finding quasitoric representatives in complex cob ordism classes).
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(

E.

)


2nd metho d: Have 1a 1 + . . . + ma m = 1 with i > 0. In fact, i = Vol Fi if |a i| = 1. By scaling the a i's can always achieve a 1 + . . . + a m = 1 and so take ( ) C1 C= where C1 is (m - n - 1) â m. 1···1 By moving the origin 0 into Int P , get bi > 0. By scaling P get b1 + . . . + bm = 1, so the last quadratic equation defining ZP is |z1|2 + . . . + |zm|2 = 1. Subtracting this from the first m - n - 1 equations, finally get
z Cm : ZP =

C|z |2 = 0, |z1|2 + . . . + |zm|2 = 1
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where C is (m - n - 1) â m.


Ex 1. P=

{

x Rn :

xi 0, i = 1, . . . , n, -x1 - . . . - xn + 1 0

}

:

n-simplex.

So m = n + 1, a i = e i for i = 1, . . . , n, a n+1 = -e 1 - . . . - e n. E AP = , C = (1 · · · 1), -1 · · · - 1 ZP = {z : |z1|2 + . . . + |zm|2 = 1} = S 2n+1, and C = .
( )

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2. From quadrics to p olytop es. After [Bosio­Meersseman]. C = (c 1, . . . , c m): p â m matrix, 0 Set LC =
z Cm :

p < m (later set p = m - n - 1).

2 + . . . + c |z |2 = 0 c 1|z1| mm " link of p sp ecial : real quadrics in Cm". |z1|2 + . . . + |zm|2 = 1

C is admissible if LC is nonempty and nondegenerate. (So LC is a (2m - p - 1)-dimensional manifold.) Conv(C) := convex hull of c 1, . . . , c m in Rp. Lemma 1. C is admissible iff 1) 0 Conv(C), and 2) 0 Conv(ci, i I ) implies |I | > p.
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Ex 2. p = 1, so c i R. Then (2) implies c i = 0. Assume k of c i are p ositive and l = m - k are negative. Then (1) implies k > 0, l > 0. Get like |z1|2 + . . . + |zk |2 - |zk+1|2 - . . . - |zm|2 = 0: cone over S 2k-1 â S 2l-1, and |z1|2 + . . . + |zm|2 = 1, so LC = S 2k-1 â S 2l-1. Ex 3. p = 2, so c i R2. Then by (1), 0 Conv(c 1, . . . , c m). (2) says that no segment joining c i contains 0. Lemma 2. Can always achieve o dd numb er of p oints on a circle with p ositive weights assigned. Set k(C ) = numb er of i such that LC {zi = 0} = . Lemma 3. LC = LC â T k(C ), where LC Cm-k(C ) intersects every co ordinate hyp erplane.
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How to get a p olytop e out of LC ? LC /T m is given by the nonnegative solutions of C y = 0 , y1 + . . . + ym = 1.

By nondegeneracy condition (2), this system has maximal rank. So may write its general solution as yi = (a i, x ) + bi, Therefore, LC /T m = {x Rm-p-1 : (a i, x ) + bi 0}. x Rm-p-1, with bi > 0.

Lemma 4. This is a simple (m - p - 1)-dimensional p olytop e P with m - k(C) facets. So every {zi = 0} LC = gives a redundant inequality.
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( ) = Conv a 1 , . . . , a m : p olar (or dual) simplicial p olytop e. P b1 bm Denote a = a i . i bi

Lemma 5. 0 Int Conv(ci, i Conv(

I) a , i [m] \ I ) is a prop er face of P . i

In other words, (c 1, . . . , c m) is the Gale diagram of (a , . . . , a ). m 1

Finally, we get Thm 1. Every LC with k(C ) = 0 is ZP for some P .

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3. Top ology of ZP . F1, . . . , Fm: facets of P . Given I [m], set PI =

iI Fi P .

k (Z ) = k-|I |-1 (P ). Thm 2. H P I I [m] H

Ex 4. P a 5-gon. Then dim ZP = 7, and the Betti vector is (1, 0, 0, 5, 5, 0, 0, 1). In fact, ZP = (S 3 â S 4)#5.

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Pro of of Thm 2. ZP generalises to ZK , the moment-angle complex defined for an arbitrary simplicial complex K on m vertices. Given P as ab ove, set KP = = {i1, . . . , ik } : Fi1 . . . Fik = in P , the b oundary complex of P . It is a sphere triangulation: |KP | = S n-1. Then ZKP = ZP . By [Buchstab er-P], there is an isomorphism of (bi)graded algebras
, H (ZK ) = TorZ[v ,...,v ](Z[K ], Z) m 1 [ ] H [u , . . . , u ] Z[K ]; d , = m 1

{

}

where Z[K ] is the face ring (or the Stanley­Reisner ring) of K , dui = vi, dvi = 0 for 1 i m.
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From this description follows Ho chster's calculation of Tor mo dules in terms of full sub complexes of K :
-i,2j H j -i-1(K ), TorZ[v ,...,v ](Z[K ], Z) = J m 1 | J | =j

where KJ is the restriction of K to the subset J [m]. This dualises to the required description of the cohomology in the case K = KP (b ecause PJ retracts onto KJ ).


4. Quasitoric manifolds and cob ordism. Assume given P as ab ove, and an integral n â m matrix
=

1 0 . . . 0

0 1 . . . 0

. . .

. . . ..

. 0 1,n+1 . 0 2,n+1 . . . .. . . . 1 n,n+1

. . .

. . . ..

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



satisfying the condition the column vectors j1 , . . . , jn corresp onding to any vertex v = Fj1 · · · Fjn form a basis of Zn. We refer to (P, ) as a combinatorial quasitoric pair.
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Define K = K () := ker( : T m T n) = T m-n. Prop 2. K () acts freely on ZP . The quotient M = M (P, ) := ZP /K () is the quasitoric manifold corresp onding to (P, ). It has a residual T n-action (T m/K () = T n) satisfying the two Davis­Januszkiewicz conditions: a) the T n-action is lo cally standard; b) there is a projection : M P whose fibres are orbits of the T n-action.
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Algebraic and symplectic geometers would recognise in the ab ove construction of a quasitoric manifold M from ZP a generalisation of the symplectic reduction construction of a Hamiltonian toric manifold. In the latter case we take = At ; then M is a toric manifold P corresp onding to the Delzant p olytop e P = {x Rn : (a i, x ) + bi 0 for 1 i m} , a i Z n , bi R .

Here we additionally assume the normal vectors a i to b e integer, and the Delzant condition: for every vertex v = Fi1 . . . Fin of P , the corresp onding normal vectors a i1 , . . . , a in form a basis of Zn to b e satisfied. Then ZP is the level set for the moment map µ : Cm Rm-n corresp onding to the Hamiltonian action of K = Ker = Ker At on Cm.
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Define complex line bundles i : ZP âK Ci M , 1 i m,

where Ci is the 1-dim complex T m-representation defined via the quotient projection Cm Ci onto the ith factor.

Thm 3 (Davis­Januszkiewicz). There is an isomorphism of real vector bundles
= 2(m-n) - · · · . M R 1 m

This endows M with the canonical equivariant stably complex structure. So we may consider its complex cob ordism class [M ] U .
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Thm 4 (Buchstab er-P-Ray). Every complex cob ordism class in dim > 2 contains a quasitoric manifold. The complex cob ordism ring U is multiplicatively generated by the cob ordism classes [Hij ], 0 i j , of Milnor hyp ersurfaces Hij = {(z0 : . . . : zi)â(w0 : . . . : wj ) CP i â CP j : z0w0 + . . . + ziwi = 0}. But Hij is not a quasitoric manifold if i > 1. Idea of pro of of Thm 4 1) Replace each Hij by a quasitoric (in fact, toric) manifold Bij so that {Bij } is still a multiplicative generator set for U . Therefore, every stably complex manifold is cob ordant to the disjoint union of pro ducts of Bij 's. Every such pro duct is a q-t manifold, but their disjoint union is not. 2) Replace disjoint unions by certain connected sums. This is tricky, b ecause you need to take account of b oth the torus action and the stably complex structure.
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[1] Fr´ ´ ederic Bosio and Laurent Meersseman. Real quadrics in Cn, complex manifolds and convex p olytop es. Acta Math. 197 (2006), no. 1, 53­127. [2] Victor M Buchstab er and Taras E Panov. Torus Actions and Their Applications in Top ology and Combinatorics. University Lecture Series, vol. 24, Amer. Math. So c., Providence, R.I., 2002. [3] Victor M. Buchstab er, Taras E. Panov and Nigel Ray. Spaces of p olytop es and cob ordism of quasitoric manifolds. Moscow Math. J. 7 (2007), no. 2; arXiv:math.AT/0609346. [4] Victor M. Buchstab er, Taras E. Panov and Nigel Ray. Toric genera. Preprint (2009), to app ear in the arXiv so on. [5] Michael W. Davis and Tadeusz Januszkiewicz. Convex p olytop es, Coxeter orbifolds and torus actions. Duke Math. J., 62 (1991), no. 2, 417­451.
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