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Torus Actions and Complex Cob ordism
Taras Panov Moscow State University

joint with Victor Buchstab er and Nigel Ray

, . . .


Thm 1. Every complex cob ordism class in dim > 2 contains a quasitoric manifold. In other words, every stably complex manifold is cob ordant to a manifold with a nicely b ehaving torus action. In cob ordism theory, all manifolds are smo oth and closed.
n n M1 M2 (co)b ordant if there is a manifold with b oundary W n+1 such that W n+1 = M1 M2.

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Complex cob ordism: work with complex manifolds. complex mflds almost complex mflds stably complex mflds Stably complex structure on M is determined by a choice of isomorphism
= n- M R

where is a complex vector bundle. Complex cob ordism classes [M ] form the complex cob ordism ring U with resp ect to the disjoint union and pro duct. U = Z[a1, a2, ...], dim ai = 2i Novikov'60.

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Quasitoric manifolds: torus T n;

manifolds M 2n with a "nice" action of the

· the T n-action is lo cally standard (lo cally lo oks like the standard T n-representation in Cn); · the orbit space M 2n/T n is an n-dim simple p olytop e P n.

Examples include projective smo oth toric varieties and symplectic manifolds M 2n with Hamiltonian actions of T n (also known as toric manifolds).

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Quasitoric manifolds from combinatorial data. Rn Euclidean vector space. Consider a convex p olyhedron P = {x Rn : (a i, x ) + bi Assume: a) dim P = n; b) no redundant inequalities (cannot remove any inequality without changing P ); c) P is b ounded; d) b ounding hyp erplanes Hi = {(a i, x ) + bi = 0}, 1 i m, intersect in general p osition at every vertex, i.e. there are exactly n facets of P meeting at each vertex.
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0 for 1

i

m} ,

a i Rn, bi R.


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 i m.

The faces of P form a p oset with resp ect to the inclusion. Two p olytop es are said to b e combinatorially equivalent if their face p osets are isomorphic. The corresp onding equivalence classes are called combinatorial p olytop es.

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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 в 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 2. ZP is a smo oth T m-manifold with the canonical trivialisation of the 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 - n linear equations in yi, 1 i m; 2) replace every yi by |zi|2 to get a representation of ZP as an intersection of m - n real quadratic hyp ersurfaces; 3) check that 2) is a "complete" intersection, i.e. the gradients are linearly indep endent at each p oint of ZP .

ZP is called the moment-angle manifold corresp onding to P . It can b e proved that the equivariant smo oth structure on ZP dep ends only on the combinatorial typ e of P .
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Assume given P as ab ove, and an integral n в 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 columns of j1 , . . . , jn corresp onding to any vertex p = Fj1 · · · Fjn form a basis of Zn. We refer to (P, ) as the combinatorial quasitoric pair.

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Define K = K () := ker( : T m T n) = T m-n. Prop 3. 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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Define complex line bundles i : ZP вK 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 4. There is an isomorphism of real vector bundles
= 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 1. 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) в (w0 : . . . : wj ) CP i в CP j : z0w0 + . . . ziwi = 0}. But Hij is not a quasitoric manifold if i > 1. Idea of pro of of the main theorem. 1) Replace each Hij by a quasitoric 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 the disjoint unions by the 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] 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, R.I., 2002.

[2] . . , . . . . , , 2004 (in Russian).

[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, 219­242; arXiv:math.AT/0609346.

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