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TORUS ACTIONS, EQUIVARIANT MOMENT-ANGLE
COMPLEXES, AND COORDINATE SUBSPACE
ARRANGEMENTS
VICTOR M. BUCHSTABER AND TARAS E. PANOV
Abstract. We show that the cohomology algebra of the complement
of a coordinate subspace arrangement in m-dimensional complex space
is isomorphic to the cohomology algebra of Stanley--Reisner face ring of
a certain simplicial complex on m vertices. (The face ring is regarded as
a module over the polynomial ring on m generators.) Then we calculate
the latter cohomology algebra by means of the standard Koszul resolu-
tion of polynomial ring. To prove these facts we construct an equivariant
with respect to the torus action homotopy equivalence between the com-
plement of a coordinate subspace arrangement and the moment-angle
complex defined by the simplicial complex. The moment-angle complex
is a certain subset of a unit poly-disk in m-dimensional complex space
invariant with respect to the action of an m-dimensional torus. This
complex is a smooth manifold provided that the simplicial complex is a
simplicial sphere, but otherwise has more complicated structure. Then
we investigate the equivariant topology of the moment-angle complex
and apply the Eilenberg--Moore spectral sequence. We also relate our
results with well known facts in the theory of toric varieties and sym-
plectic geometry.
1. Introduction
In this paper we apply the results of our previous paper [BP2] to de-
scribing the topology of the complement of a complex coordinate subspace
arrangement. A coordinate subspace arrangement A is a set of coordinate
subspaces L of a complex space C m , and its complement is the set U(A) =
C m n
S
L2A L. The complement U(A) decomposes as U(A) = U(A 0 ) \Theta (C \Lambda ) k ,
were A 0 is a coordinate arrangement in C m\Gammak that does not contain any
hyperplane. There is a one-to-one correspondence between coordinate sub-
space arrangements in C m without hyperplanes and simplicial complexes
on m vertices v 1 ; : : : ; v m : each arrangement A defines a simplicial complex
K(A) and vice versa. Namely let jAj denotes the support
S
L2A L of the
coordinate subspace arrangement A; then a subset v I = fv i 1 ; : : : ; v i k g is a
(k \Gamma 1)-simplex of K(A) if and only if the (m \Gamma k)-dimensional coordinate
subspace L I ae C m defined by equations z i 1 = : : : = z i k = 0 does not belong
1991 Mathematics Subject Classification. 55N91, 05B35 (Primary) 13D03 (Secondary).
Partially supported by the Russian Foundation for Fundamental Research, grant no.
99-01-00090, and INTAS, grant no. 96-0770.
1

2 VICTOR M. BUCHSTABER AND TARAS E. PANOV
to jAj. An arrangement A is obviously recovered from its simplicial com-
plex K(A); that is why we write U(K) instead of U
\Gamma A(K)
\Delta throughout this
paper. (For more information about relations between arrangements and
simplicial complexes see the beginning of Section 2.)
Subspace arrangements and their complements play a pivotal role in many
constructions of combinatorics, algebraic and symplectic geometry, mechan-
ics etc., they also arise as configuration spaces of different classical systems.
That is why the topology of complements of arrangements entranced many
mathematicians during the last two decades. The first important result here
deals with arrangements of hyperplanes (not necessarily coordinate) in C m .
Arnold [Ar] and Brieskorn [Br] shown that the cohomology algebra of the
corresponding complement U(A) is isomorphic to the algebra of differential
forms generated by the closed forms 1
2Яi
dFA
FA , where FA is a linear form defin-
ing the hyperplane A of the arrangement. Orlik and Solomon [OS] proved
that the cohomology algebra of the complement of a hyperplane arrange-
ment depends only on the combinatorics of intersections of hyperplanes
and presented H \Lambda
\Gamma
U(A)
\Delta
by generators and relations. In general situation,
the Goresky--MacPherson theorem [GM, Part III] expresses the cohomology
groups H i
\Gamma
U(A)
\Delta
(without ring structure) as a sum of homology groups
of subcomplexes of a certain simplicial complex. This complex, called the
order (or flag) complex, is defined via the combinatorics of intersections
of subspaces of A. The proof of this result uses the stratified Morse the-
ory developed in [GM]. Another way to handle the cohomology algebra of
the complement of a subspace arrangement was recently presented by De
Concini and Procesi [dCP]. They proved that the rational cohomology ring
of U(A) is also determined by the combinatorics of intersections. This result
was extended by Yuzvinsky in [Yu]. In the case of coordinate subspace ar-
rangements the order complex is the barycentric subdivision of a simplicial
complex ~
K, while the summands in the Goresky--MacPherson formula are
homology groups of links of simplices of ~
K. The complex ~
K has the same
vertex set v 1 ; : : : ; v m as our simplicial complex K and is ``dual'' to the latter
in the following sense: a set v I = fv i 1 ; : : : ; v i k g spans a simplex of ~
K if and
only if the complement fv 1 ; : : : ; v m g n v I does not span a simplex of K. The
product of cohomology classes of the complement of a coordinate subspace
arrangement was described in [dL] in combinatorial terms using the complex
~
K and the above interpretation of the Goresky--MacPherson formula.
In our paper we prefer to describe a coordinate subspace arrangement in
terms of the simplicial complex K instead of ~
K because such an approach re-
veals new connections between the topology of complements of subspace ar-
rangements, commutative algebra, and geometry of toric varieties. We show
that the complement U(K) is homotopically equivalent to what we call the
moment-angle complex ZK defined by the simplicial complex K. This ZK is
a compact subset of a unit poly-disk (D 2 ) m ae C m invariant with respect to
the standard T m -action on (D 2 ) m . At the same time ZK is a homotopy fibre

TORUS ACTIONS AND COORDINATE SUBSPACE ARRANGEMENTS 3
of cellular embedding i : ]
B T K ,! BT m , where BT m is the T m -classifying
space with standard cellular structure, and ]
B T K is a cell subcomplex whose
cohomology is isomorphic to the Stanley--Reisner face ring k(K) of sim-
plicial complex K. Then we calculate the cohomology algebra of ZK (or
U(K)) by means of the Eilenberg--Moore spectral sequence. As the result,
we obtain an algebraic description of the cohomology algebra of U(K) as the
bigraded cohomology algebra Tor k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta
of the face ring k(K).
By means of the standard Koszul resolution the latter can be expressed
as the cohomology of differential bigraded algebra
k(K)\Omega \Lambda[u 1 ; : : : ; um ],
where \Lambda[u 1 ; : : : ; um ] is an exterior algebra, and the differential sends exte-
rior generator u i to v i 2 k(K) = k[v 1 ; : : : ; v m ]=I . The rational models of De
Concini and Procesi [dCP] and Yuzvinsky [Yu] also can be interpreted as an
application of the Koszul resolution to the cohomology of the complement a
subspace arrangement, however the role of the face ring became clear only
after our paper [BP2].
If K is an (n \Gamma 1)-dimensional simplicial sphere (for instance, K is the
boundary complex of an n-dimensional convex simplicial polytope), our
moment-angle complex ZK turns to be a smooth (m + n)-dimensional
manifold (hence, U(K) is homotopically equivalent to a smooth manifold).
This important particular case of our constructions was detailedly studied
in [BP1], [BP2]. Topological properties of the above manifolds ZK are of
great interest because of their relations with combinatorics of polytopes,
symplectic geometry, and geometry of toric varieties; the last thing was the
starting point in our study of coordinate subspace arrangements. The clas-
sical definition of toric varieties (see [Da], [Fu]) deals with the combinatorial
object known as fan. However, as it have been recently shown by several au-
thors (see, for example, [Au], [Ba], [Co]), in the case when the fan defining
a toric variety M is simplicial, M can be defined as the geometric quotient
of the complement U(K) with respect to a certain action of the algebraic
torus (C \Lambda ) m\Gamman (here K is the simplicial complex defined by the fan). Our
moment-angle manifold ZK is the pre-image of a regular point in the image
of the moment map U(K) ! R m\Gamman for the Hamiltonian action of compact
torus T m\Gamman ae (C \Lambda ) m\Gamman .
In their paper [DJ] Davis and Januszkiewicz introduced the notion of toric
manifold (now also known as quasitoric manifold or unitary toric manifold),
which can be regarded as a natural topological extension of the notion of
smooth toric variety. A (quasi)toric manifold M 2n admits a smooth action
of the torus T n that locally looks like the standard action of T n on C n ;
the orbit space is required to be an n-dimensional ball, invested with the
combinatorial structure of a simple convex polytope by the fixed point sets
of appropriate subtori. Topology, geometry and combinatorics of quasitoric
manifolds are very beautiful; after the pioneering paper [DJ] many new
relations have been discovered by different authors (see [BR1], [BR2], [BP1],
[BP2], [Pa1], [Pa2], and more references there). The dual complex to the
boundary complex of a simple polytope in the orbit space of a quasitoric

4 VICTOR M. BUCHSTABER AND TARAS E. PANOV
manifold is a simplicial sphere. That is why many results from the present
paper may be considered as an extension of our previous constructions with
simplicial spheres to the case of general simplicial complex. We also mention
that some our definitions and constructions (such as the Borel construction
B T P ) firstly appeared in [DJ] in a different fashion; in this case we have
tried to retain initial notations.
The authors express special thanks to Nigel Ray for stimulating discus-
sions and fruitful collaboration which inspired some ideas and constructions
from this paper. We also grateful to Nataliya Dobrinskaya who have drawn
our attention to paper [Ba], which reveals some connections between toric
varieties and coordinate subspace arrangements, and to Sergey Yuzvinsky
who informed us about the results of preprint [dL].
2. Homotopical realization of complement of a coordinate
subspace arrangement
Let C m be a complex m-dimensional space with coordinates z 1 ; : : : ; z m .
For any index subset I = fi 1 ; : : : ; i k g denote by L I the (m \Gamma k)-dimensional
coordinate subspace defined by the equations z i 1
= : : : = z i k = 0. Note that
L f1;::: ;mg = f0g and L? = C m .
Definition 2.1. A coordinate subspace arrangement A is a set of coordinate
subspaces L I . The complement of A is the subset
U(A) = C m n
[
L I 2A
L I ae C m :
In the sequel we would distinguish the coordinate subspace arrangement
A regarded as an abstract set of subspaces and its support jAj --- the subset
S
L I 2A L I ae C m . If I ae J and L I ae jAj, then L J ae jAj. If a coordinate
subspace arrangement A contains a hyperplane z i = 0, then its complement
U(A) is represented as U(A 0 ) \Theta C \Lambda , where A 0 is a coordinate subspace
arrangement in the hyperplane fz i = 0g, and C \Lambda = C n f0g. Thus, for any
coordinate subspace arrangement A the complement U(A) decomposes as
U(A) = U(A 0 ) \Theta (C \Lambda ) k ;
were A 0 is a coordinate arrangement in C m\Gammak that does not contain any
hyperplane. Keeping in mind this remark, we restrict ourself to coordinate
subspace arrangement without hyperplanes.
A coordinate subspace arrangement A in C m (without hyperplanes) de-
fines a simplicial complex K(A) with m vertices v 1 ; : : : ; v m in the following
way: we say that a subset v I = fv i 1
; : : : ; v i k g is a (k \Gamma 1)-simplex of K(A)
if and only if L I 6ae jAj.
Example 2.2. 1) If A = ?, then K(A) is an (m \Gamma 1)-dimensional simplex
\Delta m\Gamma1 .
2) If A = f0g, then K(A) = @ \Delta m\Gamma1 is the boundary of an (m\Gamma1)-simplex.

TORUS ACTIONS AND COORDINATE SUBSPACE ARRANGEMENTS 5
On the other hand, a simplicial complex K on the vertex set fv 1 ; : : : ; v m g
defines an arrangement A(K) such that L I ae jAj if and only if v I =
fv i 1 ; : : : ; v i k g is not a simplex of K. Note that if K 0 ae K is a subcomplex,
then A(K) ae A(K 0 ). Thus, we have a reversing order one-to-one correspon-
dence between simplicial complexes on m vertices and coordinate subspace
arrangements in C m without hyperplanes.
Now let U(K) = C m n jA(K)j denote the complement of the coordinate
subspace arrangement A(K).
Example 2.3. 1) If K = \Delta m\Gamma1 is an (m \Gamma 1)-simplex, then U(K) = C m .
2) If K = @ \Delta m\Gamma1 , then U(K) = C m n f0g.
3) If K is a disjoint union of m vertices, then U(K) is obtained by re-
moving from C m all codimension-two coordinate subspaces z i = z j = 0,
i; j = 1; : : : ; m.
Suppose that k is any field, which we refer to as the ground field. Form a
polynomial ring k[v 1 ; : : : ; v m ] where the v i are regarded as indeterminates.
Definition 2.4. The face ring (or the Stanley--Reisner ring) k(K) of sim-
plicial complex K is the quotient ring k[v 1 ; : : : ; v m ]=I , where
I = (v i 1
\Delta \Delta \Delta v i s : fv i 1
; : : : ; v i s g does not span a simplex in K) :
Thus, the face ring is a quotient ring of polynomial ring by an ideal
generated by square free monomials of degree - 2. We make k(K) a graded
ring by setting deg v i = 2, i = 1; : : : ; m.
Example 2.5. 1) If K = \Delta m\Gamma1 , then k(K) = k[v 1 ; : : : ; v m ].
2) If K = @ \Delta m\Gamma1 is the boundary complex of a (m \Gamma 1)-simplex, then
k(K) = k[v 1 ; : : : ; v m ]=(v 1 \Delta \Delta \Delta v m ).
A compact torus T m acts on C m diagonally; since the arrangement A(K)
consists of coordinate subspaces, this action is also defined on U(K). Denote
by B T K the corresponding Borel construction:
B T K = ET m \Theta T m U(K);
(1)
where ET m is the contractible space of universal T m -bundle ET m ! BT m
over the classifying space BT m = (CP 1 ) m . Thus, B T K is the total space
of bundle B T K ! BT m with fibre U(K).
The space BT m has a canonical cellular decomposition (that is, each CP 1
has one cell in each even dimension). For each index set I = fi 1 ; : : : ; i k g one
may consider the cellular subcomplex BT k
I = BT k
i 1 ;::: ;i k
ae BT m homeomor-
phic to BT k .
Definition 2.6. Given a simplicial complex K with vertex set fv 1 ; : : : ; v m g,
define cellular subcomplex ]
B T K ae BT m as the union of BT k
I over all I such
that v I is a simplex of K.
Example 2.7. Let K be a disjoint union of m vertices v 1 ; : : : ; v m . Then
]
B T K is a bouquet of m copies of C P 1 .

6 VICTOR M. BUCHSTABER AND TARAS E. PANOV
The cohomology ring of BT m is isomorphic to the polynomial ring
k[v 1 ; : : : ; v m ] (all cohomologies are with coefficients in the ground field k).
Lemma 2.8. The cohomology ring of
]
B T K is isomorphic to the face ring
k(K). The embedding i : ]
B T K ,! BT m induces the quotient epimorphism
i \Lambda : k[v 1 ; : : : ; v m ] ! k(K) = k[v 1 ; : : : ; v m ]=I in the cohomology.
Proof. The proof is by induction on the number of simplices of K. If K is
a disjoint union of vertices v 1 ; : : : ; v m , then ]
B T K is a bouquet of m copies
of C P 1 (see Example 2.7). In degree zero H \Lambda ( ]
B T K) is just k, while in
degrees - 1 it is isomorphic to k[v 1 ] \Phi \Delta \Delta \Delta \Phi k[v m ]. Therefore, H \Lambda ( ]
B T K) =
k[v 1 ; : : : ; v m ]=I , where I is the ideal generated by all square free monomials
of degree - 2, and i \Lambda is the projection onto the quotient ring. Thus, the
lemma holds for dim K = 0.
Now suppose that the simplicial complex K is obtained from the simplicial
complex K 0 by adding one (k \Gamma 1)-dimensional simplex v I = fv i 1
; : : : ; v i k g.
By the inductive hypothesis, the lemma holds for K 0 , that is, i \Lambda H \Lambda (BT m ) =
H \Lambda ( '
B T K 0 ) = k(K 0 ) = k[v 1 ; : : : ; v m ]=I 0 . By Definition 2.6, ]
B T K is obtained
from '
B T K 0 by adding the subcomplex BT k
i 1 ;::: ;i k
ae BT m . Then H \Lambda ( '
B T K 0 [
BT k
i 1 ;::: ;i k
) = k[v 1 ; : : : ; v m ]=I = k(K 0 [ v I ), where I is generated by I 0 and
v i 1
v i 2
\Delta \Delta \Delta v i k .
Let I m be the standard m-dimensional cube in R m :
I m = f(y 1 ; : : : ; y m ) 2 R m : 0 ? y i ? 1; i = 1; : : : ; mg:
A simplicial complex K with m vertices v 1 ; : : : ; v m defines a cubical complex
CK embedded canonically into the boundary complex of I m in the following
way:
Definition 2.9. For each (k \Gamma 1)-dimensional simplex v J = fv j 1 ; : : : ; v j k g
of K denote by C J the k-dimensional face of I m defined by m \Gamma k equations
y i = 1; i =
2 fj 1 ; : : : ; j k g:
Then define cubical subcomplex CK ae I m as the union of C J over all sim-
plices v J of K.
Remark. Our cubical subcomplex CK ae I m is a geometrical realization of
an abstract cubical complex in the cone over the barycentric subdivision of
K (see [DJ, p. 434]). Indeed, let \Delta m\Gamma1 be an (m \Gamma 1)-dimensional simplex on
the vertex set fv 1 ; : : : ; v m g, and '
\Delta m\Gamma1 a barycentric subdivision of \Delta m\Gamma1 ,
that is, '
\Delta m\Gamma1 has a vertex for each simplex v J of \Delta m\Gamma1 . Construct a map
' : '
\Delta m\Gamma1 ! I m by sending vertex v J of '
\Delta m\Gamma1 to the vertex of I m having
coordinates y j = 0 for j 2 J and y j = 1 for j =
2 J and then extending
this map linearly on each simplex of '
\Delta m\Gamma1 . The image of '
\Delta m\Gamma1 under the
constructed map is the union of faces of I m meeting at zero. Then build a
map C' from the cone C '
\Delta m\Gamma1 over '
\Delta m\Gamma1 to I m by sending the vertex of the
cone to the vertex (1; : : : ; 1) of the cube and extending linearly on simplices

TORUS ACTIONS AND COORDINATE SUBSPACE ARRANGEMENTS 7
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of C '
\Delta m\Gamma1 . The image of C '
\Delta m\Gamma1 under the map C' is the whole cube I m .
Now let K be a simplicial complex on the vertex set fv 1 ; : : : ; v m g. Once a
numeration of vertices is fixed, we may view K as a simplicial subcomplex
of \Delta m\Gamma1 . Then our cubical complex CK ae I m from Definition 2.9 is nothing
but the image C'(C '
K) of the cone over the barycentric subdivision of K
under the map C'.
Example 2.10. The cubical complex CK in the cases when K is a disjoint
union of 3 vertices and the boundary complex of a 2-simplex is indicated on
Figure 1 a) and b) respectively.
Remark. In the case when K is the dual to the boundary complex of an
n-dimensional simple polytope P n , the cubical complex CK coincides with
the cubical subdivision of P n studied in [BP2].
The orbit space of the diagonal action of T m on C m is the positive cone
R m
+ = f(y 1 ; : : : ; y m ) 2 R m : y i - 0; i = 1; : : : ; mg:
The orbit map C m ! R m
+ can be given by (z 1 ; : : : ; z m ) ! (jz 1 j 2 ; : : : ; jz m j 2 ).
If we restrict the above action to the standard poly-disk
(D 2 ) m = f(z 1 ; : : : ; z m ) 2 C m : jz i j ? 1; i = 1; : : : ; mg ae C m ;
then the corresponding orbit space would be the standard cube I m ae R m
+ .
Let UR (K) ae R m
+ denote the orbit space U(K)=T m . Note that if we regard
R m
+ as a subset in C m , then UR (K) is the ``real part'': UR (K) = U(K) `` R m
+ .
Definition 2.11. The equivariant moment-angle complex ZK ae C m cor-
responding to a simplicial complex K is the T m -space defined from the
commutative diagram
ZK \Gamma\Gamma\Gamma! (D 2 ) m
? ? y
? ? y
CK \Gamma\Gamma\Gamma! I m ;

8 VICTOR M. BUCHSTABER AND TARAS E. PANOV
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where the right vertical arrow denotes the orbit map for the diagonal action
of T m , and the lower horizontal arrow denotes the embedding of the cubical
complex CK to I m .
Lemma 2.12. CK ae UR (K) and ZK ae U(K).
Proof. Definition 2.11 shows that the second assertion follows from the
first one. To prove the first assertion we mention that if a point a =
(y 1 ; : : : ; y m ) 2 CK has y i 1 = : : : = y i k = 0, then v I = fv i 1 ; : : : ; v i k g is a
simplex of K, hence L I 6ae A(K).
Lemma 2.13. U(K) is equivariantly homotopy equivalent to the moment-
angle complex ZK .
Proof. We construct a retraction r : UR (K) ! CK that is covered by an equi-
variant retraction U(K) ! ZK . The latter would be a required homotopy
equivalence.
The retraction r : UR (K) ! CK is constructed inductively. We start from
the boundary complex of an (m \Gamma 1)-simplex and remove simplices of positive
dimensions until we obtain K. On each step we construct a retraction, and
the composite map would be required retraction r. If K = @ \Delta m\Gamma1 is the
boundary complex of an (m \Gamma 1)-simplex, then UR (K) = R m
+ n f0g and
the retraction r is shown on Figure 2. Now suppose that the simplicial
complex K is obtained by removing one (k \Gamma 1)-dimensional simplex v J =
fv j 1
; : : : ; v j k g from simplicial complex K 0 . By the inductive hypothesis, the
lemma holds for K 0 , that is, there is a retraction r 0 : UR (K 0 ) ! C K 0 with the
required properties. Let us consider the face C J ae I m (see Definition 2.9).
Since v J is not a simplex of K, the point a having coordinates y j 1
= : : : =
y j k = 0, y i = 1, i =
2 fj 1 ; : : : ; j k g, do not belong to U(K). Hence, we may
apply the retraction from Figure 2 on the face C J , starting from the point a.
Denote this retraction by r J . Now take r = r J ffi r 0 . It is easy to see that this
r is exactly the required retraction.
Example 2.14. 1) If K = @ \Delta m\Gamma1 is the boundary complex of an (m \Gamma 1)-
simplex, then ZK is homeomorphic to (2m \Gamma 1)-dimensional sphere S 2m\Gamma1 .

TORUS ACTIONS AND COORDINATE SUBSPACE ARRANGEMENTS 9
2) If K is the dual to the boundary complex of a n-dimensional simple
polytope P n , then ZK is homeomorphic to a smooth (m + n)-dimensional
manifold. This manifold, denoted Z P , is the main object of study in [BP2].
Corollary 2.15. The Borel construction ET m \Theta T m ZK is homotopy equiv-
alent to B T K.
Proof. The retraction r : U(K) ! ZK constructed in the proof of
Lemma 2.13 is equivariant with respect to the T m -actions on U(K) and
ZK . Since B T K = ET m \Theta T m U(K), the corollary follows.
In what follows we do not distinguish the Borel constructions ET m \Theta T m
ZK and B T K = ET m \Theta T m U(K).
Theorem 2.16. The cellular embedding i : ]
B T K ,! BT m (see Defini-
tion 2.6) and the fibration p : B T K ! BT m (see (1)) are homotopically
equivalent. In particular,
]
B T K and B T K are of same homotopy type.
Proof. Let Я : ZK ! CK denote the orbit map for the torus action on
the moment-angle complex ZK (see Definition 2.11). For each subset I =
fi 1 ; : : : ; i k g ae f1; : : : ; mg denote by B I the following subset of the poly-disk
(D 2 ) m : B I = B 1 \Theta \Delta \Delta \Delta \Theta Bm ae D 2 \Theta \Delta \Delta \Delta \Theta D 2 = (D 2 ) m , where B i is the disk D 2
if i 2 I , and B i is the boundary S 1 of D 2 if i =
2 I . Thus, B I
е = (D 2 ) k \Theta T m\Gammak ,
where k = jI j. It is easy to see that if C I is a face of cubical complex CK (see
Definition 2.9) then Я \Gamma1 (C I ) = B I . Since for I ae J the B I is canonically
identified with a subset of B J , we see that those B I for which v I is a simplex
of K fit together to yield ZK . (This idea can be used to prove that ZK is a
smooth manifold provided that K is the dual to the boundary complex of a
simple polytope, see [BP2, Theorem 2.4].)
For any simplex v I ae K the subset B I ae ZK is invariant with respect to
the T m -action on ZK . Hence, the Borel construction B T K = ET m \Theta T m ZK
is patched from Borel constructions ET m \Theta T m B I (compare this with the
local construction of B T P from [DJ, p. 435]). The latter can be factorized
as ET m \Theta T m B I =
\Gamma ET k \Theta T k (D 2 ) k
\Delta \Theta ET m\Gamman , which is homotopically
equivalent to BT k
I . Hence, the restriction of the projection p : B T K ! BT m
to ET m \Theta T mB I is homotopically equivalent to the embedding BT k
I ,! BT m .
These homotopy equivalences for all simplices v I ae K fit together to yield a
required homotopy equivalence between p : B T K ! BT m and i : ]
B T K ,!
BT m .
Corollary 2.17. The complement U(K) of a coordinate subspace arrange-
ment is a homotopy fibre of the cellular embedding i : ]
B T K ,! BT m . \Lambda
Corollary 2.18. The T m -equivariant cohomology ring H \Lambda
T m
\Gamma
U(K)
\Delta
is iso-
morphic to the face ring k(K).
Proof. We have H \Lambda
T m
\Gamma
U(K)
\Delta
= H \Lambda
\Gamma
ET m \Theta T m U(K)
\Delta
= H \Lambda (B T K). Now,
the corollary follows from Lemma 2.8 and Theorem 2.16.

10 VICTOR M. BUCHSTABER AND TARAS E. PANOV
3. Cohomology ring of U(K)
Suppose that we are given a k[v 1 ; : : : ; v m ]-free resolution of the face ring
k(K) as a graded module over the polynomial ring k[v 1 ; : : : ; v m ]:
0 ! R \Gammah d \Gammah
\Gamma\Gamma\Gamma! R \Gammah+1 d \Gammah+1
\Gamma\Gamma\Gamma\Gamma! \Delta \Delta \Delta ! R \Gamma1 d \Gamma1
\Gamma! R 0 d 0
\Gamma! k(K) ! 0
(2)
(note that the Hilbert syzygy theorem shows that h ? m above). Applying
the
functor\Omega k[v 1 ;::: ;vm ] k to (2) we obtain a cochain complex:
0 \Gamma! R
\Gammah\Omega k[v 1 ;::: ;vm ] k \Gamma! \Delta \Delta \Delta \Gamma! R
0\Omega k[v 1 ;::: ;vm ] k \Gamma! 0;
whose cohomology modules are denoted Tor \Gammai
k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta . Since all
R \Gammai in (2) are graded k[v 1 ; : : : ; v m ]-modules, Tor \Gammai
k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta
=
L
j Tor \Gammai;j
k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta is a graded k-module, and
Tor k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta =
M
i;j
Tor \Gammai;j
k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta
(3)
becomes a bigraded k-module. Note that its non-zero elements have non-
positive first grading and non-negative even second grading (since deg v i =2).
The bigraded k-module (3) can be also regarded as a one-graded module
with respect to the total degree \Gammai + j. The Betti numbers
fi \Gammai
\Gamma k(K)
\Delta
= dim k Tor \Gammai
k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta
and
fi \Gammai;2j
\Gamma k(K)
\Delta = dim k Tor \Gammai;2j
k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta
are of great interest in geometric combinatorics; they were studied by dif-
ferent authors (see, for example, [St]). We mention only one theorem due
to Hochster, which reduces calculation of fi \Gammai;2j
\Gamma k(K)
\Delta to calculating the
homology of subcomplexes of K.
Theorem 3.1 (Hochster [Ho], [St]). The Hilbert series
P
j fi \Gammai;2j
\Gamma k(K)
\Delta
t 2j
of Tor \Gammai
k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta
can be calculated as
X
j
fi \Gammai;2j
\Gamma k(K)
\Delta
t 2j =
X
Iaefv 1 ;::: ;vmg
\Gamma
dim k
~
H jI j\Gammai\Gamma1 (K I )
\Delta
t 2jI j ;
where K I is the subcomplex of K consisting of all simplices with vertices
in I. \Lambda
Note that calculation of fi \Gammai;2j
\Gamma k(K)
\Delta using this theorem is very involved
even for small K.
It turns out that Tor k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta is a bigraded algebra in a natural
way, and the associated one-graded algebra is exactly H \Lambda
\Gamma U(K)
\Delta :
Theorem 3.2. The following isomorphism of graded algebras holds:
H \Lambda
\Gamma
U(K)
\Delta е = Tor k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta

TORUS ACTIONS AND COORDINATE SUBSPACE ARRANGEMENTS 11
Proof. Let us consider the commutative diagram
' U(K) \Gamma\Gamma\Gamma! ET m
? ? y
? ? y
]
B T K i
\Gamma\Gamma\Gamma! BT m ;
(4)
where the left vertical arrow is the induced fibre bundle. Corollary 2.17
shows that '
U(K) is homotopically equivalent to U(K).
From (4) we obtain that the cellular cochain algebras C \Lambda ( ]
B T K) and
C \Lambda (ET m ) are modules over C \Lambda (BT m ). It is clear from the proof of Lemma 2.8
that C \Lambda ( ]
B T K) = k(K) and i \Lambda : C \Lambda (BT m ) = k[v 1 ; : : : ; v m ] ! k(K) =
C \Lambda ( ]
B T K) is the quotient epimorphism. Since ET m is contractible, we have
a chain equivalence C \Lambda (ET m ) ! k. Therefore, there is an isomorphism
Tor C \Lambda (BT m )
\Gamma C \Lambda ( ]
B T K); C \Lambda (ET m )
\Delta е = Tor k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta :
(5)
The Eilenberg--Moore spectral sequence (see [Sm, Theorem 1.2]) of com-
mutative square (4) has the E 2 -term
E 2 = Tor H \Lambda (BT m )
\Gamma
H \Lambda ( ]
B T K); H \Lambda (ET m )
\Delta
and converges to Tor C \Lambda (BT m ) (C \Lambda ( ]
B T K); C \Lambda (ET m )). Since
Tor H \Lambda (BT m ) (H \Lambda ( ]
B T K); H \Lambda (ET m )) = Tor k[v 1 ;::: ;vm ]
\Gamma k(K); k
\Delta ;
it follows from (5) that the spectral sequence collapses at the E 2 term,
that is, E 2 = E1 . Now, Proposition 3.2 of [Sm] shows that the module
Tor C \Lambda (BT m )
\Gamma C \Lambda ( ]
B T K); C \Lambda (ET m )
\Delta is an algebra isomorphic to H \Lambda
\Gamma ' U(K)
\Delta ,
which concludes the proof.
Our next theorem gives an explicit description of the algebra H \Lambda
\Gamma
U(K)
\Delta
as the cohomology algebra of a simple differential bigraded algebra. We
consider the tensor product
k(K)\Omega \Lambda[u 1 ; : : : ; um ] of the face ring k(K) =
k[v 1 ; : : : ; v m ]=I and an exterior algebra \Lambda[u 1 ; : : : ; um ] on m generators and
make it a differential bigraded algebra by setting
bideg v i = (0; 2); bideg u i = (\Gamma1; 2);
d(1\Omega u i ) = v
i\Omega 1; d(v
i\Omega 1) = 0;
(6)
and requiring that d be a derivation of algebras.
Theorem 3.3. The following isomorphism of graded algebras holds:
H \Lambda
\Gamma U(K)
\Delta е = H
\Theta
k(K)\Omega \Lambda[u 1 ; : : : ; um ]; d
\Lambda ;
where in the right hand side stands the one-graded algebra associated to the
bigraded cohomology algebra.

12 VICTOR M. BUCHSTABER AND TARAS E. PANOV
Proof. One can make k a k[v 1 ; : : : ; v m ]-module by means of the homomor-
phism that sends 1 to 1 and v i to 0. Let us consider the Koszul resolution
(see, for example [Ma, Chapter VII, x 2]) of k regarded as a k[v 1 ; : : : ; v m ]-
module: \Theta k[v 1 ; : : : ; v m
]\Omega \Lambda[u 1 ; : : : ; um ]; d
\Lambda
;
where the differential d is defined as in (6). Since the bigraded torsion prod-
uct Tor k[v 1 ;::: ;vm ] ( ; ) is a symmetric function of its arguments, one has
Tor \Gamma
\Gamma k(K); k
\Delta = H
\Theta
k(K)\Omega \Gamma
\Gamma\Omega \Lambda[u 1 ; : : : ; u n ]; d
\Lambda =
\Theta
\Gamma\Omega \Lambda[u 1 ; : : : ; um ]; d
\Lambda ;
where we denoted \Gamma = k[v 1 ; : : : ; v m ]. Since H \Lambda
\Gamma
U(K)
\Delta е = Tor \Gamma
\Gamma
k(K); k
\Delta
by
Theorem 3.2, we obtain the required isomorphism
Note that the above theorem not only calculates the cohomology algebra
of U(K), but also makes this algebra bigraded.
Corollary 3.4. The Leray--Serre spectral sequence of the bundle
' U(K) !
]
B T K with fibre T m (see (4)) collapses at the E 3 term.
Proof. The spectral sequence under consideration converges to H \Lambda
\Gamma '
U(K)
\Delta
=
H \Lambda
\Gamma U(K)
\Delta and has
E 2 = H \Lambda ( ]
B T
K)\Omega H \Lambda (T m ) =
k(K)\Omega \Lambda[u 1 ; : : : ; um ]:
It is easy to see that the differential in the E 2 term acts as in (6). Hence, E 3 =
H [E 2 ; d] = H
\Theta
k(K)\Omega \Lambda[u 1 ; : : : ; um ]
\Lambda
= H \Lambda
\Gamma
U(K)
\Delta
by Theorem 3.3.
Proposition 3.5. Suppose that a monomial
v ff 1
i 1
: : : v ff p
i p
u j 1 : : : u jq 2
k(K)\Omega \Lambda[u 1 ; : : : ; um ];
where i 1 ! : : : ! i p , j 1 ! : : : ! i q , represents a non-trivial cohomology class
in H \Lambda
\Gamma
U(K)
\Delta
. Then ff 1 = : : : = ff p = 1, fv i 1 ; : : : ; v i p g spans a simplex of
K, and fi 1 ; : : : ; i p g `` fj 1 ; : : : ; j q g = ?.
Proof. See [BP2, Lemma 5.3].
As it was mentioned above (see Example 2.14), if K is the boundary
complex of a convex simplicial polytope (or, equivalently, K is the dual to
the boundary complex of a simple polytope) or at least a simplicial sphere,
then U(K) has homotopy type of a smooth manifold ZK . It was shown
in [BP2, Theorem 2.10] that the corresponding homotopy equivalence can
be interpreted as the orbit map U(K) ! U(K)=R m\Gamman е = ZK with respect
to a certain action of R m\Gamman on U(K).
The coordinate subspace arrangement A(K) and its complement U(K)
play important role in the theory of toric varieties and symplectic geome-
try (see, for example, [Au], [Ba], [Co]). More precisely, any n-dimensional
simplicial toric variety M defined by a (simplicial) fan \Sigma in Z n with m one-
dimensional cones can be obtained as the geometric quotient U(K \Sigma )=G. Here
G is a subgroup of the complex torus (C \Lambda ) m isomorphic to (C \Lambda ) m\Gamman and K \Sigma

TORUS ACTIONS AND COORDINATE SUBSPACE ARRANGEMENTS 13
is the simplicial complex defined by the fan \Sigma (i-simplices of K \Sigma correspond
to (i + 1)-dimensional cones of \Sigma). A smooth projective toric variety M is a
symplectic manifold of real dimension 2n. This manifold can be constructed
by the process of symplectic reduction in the following way. Let GR е = T m\Gamman
denote the maximal compact subgroup of G, and let ? : C m ! R m\Gamman be
the moment map for the Hamiltonian action of GR on C m . Then for each
regular value a 2 R m\Gamman of ? there is a diffeomorphism
? \Gamma1 (a)=GR \Gamma! U(K \Sigma )=G = M
(see [Co] for more information). In this situation it can be easily seen that
? \Gamma1 (a) is exactly our manifold ZK for K = K \Sigma .
Example 3.6. Let G е = C \Lambda be the diagonal subgroup in (C \Lambda ) n+1 and K \Sigma
be the boundary complex of an n-simplex. Then U(K \Sigma ) = C n+1 n f0g and
M = C n+1 nf0g=C \Lambda is the complex projective space C P n . The moment map
? : C m ! R takes (z 1 ; : : : ; z m ) 2 C m to 1
2 (jz 1 j 2 + : : : + jz m j 2 ) and for a 6= 0
one has ? \Gamma1 (a) е = S 2n+1 е = ZK (see Example 2.14).
In the case when K is a simplicial sphere (hence, the complement U(K)
is homotopically equivalent to the smooth manifold ZK ), there is Poincar'e
duality defined in the cohomology ring of U(K).
Proposition 3.7. Suppose that K is a simplicial sphere of dimension n \Gamma 1,
hence, U(K) is homotopically equivalent to the smooth manifold ZK . Then
1) The Poincar'e duality in H \Lambda
\Gamma
U(K)
\Delta
regards the bigraded structure de-
fined by Theorem 3.3. More precisely, if ff 2 H \Gammai;2j (U(K)
\Delta
is a cohomology
class, then its Poincar'e dual Dff belongs to H \Gamma(m\Gamman)+i;2(m\Gammaj) .
2) Let fv i 1
; : : : ; v i n g be an (n \Gamma 1)-simplex of K and let j 1 ! : : : ! j m\Gamman ,
fi 1 ; : : : ; i n ; j 1 ; : : : ; j m\Gamman g = f1; : : : ; mg. Then the value of the element
v i 1
\Delta \Delta \Delta v i n u j 1
\Delta \Delta \Delta u jm\Gamman 2 H m+n
\Gamma U(K)
\Delta е = H m+n (ZK )
on the fundamental class of ZK equals \Sigma1.
3) Let fv i 1 ; : : : ; v i n g and fv i 1 ; : : : ; v i n\Gamma1 ; v j 1 g be two (n \Gamma 1)-simplices of K
having common (n \Gamma 2)-face fv i 1 ; : : : ; v i n\Gamma1 g, and j 1 ; : : : ; j m\Gamman be as in 2).
Then
v i 1 \Delta \Delta \Delta v i n u j 1 \Delta \Delta \Delta u jm\Gamman = v i 1 \Delta \Delta \Delta v i n\Gamma1 v j 1 u i n u j 2 \Delta \Delta \Delta u jm\Gamman
in H m+n
\Gamma
U(K)
\Delta
.
Proof. For the proof of 1) and 2) see [BP2, Lemma 5.1]. To prove 3) we just
mention that
d(v i 1 \Delta \Delta \Delta v i n\Gamma1 u i n u j 1 u j 2 \Delta \Delta \Delta u jm\Gamman )
= v i 1 \Delta \Delta \Delta v i n u j 1 \Delta \Delta \Delta u jm\Gamman \Gamma v i 1 \Delta \Delta \Delta v i n\Gamma1 v j 1 u i n u j 2 \Delta \Delta \Delta u jm\Gamman
in
k(K)\Omega \Lambda[u 1 ; : : : ; um ] (see (6)).
A simplicial complex K is called Cohen--Macaulay, if its face ring k(K)
is a Cohen--Macaulay algebra, that is, k(K) is a finite-dimensional free

14 VICTOR M. BUCHSTABER AND TARAS E. PANOV
module over a polynomial ring k[t 1 ; : : : ; t n ] (here n is the maximal num-
ber of algebraically independent elements of k(K)). Equivalently, k(K) is a
Cohen--Macaulay algebra if it admits a regular sequence f- 1 ; : : : ; - n g, that
is, a set of n homogeneous elements such that - i+1 is not a zero divisor in
k(K)=(- 1 ; : : : ; - i ) for i = 0; : : : ; n \Gamma 1. If K is a Cohen--Macaulay complex
and k is of infinite characteristic, then k(K) admits a regular sequence of
degree-two elements (remember that we set deg v i = 2 in k(K)), that is,
- i = - i1 v 1 + - i2 v 2 + : : : + - im v m , i = 1; : : : ; n.
Theorem 3.8. Suppose that K is a Cohen--Macaulay complex and J =
(- 1 ; : : : ; - n ) is an ideal in k(K) generated by a regular sequence. Then the
following isomorphism of bigraded algebras holds:
H \Lambda
\Gamma U(K)
\Delta е = H
\Theta
k(K)=J\Omega \Lambda[u 1 ; : : : ; um\Gamman ]; d
\Lambda ;
where the gradings and differential in the right hand side are defined as
follows:
bideg v i = (0; 2); bideg u i = (\Gamma1; 2);
d(1\Omega u i ) = -
i\Omega 1; d(v
i\Omega 1) = 0;
Hence, in the case when K is Cohen--Macaulay, the cohomology of U(K)
can be calculated via the finite-dimensional differential algebra
k(K)=J\Omega \Lambda[u 1 ; : : : ; um\Gamman ] instead of infinite-dimensional algebra
k(K)\Omega \Lambda[u 1 ; : : : ; um ]
from Theorem 3.3.
Example 3.9. Let K be the boundary complex of an (m \Gamma 1)-dimensional
simplex. Then k(K) = k[v 1 ; : : : ; v m ]=(v 1 \Delta \Delta \Delta v m ). It easy to check that only
non-trivial cohomology classes in H
\Theta
k(K)\Omega \Lambda[u 1 ; : : : ; um ]; d
\Lambda
(see Theo-
rem 3.3) are represented by the cocycles 1 and v 1 v 2 \Delta \Delta \Delta v m\Gamma1 um or their mul-
tiples. We have deg(v 1 v 2 \Delta \Delta \Delta v m\Gamma1 um ) = 2m \Gamma 1, and Proposition 3.7 shows
that v 1 v 2 \Delta \Delta \Delta v m\Gamma1 um is the fundamental cohomological class of ZK е = S 2m\Gamma1
(see Example 2.14 1)).
Example 3.10. Let K be a disjoint union of m vertices. Then U(K) is
obtained by removing from C m all codimension-two coordinate subspaces
z i = z j = 0, i; j = 1; : : : ; m (see Example 2.3), and k(K) = k[v 1 ; : : : ; v m ]=I ,
where I is the ideal generated by all monomials v i v j , i 6= j. It is easily
deduced from Theorem 3.3 and Proposition 3.5 that any cohomology class
of H \Lambda
\Gamma U(K)
\Delta is represented by a linear combination of monomial cocycles
v i 1
u i 2
u i 3
\Delta \Delta \Delta u i k ae
k(K)\Omega \Lambda[u 1 ; : : : ; um ] such that k - 2, i p 6= i q for p 6= q.
For each k there m
\Gamma m\Gamma1
k\Gamma1
\Delta such monomials, and there
\Gamma m
k
\Delta relations between
them (each relation is obtained by calculating the differential of u i 1
\Delta \Delta \Delta u i k ).
Since deg(v i 1
u i 2
u i 3
\Delta \Delta \Delta u i k ) = k + 1, we have
dim H 0
\Gamma U(K)
\Delta = 1; H 1
\Gamma U(K)
\Delta = H 2
\Gamma U(K)
\Delta = 0;
dim H k+1
\Gamma U(K)
\Delta = m
\Gamma m\Gamma1
k\Gamma1
\Delta \Gamma
\Gamma m
k
\Delta ; 2 ? k ? m;
and the multiplication in the cohomology is trivial.

TORUS ACTIONS AND COORDINATE SUBSPACE ARRANGEMENTS 15
In particular, for m = 3 we have 6 three-dimensional cohomology classes
v i u j , i 6= j, with 3 relations v i u j = v j u i , and 3 four-dimensional cohomology
classes v 1 u 2 u 3 , v 2 u 1 u 3 , v 3 u 1 u 2 with one relation
v 1 u 2 u 3 \Gamma v 2 u 1 u 3 + v 3 u 1 u 2 = 0:
Hence, dim H 3
\Gamma
U(K)
\Delta = 3, dim H 4
\Gamma
U(K)
\Delta = 2, and the multiplication is
trivial.
Example 3.11. Let K be a boundary complex of an m-gon (m - 4). Then,
as it have been mentioned above, the moment-angle complex ZK is a smooth
manifold of dimension m+ 2, and U(K) is homotopically equivalent to ZK .
We have k(K) = k[v 1 ; : : : ; v m ]=I , where I is generated by monomials v i v j
such that i 6= j \Sigma 1. (Here we use the agreement v m+i = v i and v i\Gammam = v i .)
The cohomology rings of these manifolds were calculated in [BP2]. We have
dim H k
\Gamma
U(K)
\Delta
=
8 ? !
? :
1 if k = 0 or m+ 2;
0 if k = 1; 2; m or m+ 1;
(m \Gamma 2)
\Gamma m\Gamma2
k\Gamma2
\Delta \Gamma
\Gamma m\Gamma2
k\Gamma1
\Delta \Gamma
\Gamma m\Gamma2
k\Gamma3
\Delta
if 3 ? k ? m \Gamma 1:
For example, in the case m = 5 there 5 generators of H 3
\Gamma U(K)
\Delta represented
by the cocycles v i u i+2 2
k(K)\Omega \Lambda[u 1 ; : : : ; u 5 ], i = 1; : : : ; 5, and 5 generators
of H 4
\Gamma
U(K)
\Delta
represented by the cocycles v j u j+2 u j+3 , j = 1; : : : ; 5. As it
follows from Proposition 3.7, the product of cocycles v i u i+2 and v j u j+2 u j+3
represents a non-trivial cohomology class in H 7
\Gamma U(K)
\Delta (the fundamental co-
homology class up to sign) if and only if fi; i+2; j; j+2; j+3g = f1; 2; 3; 4; 5g.
Hence, for each cohomology class [v i u i+2 ] there is a unique (Poincar'e dual)
cohomology class [v j u j+2 u j+3 ] such that the product [v i u i+2 ] \Delta [v j u j+2 u j+3 ]
is non-trivial.
References
[Ar] V. I. Arnold, The cohomology ring of the colored braid group (Russian), Mat. Za-
metki 5 (1969), 227--231; English transl. in: Math. Notes 5 (1969), 138--140.
[Au] M. Audin, The Topology of Torus Actions on Symplectic Manifolds, Progress in
Mathematics 93, BirkhЕauser, Boston Basel Berlin, 1991.
[Ba] V. V. Batyrev, Quantum Cohomology Rings of Toric Manifolds, Journ'ees
de G'eom'etrie Alg'ebrique d'Orsay (Juillet 1992), Ast'erisque 218,
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simple polytopes (Russian), Uspekhi Mat. Nauk 53 (1998), no. 3, 195--196; English
transl. in: Russian Math. Surveys 53 (1998), no. 3, 623--625.
[BP2] V. M. Buchstaber and T. E. Panov, Torus actions and combinatorics of polytopes
(Russian), Trudy Matematicheskogo Instituta im. Steklova 225 (1999), 96--131;
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16 VICTOR M. BUCHSTABER AND TARAS E. PANOV
[BR1] V. M. Bukhshtaber and N. Ray, Toric manifolds and complex cobordisms (Russian),
Uspekhi Mat. Nauk 53 (1998), no. 2, 139--140; English transl. in: Russian Math.
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[BR2] V. M. Buchstaber and N. Ray, Tangential structures on toric manifolds, and con-
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Department of Mathematics and Mechanics, Moscow State University,
119899 Moscow, RUSSIA
E-mail address: tpanov@mech.math.msu.su buchstab@mech.math.msu.su