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HIRZEBRUCH GENERA OF MANIFOLDS WITH
TORUS ACTION
TARAS E. PANOV
Abstract. A quasitoric manifold is a smooth 2n­manifold M 2n with an ac­
tion of the compact torus T n such that the action is locally isomorphic to the
standard action of T n on C n and the orbit space is diffeomorphic, as mani­
fold with corners, to a simple polytope P n . The name refers to the fact that
topological and combinatorial properties of quasitoric manifolds are similar to
that of non­singular algebraic toric varieties (or toric manifolds). Unlike toric
varieties, quasitoric manifoldsmay fail to be complex;however, they always ad­
mit a stably (or weakly almost) complex structure, and their cobordism classes
generate the complex cobordism ring. As it have been recently shown by Buch­
staber and Ray, a stably complex structure on a quasitoric manifold is defined
in purely combinatorial terms, namely, by an orientation of the polytope and
a function from the set of codimension­one faces of the polytope to primitive
vectors of an integer lattice. We calculate the üy ­genus of a quasitoric manifold
with fixed stably complex structure in terms of the corresponding combinato­
rial data. In particular, this gives explicit formulae for the classical Todd genus
and signature. We also relate our results with well­known facts in the theory
of toric varieties.
Introduction
Manifolds with torus action arise in different areas of topology, algebraic and
differential geometry, and mathematical physics. Specific properties of torus action
or additional structures on manifolds usually allow to solve corresponding problems
by geometrical or combinatorial methods. The most well known examples here are
Hamiltonian torus actions in symplectic geometry and smooth toric varieties in
algebraic geometry. Both cases allow a natural topological generalization, namely
quasitoric manifolds, introduced by Davis and Januszkiewicz in [DJ]. (Davis and
Januszkiewicz used the term ``toric manifolds''; the term ``quasitoric manifolds''
firstly appeared in [BP2] and [BR2] because of the reasons discussed below.) A qu­
asitoric manifold is a manifold with a torus action that satisfies two natural con­
ditions. The first one is that the action locally looks like the standard torus action
on a complex space by diagonal matrices. If this condition is satisfied, then the
orbit space is a manifold with corners; the second condition is that this manifold
with corners is diffeomorphic to a convex simple polytope. These two properties
are well known for smooth algebraic toric varieties [Da], [Fu]. Though quasitoric
manifolds retain most topological and combinatorial properties of smooth toric va­
rieties, they may fail to admit a complex structure. Like toric varieties, quasitoric
manifolds can be defined in purely combinatorial terms. Namely, any quasitoric
manifold is defined by combinatorial data: the lattice of faces of a simple polytope
1991 Mathematics Subject Classification. 57R20, 57S25 (Primary) 14M25, 58G10 (Secondary).
This work was partly supported by the Russian Foundation for Fundamental Research (grant
no. 99­01­00090).
1

2 TARAS E. PANOV
and a characteristic function that assigns an integer primitive vector defined up to
sign to each facet. Despite their simple and specific definition, quasitoric manifolds
in many cases may serve as model examples (for instance, as it was shown in [BR1],
each complex cobordism class contains a quasitoric manifold). All these facts allow
to use quasitoric manifolds for solving topological problems by combinatorial meth­
ods and vice versa. A number of such relations was firstly discovered in the theory
of toric varieties. Some applications were obtained in [BP1], [BP2], where quasitoric
manifolds are studied in the general context of ``manifolds defined by simple poly­
topes''. Another example of interplay between topology and combinatorics is the
calculation of the KO­theory of quasitoric manifolds held in [BB].
Besides, it is important to mention that the term ``quasitoric manifold'' is not
common. Some authors starting from Davis and Januszkiewicz call the object of
our study just ``toric manifolds''. However, we prefer to call those manifolds ``qua­
sitoric'', reserving the term ``toric manifold'' for smooth toric varieties, as commonly
used in algebraic geometry (see, for instance, [Ba]).
In the present paper we calculate well­known cobordism invariant, the ü y ­genus,
for quasitoric manifolds in terms of the corresponding combinatorial data. The most
important particular cases here are the signature and the Todd genus, which cor­
respond to the values y = 1 and y = 0 respectively. The signature is an oriented
cobordism invariant and is defined for any oriented manifold (it equals zero in di­
mensions other than 4k). At the same time the Todd genus, as well as the general
ü y ­genus, requires the Chern classes of manifold to be defined. As it is mentioned
above, a quasitoric manifold is not necessarily complex, however as it was recently
shown by Buchstaber and Ray [BR2], it always admits a stably (or weakly almost)
complex structure. Moreover, stably complex structures (i.e. complex structures in
the stable tangent bundle) on quasitoric manifolds are also defined combinatorially,
namely, by specifying an orientation of the simple polytope and choosing signs for
vectors corresponding to facets. This in turn is equivalent to a choice of orienta­
tions for the manifold and all submanifolds corresponding to facets. A quasitoric
manifold with such additional structure was called in [BR2] multioriented. Hence, a
multioriented quasitoric manifold determines a complex cobordism class, for which
one can define characteristic numbers and complex Hirzebruch genera. We calculate
the ü y ­genus of a multioriented (i.e. with fixed stably complex structure) quasitoric
manifold in terms of its combinatorial data. To do this, we construct a circle ac­
tion with only isolated fixed points and then apply Atiyah and Bott's generalized
Lefschetz fixed point theorem [AB]. The obtained formula allows to calculate the
ü y ­genus as a sum of contributions corresponding to the vertices of polytope. These
contributions depend only on the ``local combinatorics'' near the vertex. Our for­
mula contains one external parameter (an integer primitive vector š), which does
not affect the answer. In the particular case of smooth toric varieties the Todd genus
(or the arithmetic genus) is always equal to 1 [Fu]. However, this fact fails to be
true for general quasitoric manifolds, since the Todd genus is a complex cobordism
invariant and each cobordism class contains a quasitoric representative [BR1].

HIRZEBRUCH GENERA OF MANIFOLDS WITH TORUS ACTION 3
1. Quasitoric manifolds and polytopes, facet and edge vectors, and
stably complex structures
Here we briefly discuss the definition of quasitoric manifolds and introduce stably
complex structures on them, following [DJ] and [BR2]. We also describe some new
relations between the combinatorial data associated to a quasitoric manifold.
A convex polytope P n of dimension n is a bounded set in R n that is obtained as
the intersection of a finite number of half­spaces:
P n = fx 2 R n : hl i ; xi – \Gammaa i ; i = 1; : : : ; mg
(1)
for some l i 2 (R n ) \Lambda , a i 2 R. A convex polytope P n is called simple if its bounding
hyperplanes are in general position at each vertex, i.e. there exactly n codimension­
one faces (or facets) meet at each vertex. It follows that any point of a simple
polytope lies in a neighbourhood that is affinely isomorphic to an open set of the
positive cone
R n
+ = f(x 1 ; : : : ; xn ) 2 R n : x 1 – 0; : : : ; xn – 0g:
This means exactly that a simple polytope P n is an n­dimensional manifold with
corners. The faces of P n of all dimensions form a partially ordered set with respect
to inclusion, which is called the lattice of faces of P n . We say that two polytopes
are combinatorially equivalent if they have same lattices of faces. Two polytopes are
combinatorially equivalent if and only if they are diffeomorphic as manifolds with
corners.
Let M 2n be a compact 2n­dimensional manifold with an action of the compact
torus T n . One can view T n as a subgroup of the complex torus (C \Lambda ) n in the standard
way:
T n = f
\Gamma
e 2úi'1 ; : : : ; e 2úi'n
\Delta
2 C n g;
where (' 1 ; : : : ; 'n ) runs over R n . We say that a T n ­action is locally isomorphic to
the standard diagonal action of T n on C n if every point x 2 M 2n lies in some T n ­
invariant neighbourhood U (x) for which there exists an equivariant homeomorphism
f : U (x) ! W with some (T n ­stable) open subset W ae C n . The last statement
means that there is an automorphism ` : T n ! T n such that f(t \Delta y) = `(t)f(y) for
all t 2 T n , y 2 U (x). The orbit space for such an action of T n is an n­dimensional
manifold with corners; we refer to M 2n as a quasitoric manifold if the orbit space is
diffeomorphic, as manifold with corners, to a simple polytope P n . Thus, the orbit
space of a quasitoric manifold is decomposed into faces in such a way that points
from the relative interior of each k­face correspond to orbits with same isotropy
subgroup of codimension k. In particular, the action of T n is free over the interior
of P n , while the vertices of P n correspond to T n ­fixed points of M 2n .
Now let M 2n be a quasitoric manifold with orbit space P n , and let F =
fF 1 ; : : : ; Fmg be the set of codimension­one faces (facets) of the polytope P n ,
m = ]F . The interior of each facet F i consists of orbits with the same one­
dimensional isotropy subgroup GF i
. This one­parameter subgroup of T n is deter­
mined by an integer primitive vector – i = (– 1i ; : : : ; – ni ) ? in the corresponding
lattice L ' Z n :
GF i
= f
\Gamma e 2úi–1i' ; : : : ; e 2úi–ni' \Delta 2 T n g;
(2)
where ' 2 R. Of course, the vector – i is defined only up to sign. In this way one de­
fines the characteristic function – : F !Z n that takes a facet to the corresponding

4 TARAS E. PANOV
primitive vector. By the definition of quasitoric manifolds, the characteristic func­
tion satisfies the following condition: if n facets F i 1
; : : : ; F i n
meet at same vertex
p, i.e. p = F i 1
`` \Delta \Delta \Delta `` F i n
, then the integer vectors –(F i 1
); : : : ; –(F i n
) constitute an
integer basis of the lattice L ' Z n . As it was shown in [DJ], the characteristic pair
(P n ; – : F !Z n ) satisfying the above condition defines a quasitoric manifold M 2n
uniquely up to an equivariant diffeomorphism. Once the numeration of facets of P n
is fixed, the characteristic function can be viewed as the integer (n \Theta m)­matrix \Lambda
whose i­th column is the vector –(F i ). Each vertex p of P n can be represented as
the intersection of n facets: p = F i 1
`` \Delta \Delta \Delta `` F i n
; we denote by \Lambda (p) = \Lambda (i 1 ;::: ;i n ) the
minor matrix of \Lambda formed by the columns i 1 ; : : : ; i n . It follows from the above that
det \Lambda (p) = det(– i 1
; : : : ; – i n
) = \Sigma1:
(3)
In the sequel we refer to the vectors – i = –(F i ) as facet vectors. Note again that
for now they are defined only up to sign.
The next things we need in order to define genera of a quasitoric manifold is
an orientation and stably complex structures. We fix an orientation of T n and
specify an orientation of the polytope P n by orienting the ambient space R n . These
orientation data define an orientation of the quasitoric manifold M 2n . The above
construction of the characteristic function – involves arbitrary choice of signs for
the vectors –(F i ). The inverse image ú \Gamma1 (F i ) of the facet F i under the orbit map
ú : M 2n ! P n is a quasitoric submanifold M i ae M 2n of codimension 2. This facial
submanifold M i belongs to the fixed point set of the circle subgroup GF i ae T n
(see (2)), which therefore acts on the normal bundle š i := š(M i ae M 2n ). Hence,
one can invest š i with a complex structure (and orientation) by specifying a sign of
the primitive vector –(F i ). Given an orientation of M 2n (i.e. an orientation of P n )
and an orientation of š i (i.e. a sign of – i ), one can orient the facial submanifold M i
(or, equivalently, the facet F i ae P n ). Now, the oriented submanifold M i ae M 2n
of codimension 2 gives rise, by the standard topological construction, to a complex
line bundle oe i over M 2n that restricts to š i over M i . The following theorem was
proved by Buchstaber and Ray in [BR2]:
Theorem 1.1. The following isomorphism of real oriented 2m­bundles holds for
any quasitoric manifold M 2n :
Ü (M 2n ) \Phi R 2(m\Gamman) ' oe 1 \Phi \Delta \Delta \Delta \Phi oe m :
Here Ü (M 2n ) is the tangent bundle and R 2(m\Gamman) denotes the trivial 2(m \Gamma n)­bundle
over M 2n .
An oriented quasitoric manifold M 2n together with fixed orientations of facial sub­
manifolds M i = ú \Gamma1 (F i ) was called in [BR2] multioriented. It follows that signs for
the facet vectors f– i g of a multioriented quasitoric manifold are defined unambigu­
ously. Theorem 1.1 shows that a multioriented quasitoric manifold can be invested
with a canonical stably complex structure. Thus, an oriented simple polytope P n
with characteristic matrix \Lambda not only define a (multioriented) quasitoric manifold,
but also specify a cobordism class in the complex cobordism
ring\Omega U .
Given an edge (1­dimensional face) E j of P n , we see that points from its relative
interior in P n correspond to orbits with the same (n \Gamma 1)­dimensional isotropy
subgroup GE j
. This subgroup can be written as
GE j
= f
\Gamma
e 2úi'1 ; : : : ; e 2úi'n
\Delta
2 T n : ¯ 1j ' 1 + : : : + ¯ nj 'n = 0g;
(4)

HIRZEBRUCH GENERA OF MANIFOLDS WITH TORUS ACTION 5
i.e. it is determined by a primitive (co)vector ¯ j = (¯ 1j ; : : : ; ¯ nj ) ? in the dual
(or weight) lattice W = L \Lambda . We refer to this ¯ j as edge vector; again it is defined
only up to sign. The edge vectors satisfy the condition similar to that for the facet
vectors: if E j1 ; : : : ; E jn are edges that meet at same vertex p, then
det M (p) = det(¯ j1 ; : : : ; ¯ jn ) = \Sigma1:
(5)
Here M (p) is the square matrix with columns ¯ j1 ; : : : ; ¯ jn . The above equality means
that the vectors ¯ j1 ; : : : ; ¯ jn constitute a basis of W ' Z n .
The following lemma enables to choose signs of edge vectors for a multioriented
quasitoric manifold unambiguously ``locally'' at each vertex.
Lemma 1.2. Given a vertex p of P n , one can choose signs for edge vectors
¯ j1 ; : : : ; ¯ jn meeting at p in such a way that
M ?
(p) \Delta \Lambda (p) = E;
where E denotes the identity matrix, and the matrices \Lambda (p) , M (p) are those
from (3), (5).
Proof. Since we are interested only in facet and edge vectors meeting at the vertex
p = F i 1
`` \Delta \Delta \Delta``F i n
, we may renumerate the edge vectors ¯ j1 ; : : : ; ¯ jn at p by the index
set fi 1 ; : : : ; i ng of the facet vectors at p. To do this we just set j k = i k if E jk 6ae F i k
(i.e. a facet vector and an edge vector have same index if the corresponding facet
and edge of P n span the whole R n ). Then for k 6= l one has E i k
ae F i l
. Hence,
GF i l
ae GE i k
and
h¯ i k
; – i l
i = 0; k 6= l;
(6)
(see (2) and (4)). Now, since ¯ i k
is a primitive vector, it follows from (6) that
h¯ i k
; – i k
i = \Sigma1. Changing the sign of ¯ i k
if necessary, we obtain
h¯ i k
; – i k
i = 1;
(7)
which together with (6) gives M ?
(p) \Delta \Lambda (p) = E, as needed.
Throughout the rest of this paper we assume that once a stably complex structure
of quasitoric manifold is fixed (i.e. signs for columns of the characteristic (n \Theta m)­
matrix \Lambda are specified), signs of edge vectors for each vertex are chosen as in
Lemma 1.2.
We fix an orientation of the torus T n once and forever; then a choice of orientation
for M 2n is equivalent to a choice of orientation for the polytope P n ae R n . Hence,
edges of P n meeting at the same vertex p can be ordered canonically in such a
way that the ordered set of vectors along the edges pointing out of p constitute a
positively oriented basis of R n (i.e. an orientation of P n defines an ordering of edges
at each vertex). In the sequel we assume that once an orientation of M 2n is fixed,
the edge vectors ¯ i 1
; : : : ; ¯ i n
meeting at p are ordered in accordance with the above
ordering of edges at p. In this situation the edge vectors ¯ i 1
; : : : ; ¯ i n
themselves may
constitute a positively or negatively oriented basis of R n . (We assume that signs
for ¯ i 1
; : : : ; ¯ i n
are determined by Lemma 1.2.) Thus, we come to the following
Definition 1.3. Given a multioriented quasitoric manifold M 2n with orbit space
P n , the sign of a vertex p 2 P n is
oe(p) = det M (p) = det(¯ i 1
; : : : ; ¯ i n
);
where ¯ i 1
; : : : ; ¯ i n
are canonically ordered edge vectors meeting at p.

6 TARAS E. PANOV
Obviously, Lemma 1.2 shows that
oe(p) = det \Lambda (p) = det(– i 1
; : : : ; – i n
);
where – i 1
; : : : ; – i n
are the facet vectors at p. We mention that the above definition
of the sign of vertex already appeared in [Do] while studying characteristic functions
and quasitoric manifolds over given simple polytope.
As it is mentioned in the introduction, quasitoric manifolds can be viewed as
a topological generalization of algebraic smooth toric varieties. Toric variety [Da]
is a normal algebraic variety on which the complex torus (C \Lambda ) n acts with a dense
orbit. Though any smooth toric variety (also called toric manifold) is quasitoric
manifold, we illustrate our above constructions in the more restricted case of smooth
projective toric varieties arising from polytopes. Any such toric variety is obtained
via the standard procedure (see [Fu]) from a simple polytope (1) with vertices
in the integer lattice Z n ae R n (we refer to such polytope as integral). Normal
covectors l i of facets of an integral polytope P n can be chosen integer and primitive.
The toric variety MP obtained from an integral simple polytope P n is necessarily
projective and has complex dimension n. The compact torus T n ae (C \Lambda ) n acts
on MP with orbit space P n ; moreover, there is a map from MP to R n which is
constant on T n ­orbits and has image P n (the moment map). The variety MP is
smooth whenever for each vertex p 2 P n the normal covectors l i k
, k = 1; : : : ; n,
of facets containing p constitute an integer basis of the dual lattice. This means
exactly that the map F i ! l i defines a characteristic function. It can be easily seen
that it is exactly the characteristic function of MP regarded as a quasitoric manifold
(or, conversely, the quasitoric manifold corresponding to the above characteristic
function is equivariantly diffeomorphic to MP ). Hence, facet vectors for MP are just
normal (co)vectors for the corresponding polytope P n . Edge vectors are primitive
integer vectors along the edges of P n . In short, this means that in the case of toric
MP ``facet vectors'' are ``vectors normal to facets'' and ``edge vectors'' are ``vectors
along edges''. Lemma 1.2 in this case says that if an edge is contained in a facet,
then the vector along the edge is orthogonal to the normal vector of the facet, while
if an edge is not contained in a facet, then signs of the corresponding vectors can be
chosen in such a way that their scalar product is equal to 1. It can be shown that the
canonical complex structure on a toric variety MP , regarded as a stably complex
structure on a quasitoric manifold, corresponds to orienting the facet vectors l i in
such a way that all of them are ``pointing inside the polytope P n ''. As it follows
from Lemma 1.2, edge vectors locally should be oriented in such a way that all of
them are ``pointing out of the vertex''. Note that globally Lemma 1.2 provides two
signs for an edge, one for each of its ends. These signs are always different in the
case of toric varieties, however this is not true in general. Note also that since an
edge vector for a toric variety is a vector along the edge pointing out of the vertex,
one has oe(p) = 1 (see Definition 1.3) for any vertex p.
2. Circle action with isolated fixed points on a quasitoric manifold
In this section we show that there exists a circle subgroup of T n that acts on
a quasitoric manifold with only isolated fixed points corresponding to vertices of
the underlying simple polytope. This action will be used in the next section for
calculating the ü y ­genus via contributions of fixed points.

HIRZEBRUCH GENERA OF MANIFOLDS WITH TORUS ACTION 7
So, we start with a multioriented quasitoric manifold M 2n with orbit space P n
and edge vectors f¯ i g. Hence, M 2n is endowed with a stably complex structure, as
described in the previous section.
Theorem 2.1. Suppose that š 2 Z n is an integer primitive vector such that
h¯ i ; ši 6= 0 for all edge vectors ¯ i . Then the circle subgroup S 1 ae T n defined by
š acts on M 2n with isolated fixed points corresponding to vertices of P n . In the
tangent space T p M 2n at fixed point corresponding to the vertex p = F i 1
`` \Delta \Delta \Delta `` F i n
this action induces a representation of S 1 with weights h¯ i 1
; ši; : : : ; h¯ i n
; ši.
Proof. Let us consider the action of T n near fixed point p 2 M 2n corresponding to
a vertex p = F i 1
`` \Delta \Delta \Delta `` F i n
of P n . Let ¯ i 1
; : : : ; ¯ i n
be the edge vectors at p. The
T n ­action induces a unitary T n ­representation in the tangent space T p M 2n . We
choose complex coordinates (x 1 ; : : : ; xn ) in T p M 2n such that the tangent space of
two­dimensional submanifold ú \Gamma1 (E i k ) ae M 2n at p is given by equations x 1 = : : : =
c x k = : : : = xn = 0 (x k is dropped). Then, the corresponding isotropy subgroup GE i k
is given by the equation h¯ i k
; 'i = 0 in T n (see (4)). Hence, the weights of the T n ­
representation are ¯ i 1
; : : : ; ¯ i n
, i.e. an element t =
\Gamma e 2úi'1 ; : : : ; e 2úi'n
\Delta
2 T n acts
on T p M 2n as
t \Delta (x 1 ; : : : ; xn ) =
\Gamma e 2úi(¯1i 1
'1+\Delta\Delta\Delta+¯ ni 1
'n ) x 1 ; : : : ; e 2úi(¯1in '1+\Delta\Delta\Delta+¯ nin 'n ) xn
\Delta
(8)
=
\Gamma
e 2úih¯ i 1
;\Phii x 1 ; : : : ; e 2úih¯ i n ;\Phii xn
\Delta
;
where (x 1 ; : : : ; xn ) 2 T p M 2n , \Phi = (' 1 ; : : : ; 'n ).
A primitive vector š = (š 1 ; : : : ; šn ) ? 2 Z n defines a one­parameter circle sub­
group f
\Gamma e 2úiš1' ; : : : ; e 2úišn'
\Delta ; ' 2 Rg ae T n . It follows from (8) that this circle acts
on T p M 2n as
s \Delta (x 1 ; : : : ; xn ) =
\Gamma
e 2úih¯ i 1 ;ši' x 1 ; : : : ; e 2úih¯ i n ;ši' xn
\Delta
;
where s = e 2úi' 2 S 1 . The fixed point p is isolated if all the weights h¯ i 1
; ši; : : : ;
h¯ i n ; ši of the S 1 ­action are non­zero. Thus, if h¯ i ; ši 6= 0 for all edge vectors, then
the S 1 ­action on M 2n defined by š has only isolated fixed points.
Note that the condition h¯ i ; ši 6= 0 for all ¯ i from the above theorem means that
the primitive vector š is ``of general position''. In the case of smooth toric variety
constructed from integral simple polytope P n the above condition is that the vector
š is not orthogonal to any edge of P n (or, equivalently, no hyperplane normal to š
intersects P n by a face of dimension ? 0).
In the next section we will need the following
Definition 2.2. Given the S 1 ­action on M 2n defined by a primitive vector š, the
index of the vertex p = F i 1
`` \Delta \Delta \Delta `` F i n
is
ind š (p) = f]k : h¯ i k
; ši ! 0g;
i.e. ind š (p) equals the number of negative weights at p.
The following lemma shows that the index of any vertex p can be defined directly
by means of facet vectors at p without calculating edge vectors.
Lemma 2.3. Let p = F i 1
`` \Delta \Delta \Delta `` F i n
be a vertex of P n . Then the index ind š (p)
equals the number of negative coefficients in the representation of š as a linear
combination of basis vectors – i 1
; : : : ; – i n .

8 TARAS E. PANOV
Proof. It follows from Lemma 1.2 that for any vertex p = F i 1
`` \Delta \Delta \Delta `` F i n
one can
write
š = h¯ i 1
; ši– i 1
+ : : : + h¯ i n ; ši– i n :
Then our lemma follows from the definition of ind š (p).
3. Formulae for ü y ­genus, signature and Todd genus
Here we calculate the ü y ­genus of a multioriented quasitoric manifold in terms
of its characteristic pair (P n ; \Lambda). This is done by applying the Atiyah--Hirzebruch
formula for the S 1 ­action constructed in the previous section. We also stress upon
the most important particular cases of signature and Todd genus.
Theorem 3.1. Let M 2n be a multioriented quasitoric manifold, and let š 2 Z n be
an integer primitive vector such that h¯ i ; ši 6= 0 for all edge vectors ¯ i . Then
ü y (M 2n ) =
X
p2P n
(\Gammay) ind š (p) oe(p);
where the sum is taken over all vertices of P n , oe(p) and ind š (p) are as in defini­
tions 1.3 and 2.2.
Proof. The Atiyah--Hirzebruch formula ([AH]; see also [Kr], where it was deduced
within the cobordism theory) states that the ü y ­genus of a stably complex manifold
X with a S 1 ­action can be calculated as
ü y (X) =
X
i
(\Gammay) n(F i ) ü y (F i );
(9)
where the sum is taken over all S 1 ­fixed submanifolds F i ae X, and n(F i ) denotes the
number of negative weights of the S 1 ­representation in the normal bundle of F i ae X.
In our case all fixed submanifolds are isolated fixed points corresponding to vertices
p 2 P n . Hence, ü y (F i ) = ü y (p) = \Sigma1 depending on whether or not the orientation
of R n defined by ¯ i 1
; : : : ; ¯ i n
coincides with that defined by P n . Thus, for quasitoric
M 2n we may substitute oe(p) for ü y (F i ) in (9). Theorem 2.1 shows that the weights
of induced S 1 ­representation in T p M 2n equal h¯ i 1
; ši; : : : ; h¯ i n
; ši, therefore n(F i )
in (9) is exactly ind š (p) (see Definition 2.2), and the required formula follows.
The ü y ­genus ü y (M 2n ) at y = \Gamma1 equals the nth Chern number c n [M 2n ] for any
2n­dimensional stably complex manifold M 2n . Theorem 3.1 gives
c n [M 2n ] =
X
p2P n
oe(p):
(10)
If M 2n is a complex manifold (e.g., M 2n is a smooth toric variety) one has oe(p) = 1
for all vertices p 2 P n and c n [M 2n ] equals the Euler number e(M 2n ). Hence, for
complex M 2n the Euler number equals the number of vertices of P n (which is well
known for toric varieties). For general quasitoric M 2n the Euler number is also
equal to the number of vertices of P n (since the Euler number of any S 1 ­manifold
equals the sum of Euler numbers of fixed submanifolds), however this number may
be different from c n [M 2n ].
The ü y ­genus at y = 1 equals the signature (or the L­genus). Theorem 3.1 gives
in this case

HIRZEBRUCH GENERA OF MANIFOLDS WITH TORUS ACTION 9
Corollary 3.2. The signature of a multioriented quasitoric manifold M 2n can be
calculated as
sign(M 2n ) =
X
p2P n
(\Gamma1) indš (p) oe(p);
where the sum is taken over all vertices of P n .
Being an invariant of an oriented cobordism class the signature of a quasitoric
manifold M 2n does not depend on a stably complex structure (i.e. on a choice of
signs for facet vectors) and is determined only by an orientation of M 2n (or P n ).
This can be seen directly from our above considerations. Indeed, we have chosen
signs of edge vectors locally at each vertex as described in Lemma 1.2. Then Corol­
lary 3.2 states that the signature can be calculated as the sum over vertices of
values (\Gamma1) ind š (p) oe(p), where ind š (p) is the number of negative scalar products
h¯ i k ; ši and oe(p) = det(¯ i 1
; : : : ; ¯ i n
) determines the difference between orienta­
tions of T p M 2n defined by the T n ­representation with weights ¯ i 1
; : : : ; ¯ i n
and the
standard T n ­representation. Instead of this, we may choose signs for edge vectors
in such a way that all scalar products with š are positive. Then one obviously has
(\Gamma1) indš (p) oe(p) = det(~¯ i 1
; : : : ; ~
¯ i n
);
where ~
¯ i k
= \Sigma¯ i k
and h~¯ i k
; ši ? 0, k = 1; : : : ; n. Now, the right hand side of the
above equality is independent of particular choice of signs for facet vectors (i.e.
independent of a stably complex structure). Thus, we come to the following
Corollary 3.3. The signature of an oriented quasitoric manifold M 2n can be cal­
culated as
sign(M 2n ) =
X
p2P n
det(~¯ i 1
; : : : ; ~
¯ i n
);
where f~¯ i k
g are edge vectors at p with signs chosen to satisfy h~¯ i k
; ši ? 0, k =
1; : : : ; n.
In the case of smooth toric variety MP we have oe(p) = 1 for all vertices p 2 P n ,
and Corollary 3.2 gives
sign(MP ) =
X
p2P n
(\Gamma1) ind š (p) :
(11)
The scalar product with vector š can be viewed as an analog of Morse height func­
tion on the polytope P n ae R n (see [Br], [Kh], and [DJ]). Since š is not orthogonal
to any edge of P n , we may orient the edges of P n in such a way that the scalar
product with š increases along the edges. Then the index of vertex p is just the
number of edges pointing in (i.e. towards p). An easy combinatorial argument [Br,
p. 115] shows that the number of vertices with exactly k edges pointing in equals
h k . Here (h 0 ; h 1 ; : : : ; hn ) is the so­called h­vector of the polytope P n . It is defined
from the equation
h 0 t n + : : : + hn\Gamma1 t + hn = (t \Gamma 1) n + f 0 (t \Gamma 1) n\Gamma1 + : : : + f n\Gamma1 ;
where f k is the number of faces of P n of codimension (k + 1). It is well known [Fu]
that h k equals 2k­th Betti number of the toric variety MP (this is also true for a
general quasitoric manifold [DJ], however we do not use this fact here). Now, using
formula (11) we obtain that the signature of a smooth toric variety MP is
sign(MP ) =
n
X
k=1
(\Gamma1) k h k :

10 TARAS E. PANOV
This formula can be also deduced from the Hodge structure in the cohomology of
MP . However, we believe that obtaining this result from our general formula for
quasitoric manifolds could be of interest.
The next important particular case of the ü y ­genus is the Todd genus correspond­
ing to the value y = 0. In this case summands in the formula from Theorem 3.1
are not defined for those vertices having index 0 , so it requires some additional
analysis.
Theorem 3.4. The Todd genus of a multioriented quasitoric manifold can be cal­
culated as
td(M 2n ) =
X
p2P n
indš (p)=0
oe(p)
(the sum is taken over all vertices of index 0). Here š is any primitive vector such
that h¯ i ; ši 6= 0 for all ¯ i .
Proof. For each vertex p = F i 1
`` \Delta \Delta \Delta`` F i n
of P n the stably complex structure on M 2n
determined by (P n ; \Lambda) defines a complex structure on T p M 2n via the isomorphism
T p M 2n ' oe i 1
\Phi \Delta \Delta \Delta \Phi oe i n
(12)
(see Theorem 1.1). The S 1 ­action on M 2n determined by š 2 Z n induces the
complex S 1 ­representation on T p M 2n with weights w 1 (p) = h¯ i 1
; ši; : : : ; wn (p) =
h¯ i n
; ši (signs of edge vectors are determined by Lemma 1.2). Atiyah and Bott's
generalized Lefschetz fixed point formula ([AB], see also [HBY]) gives the following
expression for the equivariant ü y ­genus of M 2n
ü y (q; M 2n ) =
X
p2P n
k
Y
i=1
1 + yq wk (p)
1 \Gamma q wk (p) oe(p);
(13)
where q = e 2úi' 2 S 1 , and oe(p) = det(¯ i 1
; : : : ; ¯ i n
) = \Sigma1 depending on whether
or not the orientation of T p M 2n defined by (12) coincides with that defined by the
original orientation of M 2n . (We note again that this sign oe(p) is equal 1 for all
vertices if M 2n is a true complex manifold, e.g., a smooth toric variety.) Atiyah
and Hirzebruch's theorem [AH] states that the above expression for ü y (q; M 2n ) is
independent of q and equals ü y (M 2n ). Taking the limit of the right hand side of (13)
as q ! 0, one obtains the Atiyah--Hirzebruch formula (9) (since the lim q!0 of each
summand in (13) is exactly (\Gammay) ind(p) oe(p)). In the case y = 0 corresponding to the
Todd genus the same limit for the summand corresponding to a vertex p equals 0
if there is at least one w k (p) ! 0 and equals 1 otherwise. This is exactly what is
stated in the theorem.
As for the Todd genus of a smooth toric variety, it is easy to see that there is
only one vertex of index 0 in this case. Indeed, if we orient edges of the polytope
by means of the scalar product with š (see the above considerations concerning
the signature of toric varieties), then only one ``bottom'' vertex will have all edges
pointing out, i.e. index 0. Since one has oe(p) = 1 for all vertices p 2 P n , Theorem 3.4
gives td(MP ) = 1, which is well known (see e.g. [Fu]).
At the end we consider some examples illustrating our results. All our examples
are multioriented four­dimensional quasitoric manifolds. Thus, all facet vectors are
defined unambiguously, while signs for edge vectors are defined ``locally'' at each

HIRZEBRUCH GENERA OF MANIFOLDS WITH TORUS ACTION 11
@
@
@
@
@
@
@
@
@
@
@
@
@
''!
/
oe
p 1 (1; 0) (\Gamma1; 0) p 2
– 3 = (0; 1)
(0; 1)
– 2 = (1; 0)
(0; \Gamma1)
p 3
(1; \Gamma1)
(\Gamma1; 1)
– 1 = (\Gamma1; \Gamma1)
Figure 1. Ü (CP 2 ) \Phi C ' ¯
j \Phi ¯
j \Phi ¯ j
vertex as described in Lemma 1.2. Our pictures contain a polytope (a polygon in
our case), facet and edge vectors, and an orientation of the polytope (which is
determined in our case by a cyclic order of vertices).
Figure 1 corresponds to CP 2 regarded as a toric variety. Hence, a stably complex
structure is determined by the standard complex structure on CP 2 , i.e. via the
isomorphism of bundles Ü (C P 2 ) \Phi C ' ¯
j \Phi ¯
j \Phi ¯
j, where C is the trivial complex line
bundle and j is the Hopf line bundle over CP 2 . The orientation is defined by the
complex structure. As we have pointed out above, the toric variety CP 2 arises from
an integral polytope (2­dimensional simplex with vertices (0; 0), (1; 0) and (0; 1) in
this case). Facet vectors here are primitive normal vectors to facets pointing inside
the polytope, and edge vectors are primitive vectors along edges pointing out of
a vertex. This can be seen on Figure 1. Let us calculate the Todd genus and the
signature by means of Corollary 3.2 and Theorem 3.4. We have oe(p 1 ) = oe(p 2 ) =
oe(p 3 ) = 1. Take š = (1; 2), then ind(p 1 ) = 0, ind(p 2 ) = 1, ind(p 3 ) = 2 (remember
that the index is the number of negative scalar products of edge vectors with š).
Thus, sign(C P 2 ) = sign(C P 2 ; ¯ j \Phi ¯
j \Phi ¯
j) = 1, td(C P 2 ) = td(C P 2 ; ¯ j \Phi ¯
j \Phi ¯
j) = 1.
Changing the orientation of polytope on Figure 1 causes changing of signs of
the vertices: oe(p 1 ) = oe(p 2 ) = oe(p 3 ) = \Gamma1, while the indices of vertices remain un­
changed. This corresponds to reversing the canonical orientation of CP 2 . Denoting
the resulting manifold CP 2 , we obtain sign(C P 2 ) = sign(C P 2 ; ¯
j \Phi ¯
j \Phi ¯
j) = \Gamma1,
td(C P 2 ) = td(C P 2 ; ¯
j \Phi ¯
j \Phi ¯ j) = \Gamma1. The same stably complex structure can be ob­
tained from the initial orientation of the polytope by taking another facet vectors,
as it is shown on Figure 2. In this case one has
oe(p 1 ) =
fi fi fi fi \Gamma1 0
0 1
fi fi fi fi = \Gamma1; oe(p 2 ) =
fi fi fi fi 1 1
1 0
fi fi fi fi = \Gamma1; oe(p 3 ) =
fi fi fi fi 0 \Gamma1
\Gamma1 \Gamma1
fi fi fi fi = \Gamma1:
Again, we can take š = (1; 2), then ind(p 1 ) = 1, ind(p 2 ) = 0, ind(p 3 ) = 2. Thus, we
see again that sign(C P 2 ) = sign(C P 2 ; ¯ j \Phi ¯
j \Phi ¯
j) = \Gamma1, td(C P 2 ) = td(C P 2 ; ¯ j \Phi ¯
j \Phi
¯
j) = \Gamma1.
Our third example (Figure 3) is CP 2 with the standard orientation and stably
complex structure determined by the isomorphism Ü (C P 2 ) \Phi C ' j \Phi ¯
j \Phi ¯ j (this is

12 TARAS E. PANOV
@
@
@
@
@
@
@
@
@
@
@
@
@
''!
/
oe
p 1 (\Gamma1; 0) (1; 0) p 2
– 3 = (0; 1)
(0; 1)
– 2 =(\Gamma1; 0)
(0; \Gamma1)
p 3
(\Gamma1; \Gamma1)
(1; 1)
– 1 = (1; \Gamma1)
Figure 2. Ü (CP 2 ) \Phi C ' ¯
j \Phi ¯
j \Phi ¯ j
@
@
@
@
@
@
@
@
@
@
@
@
@
''!
/
oe
p 1 (1; 0) (1; 0) p 2
– 3 = (0; 1)
(0; 1)
– 2 = (1; 0)
(0; 1)
p 3
(1; \Gamma1)
(\Gamma1; 1)
– 1 = (1; 1)
Figure 3. Ü (CP 2 ) \Phi C ' j \Phi ¯
j \Phi ¯ j
obtained from Figure 1 by changing the sign of first facet vector). Then we have
oe(p 1 ) =
fi fi fi fi 1 0
0 1
fi fi fi fi = 1; oe(p 2 ) =
fi fi fi fi \Gamma1 1
1 0
fi fi fi fi = \Gamma1; oe(p 3 ) =
fi fi fi fi 0 1
1 \Gamma1
fi fi fi fi = \Gamma1:
Taking š = (1; 2), we find ind(p 1 ) = 0, ind(p 2 ) = 0, ind(p 3 ) = 1. Thus, sign(C P 2 ; j\Phi
¯
j \Phi ¯
j) = 1, td(C P 2 ; j \Phi ¯
j \Phi ¯
j) = 0. Note that in this case formula (10) gives
c n (C P 2 ; j \Phi ¯
j \Phi ¯
j)[CP 2 ] = oe(p 1 ) + oe(p 2 ) + oe(p 3 ) = \Gamma1 (this could be also checked
directly), while the Euler number of CP 2 is c n (C P 2 ; ¯
j \Phi ¯
j \Phi ¯
j)[C P 2 ] = 3.
The author is grateful to Prof. V.M. Buchstaber for stimulating discussions and
useful recommendations.

HIRZEBRUCH GENERA OF MANIFOLDS WITH TORUS ACTION 13
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Department of Mathematics and Mechanics, Moscow State University, Moscow
119899 RUSSIA
E­mail address: tpanov@mech.math.msu.su