TODD GENERA OF COMPLEX TORUS MANIFOLDSHIROAKI ISHIDA AND MIKIYA MASUDA
Abstract. In this paper, we prove that the Todd genus of a compact complex manifold X of complex dimension n with vanishingodd degree cohomology is one if the automorphism group of X
contains a compact n-dimensional torus Tn as a subgroup. Thisimplies that if a quasitoric manifold admits an invariant complex
structure, then it is equivariantly homeomorphic to a compact
smooth toric variety, which gives a negative answer to a problem
posed by Buchstaber-Panov.

1. Introduction
A torus manifold is a connected closed oriented smooth manifoldof even dimension, say 2n, endowed with an eﬀective action of ann-dimensional torus Tn having a fixed point. A typical example of atorus manifold is a compact smooth toric variety which we call a toric
manifold in this paper. Every toric manifold is a complex manifold.
However, a torus manifold does not necessarily admit a complex(even an almost complex) structure. For example, the 4-dimensional
sphere S4 with a natural T2 -action is a torus manifold but admits noalmost complex structure.
On the other hand, there are infinitely many nontoric torus manifolds of dimension 2n which admit Tn -invariant almost complex
structures when n ≥ 2. For instance, for any positive integer k, there
exists a torus manifold of dimension 4 with an invariant almost complex structure whose Todd genus is equal to k ([10, Theorem 5.1])
while the Todd genus of a toric manifold is always one. One canproduce higher dimensional examples by taking products of those 4dimensional examples with toric manifolds. The cohomology rings
Date: March 18, 2012.
2010 Mathematics Subject Classification. Primary 57R91, Secondary 32M05, 57S25.
Key words and phrases. Torus manifold, complex manifold, toric manifold.
The first author was supported by JSPS Research Fellowships for Young Scientists.
The second author was partially supported by Grant-in-Aid for Scientific Research 22540094.
1

2

H. ISHIDA AND M. MASUDA

of the torus manifolds in these examples are generated by its degreetwo part like toric manifolds.
In this paper, we consider a torus manifold with a Tn -invariant(genuine) complex structure. We will call such a torus manifold acomplex torus manifold. The following is our main theorem.
Theorem 1.1. If a complex torus manifold has vanishing odd degree cohomology, then its Todd genus is equal to one.
Remark 1.2. If a closed smooth manifold M has vanishing odd degree
cohomology, then any smooth Tn -action on M has a fixed point (see [2,Corollary 10.11 in p.164]). In particular, a connected closed oriented
smooth manifold M of dimension 2n with an eﬀective Tn -action isa torus manifold if M has vanishing odd degree cohomology. Thisimplies that Theorem 1.1 is equivalent to the statement in the abstract.
Other important examples of torus manifolds are 1quasitoric manifolds introduced by M. W. Davis and T. Januskiewicz ([4]). A quasitoric manifold of dimension 2n is a closed smooth manifold witha locally standard Tn -action, whose orbit space is an n-dimensional
simple polytope. It is unknown whether any toric manifold is a quasitoric manifold. However, if a toric manifold is projective, then it isa quasitoric manifold because a projective toric manifold with the restricted compact torus action admits a moment map which identifies
the orbit space with a simple polytope.
A. Kustarev ([9, Theorem 1]) gives a criterion of when a quasitoric
manifold admits an invariant almost complex structure. It also follows from his criterion that there are many nontoric quasitoric manifolds which have invariant almost complex structures. However,it has been unknown whether there is a quasitoric manifold which
admits an invariant complex structure, and V. M. Buchstaber andT. E. Panov posed the following problem ([3, Problem 5.23]), which
motivated the study in this paper.
Problem 1.3 (Buchstaber-Panov). Find an example of nontoric quasitoric
manifold that admits an invariant complex structure.
As a consequence of Theorem 1.1, we obtain the following which
gives a negative answer to Problem 1.3.
Theorem 1.4. If a quasitoric manifold admits an invariant complex structure, then it is equivariantly homeomorphic to a toric manifold.
1

Davis-Januszkiewicz [4] uses the terminology toric manifold but it was alreadyused in algebraic geometry as the meaning of (compact) smooth toric variety, so
Buchstaber-Panov [3] started using the word quasitoric manifold.

TODD GENERA OF COMPLEX TORUS MANIFOLDS

3This paper is organized as follows. In Section 2, we study simply
connected compact complex surfaces with torus actions. In Section
3, we review the notion of multi-fan and recall a result on Todd
genus. In Section 4, we define a map associated with the multifan of a complex torus manifold X and give a criterion of when the
Todd genus of X is one in terms of the map. Theorems 1.1 and
1.4 are proved in Sections 5 and 6 respectively. Throughout thispaper, all cohomology rings and homology groups are taken with
Z-coeﬃcients.
While preparing this paper, the first author and Yael Karshon
proved that a complex torus manifold is equivariantly biholomorphic to a toric manifold ([7]). Although Theorem 1.1 is contained in
the result, the argument in this paper is completely diﬀerent fromthat in [7] and we believe that this paper is worth publishing.
2. Simply connected complex surfaces with torus actions
We first recall two results on simply connected 4-manifolds.
Theorem 2.1 ([12]). If a simply connected closed smooth manifold of dimension 4 admits an eﬀective smooth action of T2 , then it is diﬀeomorphic
to a connected sum of copies of CP2 , CP2 (CP2 with reversed orientation)and S2 × S2 .
Theorem 2.2 ([5]). If a simply connected projective complex surface isdecomposed into Y1 #Y2 as oriented smooth manifolds, then either Y1 or Y2
has a negative definite cup-product form.
Let X be a simply connected compact complex surface whose automorphism group contains T2 as a subgroup. By Theorem 2.1(2.1)

X  kCP2 #ℓCP2 #m(S2 × S2 ),k, ℓ, m ≥ 0

as oriented smooth manifolds. Therefore, the Euler characteristic
χ(X) and the signature σ(X) of X are respectively given by
χ(X) = k + ℓ + 2m + 2 andσ(X) = k − ℓand hence the Todd genus Todd(X) of X is given by1
1
Todd(X) = (χ(X) + σ(X)) = (k + m + 1).
4
2
The following proposition is a key step toward Theorem 1.1.

(2.2)

Proposition 2.3. Let X be as above. Then Todd(X) = 1.

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H. ISHIDA AND M. MASUDA

Proof. Since X is simply connected, the first betti number of X is 0,
in particular, even. Thus, X is a deformation of an algebraic surface([8, Theorem 25]). Since any algebraic surface is projective (see [1,Chapter IV, Corollary 5.6]), we can apply Theorem 2.2 to our X.
Unless (k, m) = (1, 0) or (0, 1), it follows from (2.1) that X can be
decomposed into Y1 #Y2 as oriented smooth manifolds, where
Y1 = CP2 ,

Y2 = (k − 1)CP2 #ℓCP2 #m(S2 × S2 )Y1 = S2 × S2 ,if k ≥ 2,

Y2 = kCP2 #ℓCP2 #(m − 1)(S2 × S2 )Y1 = CP2 #ℓCP2 ,

Y2 = S2 × S2

if m ≥ 2,
if (k, m) = (1, 1).

In any case, neither of Y1 and Y2 has a negative cup-product form
and this contradicts Theorem 2.2. Therefore, (k, m) = (1, 0) or (0, 1)
and hence Todd(X) = 1 by (2.2).

3. Torus manifolds and multi-fans
In this section, we review the notion of multi-fans introduced in[6] and [10] and recall a result on Todd genus.
A torus manifold X of dimension 2n is a connected closed oriented
manifold endowed with an eﬀective action of Tn having a fixed point.
In this paper, we are concerned with the case when X has a complex
structure invariant under the action. We will call such a torus manifold a complex torus manifold.
Throughout this section, X will denote a complex torus manifoldof complex dimension n unless otherwise stated. We define a combinatorial object ∆X := (ΣX , CX , wX ) called the multi-fan of X. A characteristic submanifold of X is a connected complex codimension 1 holomorphic submanifold of X fixed pointwise by a circle subgroup of Tn .
Characteristic submanifolds are Tn -invariant and intersect transversally. Since X is compact, there are only finitely many characteristic
submanifolds, denoted X1 , . . . , Xm . We set




∩




,
I
∈
{1,
2,
.
.
.
,
m}
|
X
:=
X
,
∅
ΣX := 

I
i




i∈I

which is an abstract simplicial complex of dimension n − 1.
Let S1 be the unit circle group of complex numbers and Ti the circle
subgroup of Tn which fixes Xi pointwise. We take the isomorphismλi : S1 → Ti ⊂ Tn such that
(3.1)

λi (g)∗ (ξ) = gξ for ∀ g ∈ S1 and ∀ ξ ∈ TX|Xi /TXi

TODD GENERA OF COMPLEX TORUS MANIFOLDS

5

where λi (g)∗ denotes the diﬀerential of λi (g) and the right hand sideof (3.1) above is the scalar multiplication with the complex number gon the normal bundle TX|Xi /TXi of Xi . We regard λi as an element of
the Lie algebra Lie Tn of Tn through the diﬀerential and assign a cone
(3.2)

CX (I) := pos(λi | i ∈ I) ⊂ Lie Tn

to each simplex I ∈ ΣX , where pos(A) denotes the positive hullspanned by elements in the set A. This defines a map CX fromΣX to the set of cones in Lie Tn .
We denote the set of (n − 1)-dimensional simplices in ΣX by Σ(n).
X
For I ∈ Σ(n) , XI is a subset of the Tn -fixed point set of X. The weightfunction wX : Σ(n)→ Z>0 is given by
X
wX (I) := #XI
where #A denotes the cardinality of the finite set A.
The triple ∆X := (ΣX , CX , wX ) is called the multi-fan of X. The Todd
genus Todd(X) of X can be read from the multi-fan ∆X as follows.
Theorem 3.1 ([10]). Let v be an arbitrary vector in Lie Tn which is notcontained in CX (J) for any J ∈ ΣX \Σ(n). ThenX∑
Todd(X) =wX (I)
where the summation runs over all I ∈ Σ(n)such that CX (I) contains v.
X
The following corollary follows immediately from Theorem 3.1.
Corollary 3.2. Todd(X) = 1 if and only if the pair (ΣX , CX ) forms an
ordinary complete nonsingular fan and wX (I) = 1 for every I ∈ Σ(n).
X
Suppose X J is connected for every J ∈ ΣX . Then X J is a complex
codimension #J holomorphic submanifold of X having a Tn -fixed
point. Moreover, the induced action of the quotient torus Tn /T J on
X J is eﬀective and preserves the complex structure of X J , where T J is
the #J-dimensional subtorus of Tn generated by T j for j ∈ J. Therefore,
X J is a complex torus manifold of complex dimension n − #J with theeﬀective action of the quotient torus Tn /T J .
In this case, the multi-fan ∆XJ = (ΣXJ , CXJ , wXJ ) of X J for J ∈ Σ canbe obtained from the multi-fan ∆X of X as discussed in [6], which weshall review. We note that X J ∩ Xi is non-empty if and only if J ∪ {i}is a simplex in ΣX and each characteristic submanifold of X J can be
written as the non-empty intersection X J ∩ Xi . Hence, the simplicial
complex ΣXJ coincides with the link link(J; ΣX ) of J in ΣX andCXJ (I) = pos(λi | i ∈ I)

6

H. ISHIDA AND M. MASUDA

for I ∈ link(J; ΣX ), where λi denotes the image of λi by the quotient
map Lie Tn → Lie Tn /T J . The weight function wXJ is the constantfunction 1.
4. Maps associated with multi-fans
Let X be a complex torus manifold of complex dimension n and∆X = (ΣX , CX , wX ) the multi-fan of X. Throughout this section, we
assume that X J is connected for every J ∈ ΣX . We will define acontinuous map fX from the geometric realization |ΣX | of ΣX to theunit sphere Sn−1 of the vector space Lie Tn in which the cones CX (I)for I ∈ ΣX sit, and give a criterion of when the Todd genus of X isequal to 1 in terms of the map fX .
We set




∑
∑




m
σI := 
a
e
|
a
=
1,
a
≥
0
for I ∈ ΣX ,

i
i
i
i

⊂R


i∈I

i∈I

where ei is the i-th vector in the standard basis of Rm . The geometricrealization |ΣX | of ΣX is given by∪|ΣX | =σI .
I∈ΣX

Recall that the homomorphisms λi : S1 → Tn for i = 1, . . . , m definedin Section 3 are regarded as elements in Lie Tn through the diﬀerential.
We take an inner product on Lie Tn and denote the length of anelement v ∈ Lie Tn by |v|. We define a map fX : |ΣX | → Sn−1 , where
Sn−1 is the unit sphere of Lie Tn , by


∑
∑
ai λi


(4.1)
fX |σI 
ai ei  = ∑i∈I
.
|
aλ|
i∈I

i∈I i iClearly, fX is a closed continuous map.
Lemma 4.1. The map fX : |ΣX | → Sn−1 is a homeomorphism if and only if
Todd(X) = 1.
Proof. We note that XI is one point for any I ∈ Σ(n)because XI is
X
connected by assumption and of codimension n in X. Therefore,wX (I) = 1 for any I ∈ Σ(n)and this together with Theorem 3.1 tellsX
us that the Todd genus Todd(X) coincides with the number of cones
CX (I) containing the vector v ∈ Lie Tn in Theorem 3.1.
The above observation implies that the cones CX (I) for I ∈ ΣX
do not overlap and form an ordinary complete fan in Lie Tn if and

TODD GENERA OF COMPLEX TORUS MANIFOLDS

7

only if Todd(X) = 1 and this is equivalent to the map fX being ahomeomorphism, proving the lemma.
For each characteristic submanifold Xi , we can also define a map
fXi : |ΣXi | → Sn−2 , where Sn−2 is the unit sphere in Lie Tn /Ti  (Lie Ti )⊥ ,where (Lie Ti )⊥ denotes the orthogonal complement of a vector subspace Lie Ti in Lie Tn .
Lemma 4.2. If fXi : |ΣXi | → Sn−2 is a homeomorphism, then fX |star({i};ΣX ) :star({i}; ΣX ) → Sn−1 is a homeomorphism onto its image, where star({i}; ΣX )denotes the open star of {i} in ΣX .
Proof. It suﬃces to show the injectivity of fX |star({i};ΣX ) because fX isclosed and continuous. Let pi : Lie Tn → (Lie Ti )⊥ be the orthogonal
projection. Through pi , we identify Lie Tn /Ti with (Lie Ti )⊥ . Recall
that ΣXi = link({i}; ΣX ). For each vertex j of link({i}; ΣX ), we express
λ j asλ j = pi (λ j ) + ci, j λi , ci, j ∈ R.
By the definitions of link({i}; ΣX ) and star({i}; ΣX ), we can express anelement x ∈ star({i}; ΣX ) as
(4.2)

x = (1 − t) ei +ty,

with y ∈ | link({i}; ΣX )|, 0 ≤ t < 1.

Suppose y ∈ σ J ⊂ | link({i}; ΣX )| and write∑∑y=a j e j,a j = 1, a j ≥ 0.
j∈J

j∈J

Then, it follows from (4.1) thatfX (x) =(4.3)=(1 − t)λi + t|(1 − t)λi + t(1 − t)λi + t|(1 − t)λi + t

∑
∑
∑
∑

j∈J a j λ jj∈J a j λ j |j∈J a j (pi (λ j ) + ci, j λi )j∈J a j (pi (λ j ) + ci, j λi )|= g(t, y) fXi (y) + h(t, y)
where
t|

∑

λi
,
|λi |

j∈J a j pi (λ j )|

∑

(4.4)

g(t, y) :=and∑(1 − t + t j∈J a j ci, j )|λi |∑.
h(t, y) :=|(1 − t)λi + t j∈J a j (pi (λ j ) + ci,j λi )||(1 − t)λi + t

j∈J a j (pi (λ j ) + ci, j λi )|8

H. ISHIDA AND M. MASUDA

Since | fX (x)| = | fXi (y)| = 1 and fXi (y) is perpendicular to λi /|λi |, it
follows from (4.3) thatg2 (t, y) + h2 (t, y) = 1.

(4.5)

Take another point x′ ∈ star({i}, ΣX ) and write
x′ = (1 − t) ei +t′ y′ ,with y′ ∈ | link({i}; ΣX )|, 0 ≤ t′ < 1similarly to (4.2). SincefX (x′ ) = g(t′ , y′ ) fXi (y′ ) + h(t′ , y′ )

λi
,
|λi |

we have fX (x) = fX (x′ ) if and only if(4.6)

g(t, y) fXi (y) = g(t′ , y′ ) fXi (y′ )and h(t, y) = h(t′ , y′ ).

Both g(t, y) and g(t′ , y′ ) are non-negative by (4.4), so it follows from
(4.5) and (4.6) that if fX (x) = fX (x′ ), theng(t, y) = g(t′ , y′ )

(4.7)

and

fXi (y) = fXi (y′ ).

The latter identity in (4.7) above implies y = y′ since fXi is a homeomorphism by assumption. Therefore it follows from (4.6) and (4.7)
thatg(t, y) = g(t′ , y) and h(t, y) = h(t′ , y).
This together with (4.3) shows that∑∑
(1 − t)λi + t
a j λ j = (1 − t′ )λi + t′a jλ j.
∑

j∈J

j∈J

Here λi and j∈J a j λ j are linearly independent, so we conclude t = t′ .
It follows that fX |star({i};ΣX ) is injective, which implies the lemma.

We have the following corollary.
Corollary 4.3. If fXi : |ΣXi | → Sn−2 is a homeomorphism for all i, thenfX : |ΣX | → Sn−1 is a covering map, and hence if |ΣX | is connected andn − 1 ≥ 2 in addition, then fX is a homeomorphism.
5. Torus manifolds with vanishing odd degree cohomology
In this section, we prove Theorem 1.1 in the Introduction.
The Tn -action on a torus manifold X of dimension 2n is said tobe locally standard if the Tn -action on X locally looks like a faithfulrepresentation of Tn , to be more precise, any point of X has an invariant open neighborhood equivariantly diﬀeomorphic to an invariant
open set of a faithful representation space of Tn . The orbit space X/Tn
is a manifold with corners if the Tn -action on X is locally standard. A
manifold with corners Q is called face-acyclic if every face of Q (evenTODD GENERA OF COMPLEX TORUS MANIFOLDS

9

Q itself) is acyclic. A face-acyclic manifold with corners is called ahomology polytope if any intersection of facets of Q is connected unlessempty. The combinatorial structure of X/Tn and the topology of X
are deeply related as is shown in the following theorem.
Theorem 5.1 ([11]). Let X be a torus manifold of dimension 2n.
(1) Hodd (X) = 0 if and only if the Tn -action on X is locally standard
and X/Tn is face-acyclic.
(2) H∗ (X) is generated by its degree-two part as a ring if and only if theTn -action on X is locally standard and X/Tn is a homology polytope.
Suppose that X is a torus manifold of dimension 2n with vanishingodd degree cohomology. Then X/Tn = Q is a manifold with corners
and face-acyclic. Let π : X → X/Tn = Q be the quotient map andlet Q1 , . . . , Qm be the facets of Q. Then π−1 (Q1 ), . . . , π−1 (Qm ) are thecharacteristic submanifolds of X, denoted X1 , . . . , Xm before. If Q is ahomology polytope, i.e. any intersection of facets of Q is connected
unless empty (this is equivalent to any intersection of characteristic
submanifolds of X being connected unless empty), then the geometric realization |ΣX | of the simplicial complex ΣX is a homology sphereof dimension n − 1 (see [11, Lemma 8.2]), in particular, connected
when n ≥ 2. Unless Q is a homology polytope, intersections of facets
are not necessarily connected. However, we can change Q into ahomology polytope by cutting Q along faces of Q. This operation
corresponds to blowing-up along connected components of intersections of characteristic submanifolds of X equivariantly. We refer thereader to [11] for the details.
The results in Section 2 required the simply connectedness of acomplex surface. Here is a criterion of the simply connectedness ofa torus manifold in terms of its orbit space.
Lemma 5.2. Suppose that the Tn -action on a torus manifold X is locallystandard. Then X is simply connected if and only if the orbit space X/Tn issimply connected.
Proof. Since the group Tn is connected, the “only if” part in the lemma
follows from [2, Corollary 6.3 in p.91].
We shall prove the “if” part. Suppose that X/Tn is simply connected. Since each characteristic submanifold Xi of X is of real codimension two, the homomorphism(5.1)

π1 (X\ ∪i Xi ) → π1 (X)induced by the inclusion map from X\ ∪i Xi to X is surjective. Here,the Tn -action on X\ ∪i Xi is free since the Tn -action on X is locally10

H. ISHIDA AND M. MASUDA

standard, so that the quotient map from X\ ∪i Xi to (X\ ∪i Xi )/Tn gives
a fiber bundle with fiber Tn . The orbit space (X\ ∪i Xi )/Tn is simply
connected because X/Tn ia a manifold with corners, (X\ ∪i Xi )/Tn is
the interior of X/Tn and X/Tn is simply connected by assumption.
Therefore the inclusion map from a free Tn -orbit to X\ ∪i Xi induces
an isomorphism on their fundamental groups. But any free Tn -orbit
shrinks to a fixed point in X, so the epimorphism in (5.1) must be
trivial and hence X is simply connected.
Now, we are in a position to prove the following main theorem
stated in the Introduction.
Theorem 5.3. If a complex torus manifold X has vanishing odd degree
cohomology, then the Todd genus of X is 1.
Proof. Let n be the complex dimension of X as usual. Since Hodd (X) =0, the orbit space X/Tn is face-acyclic by Theorem 5.1. As remarked
after Theorem 5.1, one can change X into a complex torus manifold whose orbit space is a homology polytope by blowing-up X
equivariantly. Since Todd genus is a birational invariant, it remains
unchanged under blowing-up. Therefore we may assume that theorbit space of our X is a homology polytope, so that any intersectionof characteristic submanifolds of X is connected unless empty and|ΣX | is a homology sphere of dimension n − 1. Since the orbit spaceof Xi is a facet of X/Tn , it is also a homology polytope so that any intersection of characteristic submanifolds of Xi (viewed as a complex
torus manifold) is also connected unless empty and |ΣXi | is a homology sphere of dimension n − 2. Therefore, the results in Section 4 areapplicable to X and Xi ’s.
We shall prove the theorem by induction on the complex dimension
n of X. If n = 1, then X is CP1 and hence Todd(X) = 1. When n = 2, theorbit space X/T2 is contractible because X/T2 is acyclic by Theorem
5.1 and the dimension of X/T2 is 2. Therefore, X is simply connectedby Lemma 5.2 and Todd(X) = 1 by Proposition 2.3.
Assume that n ≥ 3 and the theorem holds when the complex
dimension is equal to n−1. Then, Todd(Xi ) = 1 for any Xi by induction
assumption and hence fXi : |ΣXi | → Sn−2 is a homeomorphism byLemma 4.1. Since |ΣX | is a homology sphere of dimension n − 1(≥ 2),|ΣX | is connected and hence fX : |ΣX | → Sn−1 is a homeomorphism
by Corollary 4.3. It follows from Lemma 4.1 that Todd(X) = 1. This
completes the induction step and the theorem is proved.
TODD GENERA OF COMPLEX TORUS MANIFOLDS

11

6. Proof of Theorem 1.4
A quasitoric manifold X of dimension 2n is a smooth closed manifold endowed with a locally standard Tn -action, whose orbit space
is a simple polytope Q of dimension n. Clearly, X is a torus manifold. The characteristic submanifolds X1 , . . . , Xm of X bijectively
correspond to the facets Q1 , . . . , Qm of Q through the quotient map
π : X → Q. Therefore, for I ⊂ {1, . . . , m}, XI = ∩i∈I Xi is non-empty if
and only if QI := ∩i∈I Qi is non-empty; so the simplicial complex
ΣX = {I ⊂ {1, . . . , m} | XI , ∅}introduced in Section 3 is isomorphic to the boundary complex of
the simplicial polytope dual to Q. As before, let Ti be the circle
subgroup of Tn which fixes Xi pointwise and let λi : S1 → Ti ⊂ Tn be
an isomorphism. There are two choices of λi for each i.
One can recover X from the data (Q, {λi }m) up to equivariant homei=1
omorphism as follows. Any codimension k face F of Q is written as
QI for a unique I ∈ ΣX with cardinality k and we denote the subgroup
TI by TF . For a point p ∈ Q, we denote by F(p) the face containing p
in its relative interior. Setn
X(Q, {λi }mi=1 ) := T × Q/ ∼,where (t, p) ∼ (s, q) if and only if p = q and ts−1 ∈ TF(p) . Then X andX(Q, {λi }m) are known to be equivariantly homeomorphic ([4]).
i=1Suppose that our quasitoric manifold X admits an invariant complex structure. Then, the isomorphism λi is unambiguously determined by requiring the identity (3.1), that is
λi (g)∗ (ξ) = gξ,∀g ∈ S1 and ∀ ξ ∈ TX|Xi /TXi .

The simplicial complex ΣX and the elements λi ’s are used to define
the multi-fan of X. But since the Todd genus of X is one by Theorem 5.3, the multi-fan of X is an ordinary complete non-singular fan
by Corollary 3.2 and hence it is the fan of a toric manifold. Finally, we
note that since ΣX is the boundary complex of the simplicial polytope
dual to the simple polytope Q, it determines Q as a manifold withcorners up to homeomorphism. This implies Theorem 1.4 because
the equivariant homeomorphism type of X is determined by Q andthe elements λi ’s as remarked above.
Acknowledgment. The authors thank Masaaki Ue and Yoshinori
Namikawa for their helpful comments on the automorphism groupsof compact complex surfaces.

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H. ISHIDA AND M. MASUDA

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Osaka City University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan.
E-mail address: ishida@sci.osaka-cu.ac.jp
Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka
558-8585, Japan.
E-mail address: masuda@sci.osaka-cu.ac.jp