Normat 57:3, 107–115 (2009) 107
Real Toric Surfaces
Oksana V. Znamenskaya, Alexey V. Shchuplev
alex@lan.krasu.ru
OVZnamenskaya@sfu-kras.ru
1. INTRODUCTION
Almost every course in topology starts with the notion of the manifold. The variety
of examples used to visualize this important concept includes the projective space.
This simple non-trivial example is particularly useful, because it allows to write
down explicitly formulas for all related notions such as charts, trivializations, and
transition functions. It is also important that these formulas are elementary, they
use only monomial functions. This property identifies an important class of alge-
braic manifolds called toric varieties. Apart from affine, projective, and weighted
projective spaces, they include their products, Hirzebruch surfaces and many other
interesting manifolds.
Another kind of examples used for the demonstration of basic properties of
manifolds is the so-called classical surfaces: the sphere, the torus, the Klein bottle,
the double torus, and so on. In general, a classical surface is a sphere, or a connected
sum of either tori or real projective planes. The process of triangulating or gluing
them from their fundamental polygons can be easily demonstrated to the audience,
which further enhances their educational value.
A question arises here whether we can analyze a real two-dimensional manifold
from these both angles. In other words, if there is a compact smooth real surface
which admits an atlas with monomial transition mappings, i.e., whether an example
of the real toric surface can be found.
The obvious answer is that such a surface exists and that it is the torus. The
latter is the real part of the product of two Riemann spheres. Since this product is
a toric variety, the torus is a real toric variety. However, this is the only example of
a smooth orientable compact surface admitting such an atlas. In this note we offer
an elementary proof of this fact.
2. CONSTRUCTION OF TORIC VARIETY
Definition. A complex toric variety is an irreducible variety X such that
1. The algebraic torus (C
∗
)
n
= (C r {0})
n
is a Zariski open subset of X;
2. The multiplicative action of (C
∗
)
n
on itself extends to an action of (C
∗
)
n
on
X.
As already noted, the most basic examples of toric varieties are the algebraic torus
(C
∗
)
n
, the affine space C
n
, and the projective space P
n
. In general, a toric variety
108 Oksana V. Znamenskaya, Alexey V. Shchuplev Normat 3/2009
X consists of (C
∗
)
n
plus a finite number of toric hypersurfaces, which are also toric
varieties. This multilevel structure is reflected in a combinatorial object called fan.
A fan Σ is defined as a finite collection of cones in R
n
with the following prop-
erties:
• each cone σ ∈ Σ is a strongly convex rational (with respect to the lattice
Z
n
⊂ R
n
) polyhedral cone;
• all faces of σ ∈ Σ belong to Σ;
• if faces σ, τ ∈ Σ, then σ ∩ τ is a face of both.
Every cone σ from Σ defines an n-dimensional affine toric variety U
σ
, and the toric
variety X
Σ
is obtained by gluing together U
σ
and U
τ
along U
σ∩τ
for all σ, τ ∈ Σ. It
should be stressed here that a fan completely determines a toric variety, therefore,
the question of varieties can be translated into the question of fans and solved by
elementary means.
In this note we choose a different construction of toric varieties from the one
mentioned above. This approach was developed independently by several people
(see [1]). It generalizes the quotient representation of the projective space as a set
of lines passing through the origin in an affine space. A toric variety in this case
is a set of certain monomial surfaces in an affine space passing through a special
collection of coordinate planes. For later use, we outline below the main points of
this construction.
Let v
1
, . . . , v
d
∈ Z
n
be the primitive elements (minimal integer generators) of
one-dimensional cones of the fan Σ in R
n
. Assign a variable t
i
to each vector v
i
. In
the space C
d
of variables t = (t
1
, . . . , t
d
) we consider the variety
Z(Σ) =
t ∈ C
d
:
Y
v
i
6∈σ
t
i
= 0 ∀ σ ∈ Σ
.
In effect, to define this variety we may restrict ourselves to monomials associated
with maximal cones of Σ. The function of this union of coordinate planes is parallel
to that of the origin in the case of the projective space.
The monomial surfaces in C
d
are the orbits of the multiplicative action of the
group G on C
d
r Z(Σ). The group G is a subgroup of (C
∗
)
d
and is defined by
G =
(
g = (µ
1
, . . . , µ
d
) ∈ C
d
:
d
Y
i=1
µ
hm, v
i
i
i
= 1 ∀m ∈ Z
n
)
,
although it suffices to take only basis elements of the lattice for m. Thus, we have
only n relations, and G is isomorphic to (C
∗
)
d−n
.
The complex toric variety is defined as a quotient (C
d
r Z(Σ))/G, but this
quotient is a smooth manifold if and only if the fan Σ is simplicial and primitive.
In other words, if each k-dimensional cone is generated by k + 1 one-dimensional
cones and the list of their primitive generators may be completed to a basis of Z
n
.
Furthermore, X
Σ
is compact if the supports of all cones of Σ cover the whole space
R
n
. We refer to the tutorial [2] for the details of both constructions and numerous
examples.
Normat 3/2009 Oksana V. Znamenskaya, Alexey V. Shchuplev 109
Either construction can be transposed to the field of real numbers R producing
a real toric variety, which is a real part of the corresponding complex one.
We reformulate now our question in terms of fans. We are looking for a complete
simplicial primitive fan Σ in R
2
such that the corresponding real toric variety is
an orientable surface. Order the generators of one-dimensional cones of Σ counter-
clockwise. A two-dimensional cone generated by a pair of adjacent vectors v
i
, v
j
is
primitive if and only if the determinant
∆
ij
:= v
i1
v
j2
− v
i2
v
j1
= 1. (1)
It should be pointed out that a fan in R
2
with one-dimensional generators v
1
, . . . , v
d
satisfying (1) is already complete and simplicial. Moreover, it can be also shown
that the resulting toric variety is orientable if and only if
for every generator v
i
= (v
i1
, v
i2
) the number |v
i
| = v
i1
+ v
i2
is odd. (2)
A justification of this fact may be found in [3], where the real toric varieties were
successfully utilized.
Condition (2) places a severe restriction on the structure of fans in R
2
and plays
a crucial role in the next section. At the same time, without this assumption our
question presents a difficult problem, which is not yet solved.
3. THE FAN OF THE ORIENTABLE TORIC SURFACE
Let us start by introducing the following integer vectors
e
1
= (1, 0), e
2
= (0, 1), e
3
= (−1, 0), e
4
= (0, −1),
e
5
= (1, 1), e
6
= (−1, 1), e
7
= (−1, −1), e
8
= (1, −1),
and open half-planes
Π
k
= {v ∈ R
2
: hv, e
k
i > 0}, k = 1, . . . , 8.
Lemma 1. Let Σ be a fan in R
2
satisfying (1) and (2). Suppose that e
k
lies in the
interior of the cone σ = σ(v
i
, v
j
) generated by v
i
and v
j
. Then either σ 6⊂ Π
k
, or
σ is a coordinate quadrant.
Proof. Consider first the case when k = 1, . . . , 4. The vector e
k
lies on an axis,
from which necessarily follows that σ 6⊂ Π
k
. Otherwise, the generators v
i
and v
j
of
σ would lie in adjacent quadrants, hence the products v
i1
v
j2
and v
i2
v
j1
differ from
zero and have different signs. This implies that the absolute value |∆
ij
| > 1, which
contradicts (1).
Examine now the case when vector e
k
lies on the bisector of a quadrant, i.e.,
k = 5, . . . , 8. Obviously, if σ is a quadrant containing e
k
, it satisfies conditions (1),
(2) and lies entirely in Π
k
.
Suppose now that σ is not a quadrant and σ ⊂ Π
k
. Then there are three possible
ways to arrange the generators v
i
and v
j
and the coordinate quadrant I
k
containing
e
k
:
110 Oksana V. Znamenskaya, Alexey V. Shchuplev Normat 3/2009
Figure 1.
a) the generators v
i
and v
j
lie in I
k
;
b) the quadrant I
k
lies entirely in σ;
c) one of the generators of σ lies inside I
k
, while the other does not.
It is easily seen that the last case has already been ruled out, since the cone σ
contains the vector e
l
, 1 ≤ l ≤ 4 and lies entirely in the corresponding open half-
plane Π
l
.
For the cases a) and b), notice that
∆
ij
=
v
i1
v
j2
− v
i2
v
j1
=
|v
i1
v
j2
| − |v
i2
v
j1
|
.
This follows from the fact that the products v
i1
v
j2
and v
i2
v
j1
have the same sign,
because the generators lie either in the same quadrant or in quadrants symmetrical
about the origin. More precisely, in these cases one of the generators of σ(v
i
, v
j
)
belongs to the set D
1
= {(x, y) : |x| > |y|}, whereas the other to the set D
2
=
{(x, y) : |x| < |y|}. Even more specifically, let v
i
be in D
1
, then we may write
|v
i1
| = p + |v
i2
|,
|v
j2
| = q + |v
j1
| with positive integers p and q.
Substituting these equalities in the expression for
∆
ij
we arrive at
|∆
ij
| =
pq + |v
i2
|q + |v
j1
|p
= pq + |v
i2
|q + |v
j1
|p.
It should be taken into account that σ is not a quadrant, that is, v
i2
and v
j1
are
non-zero. For that reason |∆
ij
| > 1, which contradicts (1). This concludes the
proof.
Lemma 2. A cone σ satisfying (1) and (2), and generated by v
i
= (v
i1
, −1) and
v
j
= (v
j1
, −1) cannot be decomposed into a fan with the same properties.
Proof. As the coordinates v
i1
and v
j1
are necessarily even, we can write v
i1
= 2p,
v
j1
= 2q, p, q ∈ Z.
Assume that there exists a refinement of σ, and every cone of it satisfies (1)
and (2). Consider the vector (2p + 1, −1). It lies in σ and cannot be a generator,
Normat 3/2009 Oksana V. Znamenskaya, Alexey V. Shchuplev 111
Figure 2.
since the sum of its coordinates is even. Therefore, it lies in some cone σ
∗
(v
∗
i
, v
∗
j
)
of the decomposition (see Figure 2), from which follows that (2p +1, −1) is a linear
combination of the generators v
∗
i
and v
∗
j
with positive coefficients, that is to say,
there exist positive integers α and β such that
(
αv
∗
i1
+ βv
∗
j1
= 2p + 1,
αv
∗
i2
+ βv
∗
j2
= −1.
On the other hand, v
∗
i2
≤ v
i2
= −1 and v
∗
j2
≤ v
j2
= −1, so that the second equality
cannot hold. The consequence is that σ cannot be further decomposed.
As a next step we will impose one additional restriction on fans. The reason
for placing this restriction is that two fans encode the same variety if there is a
unimodular linear transformation of Z
2
turning one fan in R
2
into another. There
is a simple explanation for this: such a transformation of fans induces a monomial
isomorphism of the corresponding toric varieties. In order to avoid this ambiguity,
we shall consider fans only in their canonical form, i.e., fans containing the positive
quadrant as a two-dimensional cone.
This restriction does not eliminate any possible variety we are looking for, as
demonstrated by the following lemma.
Lemma 3. Let Σ be a fan satisfying (1) and (2) and σ(v
i
, v
j
) be any of its cones.
Then there exists a unimodular linear transformation turning σ into the positive
quadrant and preserving properties (1) and (2) of the fan.
Proof. It is obvious that the transformation of R
2
with the matrix
A =
v
j2
−v
j1
−v
i2
v
i1
is unimodular and turns σ into the positive quadrant. Property (1) of the trans-
formed fan follows immediately from the properties of determinants, and property
(2) is equally easy to prove.
At this stage, we are ready to prove the following key proposition that establishes
the structure of the fan satisfying (1) and (2).
112 Oksana V. Znamenskaya, Alexey V. Shchuplev Normat 3/2009
Proposition 1. The canonical form of a complete fan Σ in R
2
satisfying (1) and
(2) has only four one-dimensional cones generated by e
1
, e
2
, e
3
, and v = (a, −1),
or by e
1
, e
2
, e
4
, and v = (−1, a) with an even a (see Figure 3).
Figure 3.
Proof. Assume that there exists a fan with prescribed properties, and its one-
dimensional generators are numbered counterclockwise, starting from e
1
. The vec-
tor e
1
becomes v
1
, e
2
becomes v
2
, and the last one, preceding e
1
, turns into v
d
.
As the fan Σ is complete, it has a cone σ(v
i
, v
j
) containing e
8
. According to
Lemma 1, the following three arrangements of vectors v
i
, v
j
, and e
k
are possible
(see Figure 4):
Figure 4.
a) the cone σ(v
i
, v
j
) coincides with the quadrant generated by e
4
and e
1
;
b) the vector v
j
coincides with e
1
, and v
i
lies inside the cone σ(e
3
, e
7
);
c) the vector v
i
coincides with e
3
, and v
j
lies inside the cone σ(e
8
, e
1
).
In the first case, the fan Σ already has two two-dimensional cones σ(e
1
, e
2
) and
σ(e
1
, v
d
) with the generator v
d
coinciding with e
4
. Condition (1) imposed on these
cones yields v
3
= (−1, a) and v
d−1
= (−1, b), but the cone generated by these two
vectors does not possess this property. Moreover, according to Lemma 2, this cone
cannot be further decomposed. Thus, the half-plane Π
3
contains only one generator
v
3
= (−1, a), and cones of Σ are generated by e
1
, e
2
, e
4
, (−1, a).
In the second case, the fan Σ contains two cones σ(e
1
, e
2
) and σ(v
d
, e
1
) with
v
d
= v
i
. Consider a refinement of the cone σ(e
2
, v
d
). The vector v
d−1
cannot lie in
Normat 3/2009 Oksana V. Znamenskaya, Alexey V. Shchuplev 113
the interior of the quadrant σ(e
2
, e
3
), since this possibility is excluded by Lemma
1, and Lemma 2 states that v
d−1
cannot lie inside of σ(e
6
, e
3
). The only possibility
left is that v
d−1
is e
3
. The next step is to subdivide the quadrant σ(e
2
, e
3
) but,
as we have seen, this contradicts Lemma 2. As a result, the fan Σ has only four
generators: e
1
, e
2
, e
3
, and (a, −1).
In the third case, we are already in a position to subdivide σ(e
2
, e
3
), which
contradicts Lemma 2.
4. CONSTRUCTING A VARIETY FROM THE FAN
In this section, we show that the fans obtained in Proposition 1 always encode, up
to a homeomorphism, the torus.
Proposition 2. Any smooth compact orientable real toric surface is homeomorphic
to the torus.
Proof. First note that toric surfaces corresponding to the fans with the generators
e
1
, e
2
, e
3
, v = (a, −1) and e
1
, e
2
, e
4
, v = (−1, a) are isomorphic since they differ
only by rotation. We shall choose the first set of generators. The corresponding
toric variety is defined as a quotient of R
4
\ Z by the action of the group G. The
set Z is the union of the two planes {t
1
= t
3
= 0} and {t
2
= t
4
= 0} in R
4
and
g = (µ
1
, µ
2
, µ
3
, µ
4
) ∈ G if µ
1
/µ
3
µ
−a
4
= µ
2
/µ
4
= 1. Denoting λ
1
= µ
3
and λ
2
= µ
4
,
we may write the action of g on t = (t
1
, t
2
, t
3
, t
4
) as
g · t = (λ
1
λ
−a
2
t
1
, λ
2
t
2
, λ
1
t
3
, λ
2
t
4
), (λ
1
, λ
2
) ∈ (R \ {0})
2
. (3)
We claim that the Cartesian product of two semicircles in R
4
E = {t
2
1
+ t
2
3
= 1, t
2
2
+ t
2
4
= 1, t
3
≥ 0, t
4
≥ 0}
is a screen for the action of G, or equivalently, that every orbit of the group action
intersects this set.
Indeed, orbit (3) of t intersects E if the following system of equations and in-
equalities in λ has a real solution:
(
(λ
1
λ
−a
2
t
1
)
2
+ (λ
1
t
3
)
2
= 1, λ
1
t
3
≥ 0,
(λ
2
t
2
)
2
+ (λ
2
t
4
)
2
= 1, λ
2
t
4
≥ 0.
(4)
One can easily see that
λ
1
= ±
s
1
λ
−2a
2
t
2
1
+ t
2
3
, λ
2
= ±
s
1
t
2
2
+ t
2
4
(5)
solves the equations. Since (t
1
, t
3
) and (t
2
, t
4
) are from R
2
\ {0}, these solutions
always exist. Besides, the inequalities in (4) determine the choice of a sign of the
roots if t
3
6= 0, t
4
6= 0. This proves that if an orbit intersects the screen E in an
interior point then this point is unique.
114 Oksana V. Znamenskaya, Alexey V. Shchuplev Normat 3/2009
We shall now turn to the case when an orbit meets E at a boundary point.
Fix a point P = (t
1
, t
2
, t
3
, t
4
) ∈ ∂E. The coordinates of P satisfy t
2
1
+ t
2
3
= 1,
t
2
2
+ t
2
4
= 1, consequently, (5) implies that the orbit of P intersects the screen
E if λ
1
= ±1, λ
2
= ±1, that is, in no more than four points, including P (for
λ
1
= λ
2
= 1).
Since P ∈ ∂E, at least one of the coordinates t
3
, t
4
is equal to zero. Consider
two cases:
a) Assume that t
3
= 0, t
4
6= 0, then P = (±1, t
2
, 0, t
4
). The inequality λ
2
t
4
≥ 0
implies that system (4) has only two solutions: λ
1
= ±1, λ
2
= 1. From this
we conclude that the orbit of P also intersects the screen E in another point
P
0
= (λ
1
λ
−a
2
· (±1), λ
2
t
2
, λ
1
· 0, λ
2
t
4
) = (∓1, t
2
, 0, t
4
).
(see Figure 5). These two points must be identified.
Figure 5. Figure 6.
b) Assume now that t
3
6= 0, t
4
= 0. Analogously, the solutions to system (4) are
λ
1
= 1, λ
2
= ±1, therefore, the point E must be identified with the point
P
0
= (λ
1
λ
−a
2
t
1
, λ
2
· (±1), λ
1
t
3
, λ
2
· 0) = (t
1
, ∓1, t
3
, 0),
(see Figure 6).
Thus, in the screen, which is homeomorphic to a square, we identify the opposite
sides in such a way that the resulting surface is the torus (see Figure 7).
We may add here that if a were odd, in case b) the sides would be glued differ-
ently, resulting in the Klein bottle, a non-orientable surface (see Figure 8).
Figure 7. Figure 8.
REMARK. This results follow immediately from Proposition 1 as soon as we
notice that those fans encode complex toric varieties called Hirzebruch surfaces.
Topologically every Hirzebruch surface is a fiber product with the base CP
1
and
the fiber CP
1
. The real part of it is either the torus or the Klein bottle.
Normat 3/2009 Oksana V. Znamenskaya, Alexey V. Shchuplev 115
References
[1] D. Cox The homogeneous coordinate ring of a toric variety, J. Alg. Geometry
4 (1995), 17–50.
[2] D. Cox What is a toric variety? Contemporary Math., vol. 334, 203–224.
[3] T.O. Ermolaeva, A.K. Tsikh Integration of rational functions over R
n
by
means of toric compactifications and higher-dimensional residues, Sb. Math.
187 (1996), 9, 1301–1318.
