In this section we want to introduce the notion of a toric variety in a way one can quickly work with. We will motivate the definition by going through three different examples. We only introduce and use what we actually need later on and therefore the definitions are mathematically not fully rigorous. For a more detailed and very careful introduction to the topic we refer the reader to the book of Cox, Little and Schenck [53].
53 3.2. From projective spaces to toric varieties
(a)The complex projective space P1. All the
points of the different lines through the origin are identified with one another. The origin itself is excluded.
(b)The weighted projective spaceP1,2. All points of different parabola through the ori- gin are identified with one another. The origin is excluded.
Figure 3.1.: Two examples of a toric variety both having only one C∗-action.
3.2.1
Complex projective spaces
Let us start off with a very simple toric variety namely the complex projective space. Considerncopies ofCas a starting point. The idea of acomplex projective space, denoted by CPd or
Pd is to consider a set of equivalence classes inside Cn
where for the projective spaces we always havep=n−1. The equivalence relation identifies all complex straight lines through the origin in this space. For instance consider a point x ∈ Cn as an element of
Pd, then an arbitrary multiple of this
point by a non-zero constantλ 6= 0 corresponds to one and the same element of
Pd. Hence the equivalence relation ∼ is defined by
(x1, ..., xn)∼(λx1, ..., λxn) ∀λ ∈C∗ :=C\{0}. (3.1) So really all points on a straight line are identified by one another with one exception, the origin. Since all these lines intersect at the origin we have to exclude it. Otherwise we would identify all points in our space with the origin and we would never obtain anything non-trivial. For the case n = 2 we can actually visualize the space P1 as shown in figure 3.1a . The action that multiplies every coordinate of a point with a non-zero constant complex number is called a C∗- action for obvious reasons. Since algebraic geometers refer toC∗ as an algebraic torus it is also called a torus action. When physicists talk to each other, this action can also take the role of an abelian gauge group acting on some two-
3. Toric Geometry 54
dimensional field theory as was explained in chapter 2and may therefore also be called the U(1)action. Taking everything we just said into account we can quote the proper definition of Pd.
Definition 3.2.1 (Complex Projective Space). A complex projective space Pd is
defined as the quotient space
Pd = C
n− {0}
C∗
, (3.2)
where the C∗-action is defined via the equivalence relation ∼:
(x1, ..., xn)∼(λ1 x1, ..., λ1 xn) ∀λ ∈C∗ =C\{0}. (3.3) A point in Pd can be written by(x
1 :...:xn) (sometimes also simply denoted as
(x1, ..., xn), keeping (3.3) in mind) where x1, ..., xn are then called the homoge-
neous coordinates of Pd.
3.2.2
Weighted projective spaces
After defining the complex projective spaces it is quite straight forward to define more general spaces having a C∗-action different from the one of a complex pro- jective space. The picture is more or less the same with just one difference. As we have identified straight lines through the origin to define a complex projective space, the spaces we are considering now are defined by identifying point sets that correspond to, not necessarily linear, polynomials through the origin. This means that we identify points with one another that are related by multiplying every component with the same constant but a different power of this constant, i.e.
(x1, ..., xn)∼(λQ1x1, ..., λQnxn)∀ λ∈C∗, (3.4) where the Qi are arbitrary numbers in Z that are called weights. Hence such a space is called a weighted projective space. For different choices ofQi’s we obtain different weighted projective spaces. One simple example for the one-dimensional case is given where Q1 = 1 and Q2 = 2. Then the identification reads for the
point (1,1)
55 3.2. From projective spaces to toric varieties
This point set is a complex parabola inC2 and similarly for every starting point
different from(1,1)we get a differently shaped parabola which is drawn in figure
3.1b. The formal definition is very much like the one for the complex projective space:
Definition 3.2.2 (Weighted Projective Space). A weighted projective space, de-
noted by PQ1,...,Qn for Qi ∈Z∀ 1≤i≤n is defined as the quotient space
Pd= C
n− {0}
C∗
, (3.6)
where the C∗-action is defined via the equivalence relation∼:
(x1, ..., xn)∼(λQ1x1, ..., λQnxn) ∀ λ∈C∗. (3.7)
Now we have already a large number of toric varieties that we can construct using definitions 3.2.1 and 3.2.2. So far we only motivated the construction and visualized their definition. We have not shown any relation to algebraic vari- eties or smooth manifolds here. In fact one can show that these are algebraic varieties and furthermore that the complex projective spaces are even smooth al- gebraic varieties and hence they are algebraic manifolds. The weighted projective spaces on the other hand can be shown to be singular spaces and hence are not represented by some smooth manifold. Nevertheless they might contain smooth subvarieties as we will see in section 6. Also, we may perform a blowup of points in the weighted projective space in order to resolve singularities.
3.2.3
Toric varieties
The step to a toric variety is not very big now and simply given by an introduction of variousC∗-actions inside the same spaceCn. An example for a one-dimensional toric variety inC3 is given by the
C∗-actions in the rows of the matrix
(Qαi) = 1 1 0 0 2 1 (3.8)
3. Toric Geometry 56
Figure 3.2.: The one dimensional toric variety corresponding to the charge matrix in equation (3.8). Since r = 2, points on sheets, i.e. complex two-dimensional spaces are identified with one another. Here the exceptional set which has to be removed corresponds to the union of the two horizontal axes.
which gives the following two equivalence relations:
(x1, x2, x3) = (λx1, λx2, x3), ∀ λ∈C∗, (3.9)
(x1, x2, x3) = (x1, µ2x2, µx3), ∀µ∈C∗. (3.10)
Plugging in any values for x1, x2 and x3 one can see that these equations just
parametrize complex two-dimensional surfaces that span inside C3 and are built out of two kinds of curves. The curves that correspond to the first row of the matrix (3.8) intersect the horizontal plane at the axis where x1 and x2 equal to
zero which is thex3 axis. Similarly intersect the curves that belong to the second
row of (3.8) the horizontal plane at thex1 axis. This scenario is plotted in figure
3.2where the horizontal axes arex1 andx3. Since all the points on these complex
surfaces are identified with one another we have to exclude the sets where they intersect in order to get something non-trivial. This set, which is here given by the x1 and the x3 axis is called the exceptional set. It is closely related to the
so-called Stanley-Reisner ideal which will be defined in a second. Let us first quote the general definition of a toric variety which is analog to the one of the projective and weighted projective space:
57 3.2. From projective spaces to toric varieties
Definition 3.2.3 (Toric Variety). Letd, r∈N and n=d+r. A ddimensional
toric variety PΣ is defined as the quotient space
PΣ = C
n−Z
(C∗)r , (3.11)
whereZ is the exceptional set. The (C∗)r actions correspond to the equivalence relations that are given by a matrix(Qαi)which is defined according to
(x1, ..., xn)∼(λQ α 1x 1, ..., λQ α nx n) ∀ α= 1, ..., r , ∀ λ∈C∗, (3.12) which identifies points on r-dimensional subspaces in Cn.
Remark. The exceptional set is a crucial part of the toric variety that encodes a lot of information on its topology. It basically tells us which combination of homogeneous coordinates are not allowed to vanish simultaneously. In the framework of the GLSM this task corresponds to minimizing the Bosonic potential which was done by solving equations (2.58). The exceptional set is here given by the solution of equation three in (2.58) and this relation between toric varieties and the classical vacuum of a two-dimensional field theory helps many physicists to loose their fear of these spaces.
What we specifically usually need is the quite closely related notion of the following.
Definition 3.2.4(Stanley-Reisner Ideal). LetZ be the exceptional set of a toric
variety PΣ. The Stanley-Reisner ideal IΣ is the minimal ideal containing square-
free monomials corresponding to the different subsets of the exceptional set:
IΣ :=hxτ | {xτ = 0} ⊂Zi , (3.13) where we defined τ :={i1, ..., ik} along with
xτ :=
k Y
j=1
xij and xτ :={xi1, ..., xik} (3.14)
Example 3.2.1. For the complex projective space and the weighted projective
3. Toric Geometry 58
one defining subset:
Z ={(x1, x2, ..., xn)∈Cn | (x1, x2, ..., xn) = (0,0, ...,0)}. (3.15) Therefore we get a Stanley-Reisner Ideal which contains only one monomial:
IΣ =hx1·x2·...·xni . (3.16)
Example 3.2.2. As a second example let us consider the toric variety defined
by the matrix in (3.8) above. As we saw, we had to remove the x1 and the x3
axes. Therefore the exceptional set of this example is
Z ={(x1, x2, x3)∈C3 : (x2, x3) = (0,0)or(x1, x2) = (0,0)}, (3.17)
and from that we obtain the Stanley-Reisner ideal which has two elements:
IΣ =hx2·x3, x1·x2i . (3.18)
As one can imagine, these sets might get very complicated to derive once we choose a matrix Q that does not look as nice as the one in (3.8) and at some point it is impossible to simply read off the exceptional set and hence the Stanley-Reisner ideal. But there is a very convenient way to calculate these sets systematically in terms ofd dimensional polytopes which we are going to explain in the next section.
Remark. We have not said anything about smoothness of a generic toric variety so far. While we know that the projective spaces are always smooth and the weighted projective spaces are always singular, for a generic toric variety it is a priory not clear. There are smooth ones that are in particular no products of complex projective spaces and there are also singular ones which are not products of weighted projective spaces. How one can check for smoothness of a toric variety in a combinatorial way will also be content of the next section.