We review the construction of families of projective varieties, in particular Calabi-Yau threefolds, as quasilinear sections in weighted flag varieties. We also describe a construction of tautological orbibundles on these varieties, which may be of interest in heterotic model building.

1. Introduction

The classical flag varieties are projective varieties which are homogeneous spaces under complex reductive Lie groups ; the stabilizer of a point in is a parabolic subgroup of . The simplest example is projective space itself, which is a homogeneous space under the complex Lie group . Weighted flag varieties , which are the analogues of weighted projective space in this more general context, were defined by Corti and Reid [1] following unpublished work of Grojnowski. They admit a Plücker-style embedding into a weighted projective space. In this paper, we review the construction of Calabi-Yau threefolds that arise as complete intersections within of some hypersurfaces of weighted projective space [13]: To be more precise, our examples are going to be quasilinear sections in , where the degree of each equation agrees with one of the . The varieties will have standard threefold singularities similar to complete intersections in weighted projective spaces; they have crepant desingularizations by standard theory.

We start by computing the Hilbert series of a weighted flag variety of a given type. By numerical considerations, we get candidate degrees for possible Calabi-Yau complete intersection families. To prove the existence of a particular family, in particular to check that general members of the family only have mild quotient singularities, we need equations for the Plücker style embedding. It turns out that the equations of in the weighted projective space, which are the same as the equations of the straight flag variety in its natural embedding, can be computed relatively easily using computer algebra [2].

The smooth Calabi-Yau models that arise from this method may be new, though it is probably difficult to tell. One problem we do not treat in general is the determination of topological invariants such as Betti and Hodge numbers of . Some Hodge number calculations for varieties constructed using a related method are performed in [4], via explicit birational maps to complete intersections in weighted projective spaces; the Hodge numbers of such varieties can be computed by standard methods. Such maps are hard to construct in general. A better route would be to first compute the Hodge structure of then deduce the invariants of their quasilinear sections and finally their resolutions . See, for example, [5] for analogous work for hypersurfaces in toric varieties. We leave the development of such an approach for future work.

We conclude our paper with the outline of a possible application of our construction: by its definition, the weighted flag variety and thus its quasilinear section carry natural orbibundles; these are the analogues of on (weighted) projective space. It is possible that these can be used to construct interesting bundles on the resolution which may be relevant in heterotic compactifications. Again, we have no conclusive results.

2. Weighted Flag Varieties

2.1. The Main Definition

We start by recalling the notion of weighted flag variety due to Corti and Reid [1]. Fix a reductive Lie group and a highest weight in the weight lattice of , the lattice of characters of the maximal torus of . Then we have a corresponding parabolic subgroup , well defined up to conjugation. The quotient is a homogeneous variety called a (generalized) flag variety, a projective subvariety of , where is the irreducible representation of with highest weight .

Let denote the lattice of one-parameter subgroups of , dual to the weight lattice . Choose and an integer such that for all elements of the Weyl group of the Lie group , where denotes the perfect pairing between and .

Consider the affine cone of the embedding . There is a -action on given by which induces an action on . Inequality (2.1) ensures that all the -weights on are positive, leading to a well-defined quotient This weighted projective space has weights where denotes the set of weights (understood with multiplicities) appearing in the weight space decomposition of the representation . Inside this weighted projective space, we consider the projective quotient We call a weighted flag variety. By definition, quasismooth, that is, its affine cone is nonsingular outside its vertex . Hence it only has finite quotient singularities.

The weighted flag variety is called well formed [6], if no of weights have a common factor, and moreover does not contain any codimension singular stratum of , where is the codimension of .

2.2. The Hilbert Series of a Weighted Flag Variety

Consider the embedding . The restriction of the line (orbi)bundle of degree one Weil divisors gives a polarization on , a -ample line orbibundle some tensor power of which is a very ample line bundle. Powers of have well-defined spaces of sections . The Hilbert series of the pair is the power series given by

Theorem 2.1 (see [2, Theorem  3.1]). The Hilbert series has the closed form Here is the Weyl vector, half the sum of the positive roots of , and depending on whether consists of an even or odd number of simple reflections in the Weyl group .

The right hand side of (2.7) can be converted into a form where, as before, denotes the set of weights of the representation . The numerator is a polynomial , the Hilbert numerator. Since (2.7) involves summing over the Weyl group, it is best to use a computer algebra system for explicit computations.

A well-formed weighted flag variety is projectively Gorenstein, which means that (i) for all ;(ii)the Hilbert numerator is a palindromic symmetric polynomial of degree , called the adjunction number of ; (iii)the canonical divisor of is given by where, as above, the are the weights of the projective space ; the integer is called the canonical weight.

2.3. Equations of Flag Varieties

The flag variety is defined by an ideal of quadratic equations generating a linear subspace of the second symmetric power of the contragradient representation . The -representation has a decomposition into irreducible direct summands, with being the highest weight of the representation . As discussed in [7, 2.1], the subspace in fact consists of all the summands except . The equations of can be readily computed from this information using computer algebra [2].

2.4. Constructing Calabi-Yau Threefolds

We recall the different steps in the construction of Calabi-Yau threefolds as quasilinear sections of weighted flag varieties.

(1) Choose Embedding
We choose a reductive Lie group and a -representation of dimension with highest weight . We get a straight flag variety of computable dimension and codimension . We choose and to get an embedding , with being the weights of the representation . The equations, the Hilbert series, and the canonical class of can be found as described above.

(2) Take Threefold Calabi-Yau Section of
We take a quasilinear complete intersection of generic hypersurfaces of degrees equal to some of the weights . We choose values so that and , thus . After relabelling the weights, this gives an embedding , with , of codimension , polarized by the ample -Cartier divisor with . More generally, as in [1], we can take complete intersections inside projective cones over , adding weight one variables to the coordinate ring which are not involved in any relation.

(3) Check Singularities
We are interested in quasismooth Calabi-Yau threefolds, subvarieties of all of whose singularities are induced by the weights of . Singular strata of correspond to sets of weights with nontrivial. If the intersection is nonempty, it has to be a singular point or a curve of quotient singularities, and we need to find local coordinates in a neighbourhood of the point of , respectively of points of , to check the local transversal structure. Since we are interested in Calabi-Yau varieties which admit crepant resolutions, singular points have to be quotient singularities of the form with divisible by , whereas the transversal singularity along a singular curve has to be of the form of type .

(4) Find Projective Invariants and Check Consistency
The orbifold Riemann-Roch formula of [4, Section  3] determines the Hilbert series of a polarized Calabi-Yau threefold with quotient singularities in terms of the projective invariants and , as well as for each curve, the degree of the polarization, and an extra invariant related to the normal bundle of in . Using the Riemann-Roch formula, we can determine the invariants of a given family from the first few values of and verify that the same Hilbert series can be recovered.

2.5. Explicit Examples

In the next two sections, we find families of Calabi-Yau threefolds admitting crepant resolutions using this programme. We illustrate the method using two embeddings, corresponding to the Lie groups of type and , leading to Calabi-Yau families of codimension 8, 6, respectively. Further examples for the Lie groups of type and , in codimensions 7 and 9, are discussed in [3].

3. The Codimension Eight Weighted Flag Variety

3.1. Generalities

Consider the simple Lie group of type . Denote by a pair of simple roots of the root system of , taking to be the short simple root and the long one. The fundamental weights are and . The sum of the fundamental weights, which is equal to half the sum of the positive roots, is . We partition the set of roots into long and short roots as . Let be the basis of the lattice dual to .

We consider the -representation with highest weight . The dimension of is 14 [8, Chapter 22]. The homogeneous variety is five-dimensional, so we have an embedding of codimension 8. To work out the weighted version in this case, take and .

Proposition 3.1. The Hilbert series of the codimension eight weighted flag variety is given by Moreover, if is well-formed, then the canonical bundle is .

The Hilbert series of the straight flag variety can be computed to be The image is defined by 28 quadratic equations, listed in the appendix of [2].

3.2. Examples

Example 3.2. Consider the following initial data. (i)Input: . (ii)Plücker embedding: . (iii)Hilbert numerator: . (iv)Canonical divisor: , as is well formed. (v)Variables on weighted projective space together with their weights: The reason for the curious ordering of the variables is that these variables are exactly those appearing in the defining equations of this weighted flag variety given in [2, Appendix]. Consider the threefold quasilinear section where the intersection is taken with general forms of degrees four and five, respectively. The canonical divisor class of is To determine the singularities of the general threefold , we need to consider sets of variables whose weights have a greatest common divisor greater than one. (i) singularities: this singular stratum is defined by setting those variables to zero whose degrees are not divisible by 4. We also have the equations of [2, Appendix]; only (A5), (A23), and (A24) from that list survive to give In this case, it is easy to see by hand (or certainly using Macaulay) that is in fact a twisted cubic curve isomorphic to . We then need to intersect this with the general ; the quintic equation will not give anything new, since are degree 4 variables, but the quartic equation will give a linear relation between them. Thus consists of three points, the three points of singularities. A little further work gives that they are all of type .(ii) singularities: the general does not intersect this singular stratum; the equations from [2, Appendix] in the degree three variables give the empty locus; this is easiest to check by Macaulay. (iii) singularities: the intersection of with this singular stratum is a rational curve containing the singular points; again, Macaulay computes this without difficulty. At each other point of the curve we can check that the transverse singularity is . Thus is a Calabi-Yau threefold with three singular points of type and a rational curve of singularities of type containing them. Comparing with the orbifold Riemann-Roch formula of [4, Section  3], feeding in the first few known values of from the Hilbert series gives that the projective invariants of this family are

Example 3.3. In this example, we consider the same initial data as in Example 3.2. To construct a new family of Calabi-Yau threefolds, we take a projective cone over . Therefore we get the embedding The canonical divisor class of is . Consider the threefold quasilinear section with ; brackets denote a general hypersurface of degree . (i) singularities: since there is no quartic equation this time, the whole twisted cubic curve , found above, is contained in the general and is a rational curve of singularities of type . (ii) singularities: the general does not intersect this singular stratum. (iii) singularities: the intersection of with this singular strata defines a further rational curve of singularities. On each point of the curve we check that local transverse parameters have odd weight. Therefore is a curve of type . Thus is a Calabi-Yau threefold with two disjoint rational curves of singularities and of type and , respectively. The rest of the invariants of this family are

Example 3.4. The next example is obtained by a slight generalization of the method described so far. The computation of the canonical class , as the basic line bundle raised to the power equal to the difference of the adjunction number and the sum of the weights on , only works if is well formed. In this example, we will make our ambient weighted homogeneous variety not well formed. We then turn it into a well formed variety by taking projective cones over it. We finally take a quasilinear section to construct a Calabi-Yau threefold .(i)Input: , . (ii)Plücker embedding: , not well formed. (iii)Hilbert numerator: . We take a double projective cone over , by introducing two new variables and of weight one, which are not involved in any of the defining equations of . We get a seven-dimensional well-formed and quasismooth variety with canonical class .
Consider the threefold quasilinear section The canonical class becomes trivial. Since is a five-dimensional variety and we are taking a complete intersection with four generic hypersurfaces of degree two inside , the singular locus defined by weight two variables defines a curve in . Thus is a Calabi-Yau threefold with a curve of singularities of type . The rest of the invariants of are given as follows:

Example 3.5. Our final initial data in this section consists of the following. (i)Input: , . (ii)Plücker embedding: . (iii)Hilbert numerator: .(iv)Canonical class: , as is well formed.
We take a projective cone over to get the embedding with . We take a complete intersection inside , with three general forms of degree seven, five, and four in . Therefore we get a threefold with trivial canonical divisor class. To work out the singularities, we work through the singular strata to find that is a polarised Calabi-Yau threefold containing three dissident singular points of type , a rational curve of singularities of type containing them, and a further isolated singular point of type . The rest of the invariants are

4. The Codimension 6 Weighted Grassmannian Variety

4.1. The Weighted Flag Variety

We take to be the reductive Lie group of type . The five simple roots are , the weight lattice with basis . The Weyl vector can be taken to be Consider the irreducible -representation , with . Then is 15-dimensional, and all of the weights appear with multiplicity one. The highest weight orbit space is eight-dimensional. This flag variety can be identified with the Grassmannian of 2-planes in a 6-dimensional vector space, a codimension 6 variety

Let be the dual basis of the dual lattice . We choose , to get the weighted version of , The set of weights on our projective space is , where are weights appearing in the -representation . As a convention we will write an element of dual lattice as row vector, that is, .

We expand formula (2.7) for the given values of to get the following formula for the Hilbert series of Here In particular, if is well formed, then its canonical bundle is , with .

The defining equations for are well known to be the Pfaffians obtained by deleting two rows and the corresponding columns of the skew symmetric matrix

4.2. Examples

Example 4.1. Consider the following data. (i)Input: . (ii)Plücker embedding: . (iii)Hilbert numerator: . (iv)Canonical class: .
Consider the threefold quasilinear section Then is trivial, and is a Calabi-Yau 3-fold with a singular point of type , lying on the intersection of two curves, of type and of type . There is an additional isolated singular point of type . The rest of the invariants of this variety are

Example 4.2. We take the following. (i)Input: , . (ii)Plücker embedding: . (iii)Hilbert numerator: . (iv)Canonical class: , as is well formed.
Consider the quasilinear section then The variety is a well-formed and quasismooth Calabi-Yau 3-fold. Its singularities consist of two rational curves and of singularities of type and , respectively. The rest of the invariants are

5. Tautological (Orbi)bundles

5.1. The Classical Story

Let be a flag variety. A representation of the parabolic subgroup gives rise to a vector bundle on as follows: In other words, the total space of consists of pairs modulo the equivalence The fiber of over each point is isomorphic to the vector space underlying .

Example 5.1. The simplest example is , a homogeneous variety with and the parabolic subgroup consisting of matrices of the form We obtain a one-dimensional representation of by mapping to . The associated line bundle is just the tautological line bundle on , the dual of the hyperplane bundle .

Example 5.2. More generally, consider , the Grassmannian of -planes in . Then and the corresponding parabolic is the subgroup of matrices of the form with of size and , respectively. The representations of defined by , , respectively, give the standard tautological sub- and quotient bundles and on the Grassmannian , fitting into the exact sequence

Example 5.3. Finally consider the -variety studied in Section 3. The smallest representations of the corresponding have dimensions 2 and 5. The corresponding tautological bundles are easiest to describe using an embedding , mapping the flag variety into the Grassmannian of 2-planes in a 7-dimensional vector space, the space of imaginary octonions. Then the tautological bundles on the -variety are the restrictions of the tautological sub- and quotient bundle from .

5.2. Orbibundles on Calabi-Yau Sections

Recall that weighted flag varieties are constructed by first considering the -covering and then dividing by a different -action given by the weights. A tautological vector bundle on pulls back to a vector bundle on . This can then be pushed forward to a weighted flag variety along the quotient map . Because of the finite stabilizers that exist under this second action, the resulting object is not a vector bundle, but an orbibundle [9, Section  4.2], which trivializes on local orbifold covers with compatible transition maps. If is a Calabi-Yau threefold inside , then we can define an orbi-bundle on by restricting to .

In the constructions of Sections 3 and 4, the Calabi-Yau sections therefore carry possibly interesting orbibundles of ranks 2 and 5, respectively 4. We have not investigated the question whether these orbibundles can be pulled back to vector bundles on a resolution , but this seems to be of some interest. If so, stability properties of the resulting vector bundles may deserve some investigation, in view of their possible use in heterotic model building [10, 11].


The first author has been supported by a grant from the Higher Education Commission (HEC) of Pakistan.