# Calabi-Yau Threefolds in Weighted Flag Varieties

Authors: Muhammad Imran Qureshi; Balázs Szendrői
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

The classical flag varietiesΣ=G/P are projective varieties which are homogeneous spaces under complex reductive Lie groups G; the stabilizer P of a point in Σ is a parabolic subgroup P of G. The simplest example is projective space ℙn itself, which is a homogeneous space under the complex Lie group GL(n+1). Weighted flag varieties wΣ, 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the analogues of weighted projective space in this more general context, were defined by Corti and Reid [1] following unpublished work of Grojnowski.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 (1.1)wΣ⊂P[w0,…,wn]
into a weighted projective space. In this paper, we review the construction of Calabi-Yau threefolds X that arise as complete intersections within wΣ of some hypersurfaces of weighted projective space [1–3]: (1.2)X⊂wΣ⊂P[w0,…,wn].
To be more precise, our examples are going to be quasilinear sections in wΣ, where the degree of each equation agrees with one of the wi. The varieties X will have standard threefold singularities similar to complete intersections in weighted projective spaces; they have crepant desingularizations Y→X by standard theory.We start by computing the Hilbert series of a weighted flag varietywΣ 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 wΣ in the weighted projective space, which are the same as the equations of the straight flag variety Σ in its natural embedding, can be computed relatively easily using computer algebra [2].The smooth Calabi-Yau modelsY 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 Y. 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 wΣ then deduce the invariants of their quasilinear sections X and finally their resolutions Y. 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 varietywΣ and thus its quasilinear section X carry natural orbibundles; these are the analogues of 𝒪(1) on (weighted) projective space. It is possible that these can be used to construct interesting bundles on the resolution Y 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 G and a highest weight λ∈ΛW in the weight lattice of G, the lattice of characters of the maximal torus T of G. Then we have a corresponding parabolic subgroup Pλ, well defined up to conjugation. The quotient Σ=G/Pλ is a homogeneous variety called a (generalized) flag variety, a projective subvariety of ℙVλ, where Vλ is the irreducible representation of G with highest weight λ.LetΛW* denote the lattice of one-parameter subgroups of T, dual to the weight lattice ΛW. Choose μ∈ΛW* and an integer u∈ℤ such that (2.1)〈wλ,μ〉+u>0
for all elements w of the Weyl group of the Lie group G, where 〈,〉 denotes the perfect pairing between ΛW and ΛW*.Consider the affine cone∑̃⊂Vλ of the embedding Σ↪ℙVλ. There is a ℂ*-action on Vλ∖{0} given by (2.2)(ε∈C*)⟼(v⟼εu(μ(ε)∘v))
which induces an action on Σ̃. Inequality (2.1) ensures that all the ℂ*-weights on Vλ are positive, leading to a well-defined quotient (2.3)wPVλ=Vλ∖{0}/C*.
This weighted projective space has weights (2.4){〈α,μ〉+u∣α∈∇(Vλ)},
where ∇(Vλ) denotes the set of weights (understood with multiplicities) appearing in the weight space decomposition of the representation Vλ. Inside this weighted projective space, we consider the projective quotient (2.5)wΣ=Σ̃∖{0}/C*⊂wPVλ.
We call wΣ a weighted flag variety. By definition, wΣ quasismooth, that is, its affine cone Σ̃ is nonsingular outside its vertex 0̲. Hence it only has finite quotient singularities.The weighted flag varietywΣ is called well formed [6], if no (n-1) of weights wi have a common factor, and moreover wΣ does not contain any codimension c+1 singular stratum of wℙVλ, where c is the codimension of wΣ.


### 2.2. The Hilbert Series of a Weighted Flag Variety

Consider the embeddingwΣ⊂wℙVλ. The restriction of the line (orbi)bundle of degree one Weil divisors 𝒪wℙVλ(1) gives a polarization 𝒪wΣ(1) on wΣ, a ℚ-ample line orbibundle some tensor power of which is a very ample line bundle. Powers of 𝒪wΣ(1) have well-defined spaces of sections H0(wΣ,𝒪wΣ(m)). The Hilbert series of the pair (wΣ,𝒪wΣ(1)) is the power series given by (2.6)PwΣ(t)=∑m≥0dim⁡H0(wΣ,OwΣ(m))tm.Theorem 2.1 (see [2, Theorem  3.1]).
The Hilbert seriesPwΣ(t) has the closed form
(2.7)PwΣ(t)=∑w∈W(-1)w(t〈wρ,μ〉/(1-t〈wλ,μ〉+u))∑w∈W(-1)wt〈wρ,μ〉.
Here ρ is the Weyl vector, half the sum of the positive roots of G, and (-1)w=1or-1 depending on whether w consists of an even or odd number of simple reflections in the Weyl group W.The right hand side of (2.7) can be converted into a form (2.8)PwΣ(t)=N(t)∏α∈∇(Vλ)(1-t〈α,μ〉+u),
where, as before, ∇(Vλ) denotes the set of weights of the representation Vλ. The numerator is a polynomial N(t), 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 isprojectively Gorenstein, which means that (i)
Hi(wΣ,𝒪wΣ(m))=0 for all mand0<i<dim(wΣ);(ii)
the Hilbert numeratorN(t) is a palindromic symmetric polynomial of degree q, called the adjunction number of wΣ;(iii)
the canonical divisor ofwΣ is given by
(2.9)KwΣ∼OwΣ(q-∑wi),
where, as above, the wi are the weights of the projective space wℙVλ; the integer k=q-∑wi is called the canonical weight.


### 2.3. Equations of Flag Varieties

The flag varietyΣ=G/P↪ℙVλ is defined by an ideal I=〈Q〉 of quadratic equations generating a linear subspace Q⊂Z=S2Vλ* of the second symmetric power of the contragradient representation Vλ*. The G-representation Z has a decomposition (2.10)Z=V2ν⊕V1⊕⋯⊕Vn
into irreducible direct summands, with ν being the highest weight of the representation Vλ*. As discussed in [7, 2.1], the subspace Q in fact consists of all the summands except V2ν. The equations of wΣ 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 groupG and a G-representation Vλ of dimension n with highest weight λ. We get a straight flag variety Σ=G/Pλ↪ℙVλ of computable dimension d and codimension c=n-1-d. We choose μ∈ΛW* and u∈ℤ to get an embedding wΣ↪wℙVλ=ℙn-1[〈αi,μ〉+u], with αi∈∇(Vλ) being the weights of the representation Vλ. The equations, the Hilbert series, and the canonical class of wΣ⊂wℙ can be found as described above.(2) Take Threefold Calabi-Yau Section ofwΣ
We take a quasilinear complete intersection(2.11)X=wΣ∩(wi1)∩⋯∩(wil)
of l generic hypersurfaces of degrees equal to some of the weights wi. We choose values so that dim(X)=d-l=3 and k+∑j=1lwij=0, thus KX~𝒪X. After relabelling the weights, this gives an embedding X↪ℙs[w0,…,ws], with s=n-l-1, of codimension c, polarized by the ample ℚ-Cartier divisor D with 𝒪X(D)=𝒪wΣ(1)∣X. More generally, as in [1], we can take complete intersections inside projective cones over wΣ, 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 ofwΣ all of whose singularities are induced by the weights of ℙs[wi]. Singular strata S of ℙs[wi] correspond to sets of weights wi0,…,wip with
(2.12)gcd(wi0,…,wip)=r
nontrivial. If the intersection X∩S is nonempty, it has to be a singular point P∈X or a curve C⊂X of quotient singularities, and we need to find local coordinates in a neighbourhood of the point of P, respectively of points of C, to check the local transversal structure. Since we are interested in Calabi-Yau varieties which admit crepant resolutions, singular points P have to be quotient singularities of the form (1/r)(a,b,c) with a+b+c divisible by r, whereas the transversal singularity along a singular curve C has to be of the form (1/r)(a,r-a) of type Ar-1.(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 (X,D) with quotient singularities in terms of the projective invariants D3 and D·c2(X), as well as for each curve, the degree degD∣C of the polarization, and an extra invariant γC related to the normal bundle of C in X. Using the Riemann-Roch formula, we can determine the invariants of a given family from the first few values of h0(nD) 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 typeG2 and A5, leading to Calabi-Yau families of codimension 8, 6, respectively. Further examples for the Lie groups of type C3 and A3, in codimensions 7 and 9, are discussed in [3].


## 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 G and a highest weight λ∈ΛW in the weight lattice of G, the lattice of characters of the maximal torus T of G. Then we have a corresponding parabolic subgroup Pλ, well defined up to conjugation. The quotient Σ=G/Pλ is a homogeneous variety called a (generalized) flag variety, a projective subvariety of ℙVλ, where Vλ is the irreducible representation of G with highest weight λ.LetΛW* denote the lattice of one-parameter subgroups of T, dual to the weight lattice ΛW. Choose μ∈ΛW* and an integer u∈ℤ such that (2.1)〈wλ,μ〉+u>0
for all elements w of the Weyl group of the Lie group G, where 〈,〉 denotes the perfect pairing between ΛW and ΛW*.Consider the affine cone∑̃⊂Vλ of the embedding Σ↪ℙVλ. There is a ℂ*-action on Vλ∖{0} given by (2.2)(ε∈C*)⟼(v⟼εu(μ(ε)∘v))
which induces an action on Σ̃. Inequality (2.1) ensures that all the ℂ*-weights on Vλ are positive, leading to a well-defined quotient (2.3)wPVλ=Vλ∖{0}/C*.
This weighted projective space has weights (2.4){〈α,μ〉+u∣α∈∇(Vλ)},
where ∇(Vλ) denotes the set of weights (understood with multiplicities) appearing in the weight space decomposition of the representation Vλ. Inside this weighted projective space, we consider the projective quotient (2.5)wΣ=Σ̃∖{0}/C*⊂wPVλ.
We call wΣ a weighted flag variety. By definition, wΣ quasismooth, that is, its affine cone Σ̃ is nonsingular outside its vertex 0̲. Hence it only has finite quotient singularities.The weighted flag varietywΣ is called well formed [6], if no (n-1) of weights wi have a common factor, and moreover wΣ does not contain any codimension c+1 singular stratum of wℙVλ, where c is the codimension of wΣ.


## 2.2. The Hilbert Series of a Weighted Flag Variety

Consider the embeddingwΣ⊂wℙVλ. The restriction of the line (orbi)bundle of degree one Weil divisors 𝒪wℙVλ(1) gives a polarization 𝒪wΣ(1) on wΣ, a ℚ-ample line orbibundle some tensor power of which is a very ample line bundle. Powers of 𝒪wΣ(1) have well-defined spaces of sections H0(wΣ,𝒪wΣ(m)). The Hilbert series of the pair (wΣ,𝒪wΣ(1)) is the power series given by (2.6)PwΣ(t)=∑m≥0dim⁡H0(wΣ,OwΣ(m))tm.Theorem 2.1 (see [2, Theorem  3.1]).
The Hilbert seriesPwΣ(t) has the closed form
(2.7)PwΣ(t)=∑w∈W(-1)w(t〈wρ,μ〉/(1-t〈wλ,μ〉+u))∑w∈W(-1)wt〈wρ,μ〉.
Here ρ is the Weyl vector, half the sum of the positive roots of G, and (-1)w=1or-1 depending on whether w consists of an even or odd number of simple reflections in the Weyl group W.The right hand side of (2.7) can be converted into a form (2.8)PwΣ(t)=N(t)∏α∈∇(Vλ)(1-t〈α,μ〉+u),
where, as before, ∇(Vλ) denotes the set of weights of the representation Vλ. The numerator is a polynomial N(t), 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 isprojectively Gorenstein, which means that (i)
Hi(wΣ,𝒪wΣ(m))=0 for all mand0<i<dim(wΣ);(ii)
the Hilbert numeratorN(t) is a palindromic symmetric polynomial of degree q, called the adjunction number of wΣ;(iii)
the canonical divisor ofwΣ is given by
(2.9)KwΣ∼OwΣ(q-∑wi),
where, as above, the wi are the weights of the projective space wℙVλ; the integer k=q-∑wi is called the canonical weight.


## 2.3. Equations of Flag Varieties

The flag varietyΣ=G/P↪ℙVλ is defined by an ideal I=〈Q〉 of quadratic equations generating a linear subspace Q⊂Z=S2Vλ* of the second symmetric power of the contragradient representation Vλ*. The G-representation Z has a decomposition (2.10)Z=V2ν⊕V1⊕⋯⊕Vn
into irreducible direct summands, with ν being the highest weight of the representation Vλ*. As discussed in [7, 2.1], the subspace Q in fact consists of all the summands except V2ν. The equations of wΣ 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 groupG and a G-representation Vλ of dimension n with highest weight λ. We get a straight flag variety Σ=G/Pλ↪ℙVλ of computable dimension d and codimension c=n-1-d. We choose μ∈ΛW* and u∈ℤ to get an embedding wΣ↪wℙVλ=ℙn-1[〈αi,μ〉+u], with αi∈∇(Vλ) being the weights of the representation Vλ. The equations, the Hilbert series, and the canonical class of wΣ⊂wℙ can be found as described above.(2) Take Threefold Calabi-Yau Section ofwΣ
We take a quasilinear complete intersection(2.11)X=wΣ∩(wi1)∩⋯∩(wil)
of l generic hypersurfaces of degrees equal to some of the weights wi. We choose values so that dim(X)=d-l=3 and k+∑j=1lwij=0, thus KX~𝒪X. After relabelling the weights, this gives an embedding X↪ℙs[w0,…,ws], with s=n-l-1, of codimension c, polarized by the ample ℚ-Cartier divisor D with 𝒪X(D)=𝒪wΣ(1)∣X. More generally, as in [1], we can take complete intersections inside projective cones over wΣ, 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 ofwΣ all of whose singularities are induced by the weights of ℙs[wi]. Singular strata S of ℙs[wi] correspond to sets of weights wi0,…,wip with
(2.12)gcd(wi0,…,wip)=r
nontrivial. If the intersection X∩S is nonempty, it has to be a singular point P∈X or a curve C⊂X of quotient singularities, and we need to find local coordinates in a neighbourhood of the point of P, respectively of points of C, to check the local transversal structure. Since we are interested in Calabi-Yau varieties which admit crepant resolutions, singular points P have to be quotient singularities of the form (1/r)(a,b,c) with a+b+c divisible by r, whereas the transversal singularity along a singular curve C has to be of the form (1/r)(a,r-a) of type Ar-1.(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 (X,D) with quotient singularities in terms of the projective invariants D3 and D·c2(X), as well as for each curve, the degree degD∣C of the polarization, and an extra invariant γC related to the normal bundle of C in X. Using the Riemann-Roch formula, we can determine the invariants of a given family from the first few values of h0(nD) 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 typeG2 and A5, leading to Calabi-Yau families of codimension 8, 6, respectively. Further examples for the Lie groups of type C3 and A3, 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 typeG2. Denote by α1,α2∈ΛW a pair of simple roots of the root system ∇ of G2, taking α1 to be the short simple root and α2 the long one. The fundamental weights are ω1=2α1+α2 and ω2=3α1+2α2. The sum of the fundamental weights, which is equal to half the sum of the positive roots, is ρ=5α1+3α2. We partition the set of roots into long and short roots as ∇=∇l∪∇s⊂ΛW. Let {β1,β2} be the basis of the lattice ΛW* dual to {α1,α2}.We consider theG2-representation with highest weight λ=ω2=3α1+2α2. The dimension of Vλ is 14 [8, Chapter 22]. The homogeneous variety Σ⊂ℙVλ is five-dimensional, so we have an embedding Σ5↪ℙ13 of codimension 8. To work out the weighted version in this case, take μ=aβ1+bβ2∈ΛW* and u∈ℤ.Proposition 3.1.
The Hilbert series of the codimension eight weightedG2 flag variety is given by
(3.1)PwΣ(t)=1-(4+2∑α∈∇st〈α,μ〉+∑α∈∇st2〈α,μ〉+∑α∈∇lt〈α,μ〉)t2u+⋯+t11u(1-tu)2∏α∈∇(1-t〈α,μ〉+u).
Moreover, if wΣ is well-formed, then the canonical bundle is KwΣ~𝒪wΣ(-3u).The Hilbert series of the straight flag varietyΣ↪ℙ13 can be computed to be (3.2)PΣ(t)=1-28t2+105t3-⋯+105t8-28t9-t11(1-t)14.
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:μ=(-1,1),u=3.(ii)
Plücker embedding:wΣ⊂ℙ13[1,24,34,44,5].(iii)
Hilbert numerator:1-3t4-6t5-8t6+6t7+21t8+⋯+6t26-8t27-6t28-3t29+t33.(iv)
Canonical divisor:KwΣ~𝒪wΣ(33-∑iwi)=𝒪(-9), as wΣ is well formed.(v)
Variables on weighted projective space together with their weightsxi:
(3.3)Variablesx1x2x3x4x5x6x7x8x9x10x11x12x13x14Weights24321242345433
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
(3.4)X=wΣ∩{f4(xi)=0}∩{g5(xi)=0}⊂P11[1,24,34,43],
where the intersection is taken with general forms f4,g5 of degrees four and five, respectively. The canonical divisor class of X is
(3.5)KX∼OX(-9+(5+4))=OX.
To determine the singularities of the general threefold X, we need to consider sets of variables whose weights have a greatest common divisor greater than one.  (i)
1/4 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
(3.6)S={19x7x10+x2x12=0-13x102+x7x12=013x72+x2x10=0}⊂Px2,x7,x10,x123.
In this case, it is easy to see by hand (or certainly using Macaulay) that S⊂ℙ3 is in fact a twisted cubic curve isomorphic to ℙ1. We then need to intersect this with the general X; the quintic equation will not give anything new, since x2,x7,x10,x12 are degree 4 variables, but the quartic equation will give a linear relation between them. Thus S∩X consists of three points, the three points of 1/4 singularities. A little further work gives that they are all of type (1/4)(3,3,2).(ii)
1/3 singularities: the general X 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)
1/2 singularities: the intersection of X with this singular stratum is a rational curve C⊂X containing the 1/4 singular points; again, Macaulay computes this without difficulty. At each other point of the curve we can check that the transverse singularity is (1/2)(1,1). Thus (X,D) is a Calabi-Yau threefold with three singular points of type (1/4)(3,3,2) and a rational curve C of singularities of type (1/2)(1,1) containing them. Comparing with the orbifold Riemann-Roch formula of [4, Section  3], feeding in the first few known values of h0(X,nD) from the Hilbert series gives that the projective invariants of this family are
(3.7)D3=98,D⋅c2(X)=21,deg⁡D∣C=94,γC=1.Example 3.3.
In this example, we consider the same initial data as in Example3.2. To construct a new family of Calabi-Yau threefolds, we take a projective cone over wΣ. Therefore we get the embedding
(3.8)CwΣ⊂P14[12,24,34,44,5].
The canonical divisor class of 𝒞wΣ is K𝒞wΣ~𝒪𝒞wΣ(-10). Consider the threefold quasilinear section
(3.9)X=CwΣ∩(5)∩(3)∩(2)⊂P11[12,23,33,44,5]
with KX~𝒪X; brackets (wi) denote a general hypersurface of degree wi. (i)
1/4 singularities: since there is no quartic equation this time, the whole twisted cubic curve C⊂ℙ3[x2,x7,x10,x12], found above, is contained in the general X and is a rational curve of singularities of type (1/4)(1,3).(ii)
1/3 singularities: the general X does not intersect this singular stratum.(iii)
1/2 singularities: the intersection of X with this singular strata defines a further rational curve E of singularities. On each point of the curve we check that local transverse parameters have odd weight. Therefore E is a curve of type (1/2)(1,1). Thus (X,D) is a Calabi-Yau threefold with two disjoint rational curves of singularities C and E of type (1/4)(1,3) and (1/2)(1,1), respectively. The rest of the invariants of this family are
(3.10)D3=2716,D⋅c2(X)=21,deg⁡D∣C=34,γC=2,deg⁡D∣E=34,γE=1.Example 3.4.
The next example is obtained by a slight generalization of the method described so far. The computation of the canonical classKwΣ, as the basic line bundle 𝒪wΣ(1) raised to the power equal to the difference of the adjunction number and the sum of the weights on wℙn, only works if wΣ 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 (X,D).(i)
Input:μ=(0,0), u=2.(ii)
Plücker embedding:wΣ⊂ℙ13[214], not well formed.(iii)
Hilbert numerator:1-28t4+105t6-162t8+84t10+84t12-162t14+105t16-28t18+t22. We take a double projective cone over wΣ, by introducing two new variables x15 and x16 of weight one, which are not involved in any of the defining equations of wΣ. We get a seven-dimensional well-formed and quasismooth variety
(3.11)CCwΣ⊂P15[12,214]
with canonical class K𝒞𝒞wΣ~𝒪𝒞𝒞wΣ(-8).
Consider the threefold quasilinear section(3.12)X=CCwΣ∩(2)4⊂P11[12,210].
The canonical class KX becomes trivial. Since wΣ is a five-dimensional variety and we are taking a complete intersection with four generic hypersurfaces of degree two inside ℙ15[12,214], the singular locus defined by weight two variables defines a curve in ℙ11[12,210]. Thus (X,D) is a Calabi-Yau threefold with a curve of singularities of type (1/2)(1,1). The rest of the invariants of (X,D) are given as follows:
(3.13)D3=92,D⋅c2(X)=42,deg⁡D∣C=9,γC=1.Example 3.5.
Our final initial data in this section consists of the following.(i)
Input:μ=(-1,1), u=5.(ii)
Plücker embedding:wΣ⊂ℙ13[3,44,54,64,7].(iii)
Hilbert numerator:1-3t8-6t9-10t10-6t11-t12+12t13+⋯+t55.(iv)
Canonical class:KwΣ~𝒪wΣ(-15), as wΣ is well formed.
We take a projective cone overwΣ to get the embedding
(3.14)CwΣ⊂P14[1,3,44,54,64,7]
with K𝒞wΣ~𝒪𝒞wΣ(-16). We take a complete intersection inside 𝒞wΣ, with three general forms of degree seven, five, and four in wℙ14. Therefore we get a threefold
(3.15)X=CwΣ∩(7)∩(5)∩(4)↪P11[1,3,43,53,64],
with trivial canonical divisor class. To work out the singularities, we work through the singular strata to find that (X,D) is a polarised Calabi-Yau threefold containing three dissident singular points of type (1/4)(1,1,2), a rational curve of singularities C of type (1/6)(1,5) containing them, and a further isolated singular point of type (1/3)(1,1,1). The rest of the invariants are
(3.16)D3=524,D⋅c2(X)=17,deg⁡D∣C=54,γC=9.


## 3.1. Generalities

Consider the simple Lie group of typeG2. Denote by α1,α2∈ΛW a pair of simple roots of the root system ∇ of G2, taking α1 to be the short simple root and α2 the long one. The fundamental weights are ω1=2α1+α2 and ω2=3α1+2α2. The sum of the fundamental weights, which is equal to half the sum of the positive roots, is ρ=5α1+3α2. We partition the set of roots into long and short roots as ∇=∇l∪∇s⊂ΛW. Let {β1,β2} be the basis of the lattice ΛW* dual to {α1,α2}.We consider theG2-representation with highest weight λ=ω2=3α1+2α2. The dimension of Vλ is 14 [8, Chapter 22]. The homogeneous variety Σ⊂ℙVλ is five-dimensional, so we have an embedding Σ5↪ℙ13 of codimension 8. To work out the weighted version in this case, take μ=aβ1+bβ2∈ΛW* and u∈ℤ.Proposition 3.1.
The Hilbert series of the codimension eight weightedG2 flag variety is given by
(3.1)PwΣ(t)=1-(4+2∑α∈∇st〈α,μ〉+∑α∈∇st2〈α,μ〉+∑α∈∇lt〈α,μ〉)t2u+⋯+t11u(1-tu)2∏α∈∇(1-t〈α,μ〉+u).
Moreover, if wΣ is well-formed, then the canonical bundle is KwΣ~𝒪wΣ(-3u).The Hilbert series of the straight flag varietyΣ↪ℙ13 can be computed to be (3.2)PΣ(t)=1-28t2+105t3-⋯+105t8-28t9-t11(1-t)14.
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:μ=(-1,1),u=3.(ii)
Plücker embedding:wΣ⊂ℙ13[1,24,34,44,5].(iii)
Hilbert numerator:1-3t4-6t5-8t6+6t7+21t8+⋯+6t26-8t27-6t28-3t29+t33.(iv)
Canonical divisor:KwΣ~𝒪wΣ(33-∑iwi)=𝒪(-9), as wΣ is well formed.(v)
Variables on weighted projective space together with their weightsxi:
(3.3)Variablesx1x2x3x4x5x6x7x8x9x10x11x12x13x14Weights24321242345433
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
(3.4)X=wΣ∩{f4(xi)=0}∩{g5(xi)=0}⊂P11[1,24,34,43],
where the intersection is taken with general forms f4,g5 of degrees four and five, respectively. The canonical divisor class of X is
(3.5)KX∼OX(-9+(5+4))=OX.
To determine the singularities of the general threefold X, we need to consider sets of variables whose weights have a greatest common divisor greater than one.  (i)
1/4 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
(3.6)S={19x7x10+x2x12=0-13x102+x7x12=013x72+x2x10=0}⊂Px2,x7,x10,x123.
In this case, it is easy to see by hand (or certainly using Macaulay) that S⊂ℙ3 is in fact a twisted cubic curve isomorphic to ℙ1. We then need to intersect this with the general X; the quintic equation will not give anything new, since x2,x7,x10,x12 are degree 4 variables, but the quartic equation will give a linear relation between them. Thus S∩X consists of three points, the three points of 1/4 singularities. A little further work gives that they are all of type (1/4)(3,3,2).(ii)
1/3 singularities: the general X 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)
1/2 singularities: the intersection of X with this singular stratum is a rational curve C⊂X containing the 1/4 singular points; again, Macaulay computes this without difficulty. At each other point of the curve we can check that the transverse singularity is (1/2)(1,1). Thus (X,D) is a Calabi-Yau threefold with three singular points of type (1/4)(3,3,2) and a rational curve C of singularities of type (1/2)(1,1) containing them. Comparing with the orbifold Riemann-Roch formula of [4, Section  3], feeding in the first few known values of h0(X,nD) from the Hilbert series gives that the projective invariants of this family are
(3.7)D3=98,D⋅c2(X)=21,deg⁡D∣C=94,γC=1.Example 3.3.
In this example, we consider the same initial data as in Example3.2. To construct a new family of Calabi-Yau threefolds, we take a projective cone over wΣ. Therefore we get the embedding
(3.8)CwΣ⊂P14[12,24,34,44,5].
The canonical divisor class of 𝒞wΣ is K𝒞wΣ~𝒪𝒞wΣ(-10). Consider the threefold quasilinear section
(3.9)X=CwΣ∩(5)∩(3)∩(2)⊂P11[12,23,33,44,5]
with KX~𝒪X; brackets (wi) denote a general hypersurface of degree wi. (i)
1/4 singularities: since there is no quartic equation this time, the whole twisted cubic curve C⊂ℙ3[x2,x7,x10,x12], found above, is contained in the general X and is a rational curve of singularities of type (1/4)(1,3).(ii)
1/3 singularities: the general X does not intersect this singular stratum.(iii)
1/2 singularities: the intersection of X with this singular strata defines a further rational curve E of singularities. On each point of the curve we check that local transverse parameters have odd weight. Therefore E is a curve of type (1/2)(1,1). Thus (X,D) is a Calabi-Yau threefold with two disjoint rational curves of singularities C and E of type (1/4)(1,3) and (1/2)(1,1), respectively. The rest of the invariants of this family are
(3.10)D3=2716,D⋅c2(X)=21,deg⁡D∣C=34,γC=2,deg⁡D∣E=34,γE=1.Example 3.4.
The next example is obtained by a slight generalization of the method described so far. The computation of the canonical classKwΣ, as the basic line bundle 𝒪wΣ(1) raised to the power equal to the difference of the adjunction number and the sum of the weights on wℙn, only works if wΣ 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 (X,D).(i)
Input:μ=(0,0), u=2.(ii)
Plücker embedding:wΣ⊂ℙ13[214], not well formed.(iii)
Hilbert numerator:1-28t4+105t6-162t8+84t10+84t12-162t14+105t16-28t18+t22. We take a double projective cone over wΣ, by introducing two new variables x15 and x16 of weight one, which are not involved in any of the defining equations of wΣ. We get a seven-dimensional well-formed and quasismooth variety
(3.11)CCwΣ⊂P15[12,214]
with canonical class K𝒞𝒞wΣ~𝒪𝒞𝒞wΣ(-8).
Consider the threefold quasilinear section(3.12)X=CCwΣ∩(2)4⊂P11[12,210].
The canonical class KX becomes trivial. Since wΣ is a five-dimensional variety and we are taking a complete intersection with four generic hypersurfaces of degree two inside ℙ15[12,214], the singular locus defined by weight two variables defines a curve in ℙ11[12,210]. Thus (X,D) is a Calabi-Yau threefold with a curve of singularities of type (1/2)(1,1). The rest of the invariants of (X,D) are given as follows:
(3.13)D3=92,D⋅c2(X)=42,deg⁡D∣C=9,γC=1.Example 3.5.
Our final initial data in this section consists of the following.(i)
Input:μ=(-1,1), u=5.(ii)
Plücker embedding:wΣ⊂ℙ13[3,44,54,64,7].(iii)
Hilbert numerator:1-3t8-6t9-10t10-6t11-t12+12t13+⋯+t55.(iv)
Canonical class:KwΣ~𝒪wΣ(-15), as wΣ is well formed.
We take a projective cone overwΣ to get the embedding
(3.14)CwΣ⊂P14[1,3,44,54,64,7]
with K𝒞wΣ~𝒪𝒞wΣ(-16). We take a complete intersection inside 𝒞wΣ, with three general forms of degree seven, five, and four in wℙ14. Therefore we get a threefold
(3.15)X=CwΣ∩(7)∩(5)∩(4)↪P11[1,3,43,53,64],
with trivial canonical divisor class. To work out the singularities, we work through the singular strata to find that (X,D) is a polarised Calabi-Yau threefold containing three dissident singular points of type (1/4)(1,1,2), a rational curve of singularities C of type (1/6)(1,5) containing them, and a further isolated singular point of type (1/3)(1,1,1). The rest of the invariants are
(3.16)D3=524,D⋅c2(X)=17,deg⁡D∣C=54,γC=9.


## 4. The Codimension 6 Weighted Grassmannian Variety

### 4.1. The Weighted Flag Variety

We takeG to be the reductive Lie group of type GL(6,ℂ). The five simple roots are αi=ei-ei+1∈ΛW, the weight lattice with basis e1,…,e6. The Weyl vector can be taken to be (4.1)ρ=5e1+4e2+3e3+2e4+e5.
Consider the irreducible G-representation Vλ, with λ=e1+e2. Then Vλ is 15-dimensional, and all of the weights appear with multiplicity one. The highest weight orbit space Σ=G/Pλ⊂ℙVλ=ℙ14 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 (4.2)Σ8=Gr(2,6)↪PVλ=P14.Let{fi,1≤i≤6} be the dual basis of the dual lattice ΛW*. We choose (4.3)μ=∑i=16aifi∈ΛW*,u∈ℤ, to get the weighted version of Gr(2,6), (4.4)wΣ(μ,u)=wGr(2,6)(μ,u)↪wP14.
The set of weights on our projective space is {〈λi,μ〉+u}, where λi are weights appearing in the G-representation Vλ. As a convention we will write an element of dual lattice as row vector, that is,  μ=(a1,a2,…,a6).We expand formula (2.7) for the given values of λ,μ to get the following formula for the Hilbert series of wGr(2,6):(4.5)PwGr(2,6)(t)=1-Q1(t)t2u+Q2(t)t3u-Q3(t)t4u-Q4(t)t5u+Q5(t)t6u-Q6(t)t7u+t3s+9u∏1≤i<j≤6(1-tai+aj+u).
Here (4.6)Q1(t)=∑1≤i<j≤6ts-(ai+aj),Q2(t)=∑1≤(i,j)≤6ts+(ai-aj)-ts,Q3(t)=∑1≤i≤j≤6ts+(ai+aj),Q4(t)=∑1≤i≤j≤6t2s-(ai+aj),Q5(t)=∑1≤(i,j)≤6t2s+(ai-aj)-t2s,Q6(t)=∑1≤i≤j≤6t2s+(ai+aj).
In particular, if wGr(2,6)↪ℙ14[〈wi,μ〉+u] is well formed, then its canonical bundle is KwGr(2,6)~𝒪wGr(2,6)(-2s-6u), with s=∑i=16ai.The defining equations forGr(2,6)⊂ℙ14 are well known to be the 4×4 Pfaffians obtained by deleting two rows and the corresponding columns of the 6×6 skew symmetric matrix (4.7)A=[0x1x2x3x4x50x6x7x8x90x10x11x120x13x140x150].


### 4.2. Examples

Example 4.1.
Consider the following data.(i)
Input:μ=(2,1,0,0,-1,-2),u=4.(ii)
Plücker embedding:wGr(2,6)⊂ℙ14[1,22,33,43,53,62,7].(iii)
Hilbert numerator:1-t5-2t6-3t7-2t8-t9+⋯+t36.(iv)
Canonical class:KwGr(2,6)~𝒪wGr(2,6)(-24).
Consider the threefold quasilinear section(4.8)X=wGr(2,6)∩(7)∩(6)∩(5)∩(4)∩(2)⊂P9[1,2,33,42,52,6].
Then KX is trivial, and X is a Calabi-Yau 3-fold with a singular point of type (1/6)(5,4,3), lying on the intersection of two curves, C of type (1/3)(1,2) and E of type (1/2)(1,1). There is an additional isolated singular point of type (1/5)(4,3,3). The rest of the invariants of this variety are
(4.9)D3=1130,D⋅c2(X)=685,deg⁡D∣C=13,γC=-152,deg⁡D∣E=12,γE=1.Example 4.2.
We take the following.(i)
Input:μ=(2,1,1,1,1,0), u=0.(ii)
Plücker embedding:wGr(2,6)⊂ℙ14[14,27,34].(iii)
Hilbert numerator:1-4t3-6t4+4t5+⋯+t18.(iv)
Canonical class:KwGr(2,6)~𝒪wGr(2,6)(-12), as wΣ is well formed.
Consider the quasilinear section(4.10)X=wGr(2,6)∩(3)2∩(2)3⊂P9[14,24,32],
then
(4.11)KX=OX(-12+(2×3+3×2))=OX.
The variety (X,D) is a well-formed and quasismooth Calabi-Yau 3-fold. Its singularities consist of two rational curves C and E of singularities of type (1/3)(1,2) and (1/2)(1,1), respectively. The rest of the invariants are
(4.12)D3=9718,D⋅c2(X)=42,deg⁡D∣C=13,γC=2,deg⁡D∣E=1,γE=1.


## 4.1. The Weighted Flag Variety

We takeG to be the reductive Lie group of type GL(6,ℂ). The five simple roots are αi=ei-ei+1∈ΛW, the weight lattice with basis e1,…,e6. The Weyl vector can be taken to be (4.1)ρ=5e1+4e2+3e3+2e4+e5.
Consider the irreducible G-representation Vλ, with λ=e1+e2. Then Vλ is 15-dimensional, and all of the weights appear with multiplicity one. The highest weight orbit space Σ=G/Pλ⊂ℙVλ=ℙ14 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 (4.2)Σ8=Gr(2,6)↪PVλ=P14.Let{fi,1≤i≤6} be the dual basis of the dual lattice ΛW*. We choose (4.3)μ=∑i=16aifi∈ΛW*,u∈ℤ, to get the weighted version of Gr(2,6), (4.4)wΣ(μ,u)=wGr(2,6)(μ,u)↪wP14.
The set of weights on our projective space is {〈λi,μ〉+u}, where λi are weights appearing in the G-representation Vλ. As a convention we will write an element of dual lattice as row vector, that is,  μ=(a1,a2,…,a6).We expand formula (2.7) for the given values of λ,μ to get the following formula for the Hilbert series of wGr(2,6):(4.5)PwGr(2,6)(t)=1-Q1(t)t2u+Q2(t)t3u-Q3(t)t4u-Q4(t)t5u+Q5(t)t6u-Q6(t)t7u+t3s+9u∏1≤i<j≤6(1-tai+aj+u).
Here (4.6)Q1(t)=∑1≤i<j≤6ts-(ai+aj),Q2(t)=∑1≤(i,j)≤6ts+(ai-aj)-ts,Q3(t)=∑1≤i≤j≤6ts+(ai+aj),Q4(t)=∑1≤i≤j≤6t2s-(ai+aj),Q5(t)=∑1≤(i,j)≤6t2s+(ai-aj)-t2s,Q6(t)=∑1≤i≤j≤6t2s+(ai+aj).
In particular, if wGr(2,6)↪ℙ14[〈wi,μ〉+u] is well formed, then its canonical bundle is KwGr(2,6)~𝒪wGr(2,6)(-2s-6u), with s=∑i=16ai.The defining equations forGr(2,6)⊂ℙ14 are well known to be the 4×4 Pfaffians obtained by deleting two rows and the corresponding columns of the 6×6 skew symmetric matrix (4.7)A=[0x1x2x3x4x50x6x7x8x90x10x11x120x13x140x150].


## 4.2. Examples

Example 4.1.
Consider the following data.(i)
Input:μ=(2,1,0,0,-1,-2),u=4.(ii)
Plücker embedding:wGr(2,6)⊂ℙ14[1,22,33,43,53,62,7].(iii)
Hilbert numerator:1-t5-2t6-3t7-2t8-t9+⋯+t36.(iv)
Canonical class:KwGr(2,6)~𝒪wGr(2,6)(-24).
Consider the threefold quasilinear section(4.8)X=wGr(2,6)∩(7)∩(6)∩(5)∩(4)∩(2)⊂P9[1,2,33,42,52,6].
Then KX is trivial, and X is a Calabi-Yau 3-fold with a singular point of type (1/6)(5,4,3), lying on the intersection of two curves, C of type (1/3)(1,2) and E of type (1/2)(1,1). There is an additional isolated singular point of type (1/5)(4,3,3). The rest of the invariants of this variety are
(4.9)D3=1130,D⋅c2(X)=685,deg⁡D∣C=13,γC=-152,deg⁡D∣E=12,γE=1.Example 4.2.
We take the following.(i)
Input:μ=(2,1,1,1,1,0), u=0.(ii)
Plücker embedding:wGr(2,6)⊂ℙ14[14,27,34].(iii)
Hilbert numerator:1-4t3-6t4+4t5+⋯+t18.(iv)
Canonical class:KwGr(2,6)~𝒪wGr(2,6)(-12), as wΣ is well formed.
Consider the quasilinear section(4.10)X=wGr(2,6)∩(3)2∩(2)3⊂P9[14,24,32],
then
(4.11)KX=OX(-12+(2×3+3×2))=OX.
The variety (X,D) is a well-formed and quasismooth Calabi-Yau 3-fold. Its singularities consist of two rational curves C and E of singularities of type (1/3)(1,2) and (1/2)(1,1), respectively. The rest of the invariants are
(4.12)D3=9718,D⋅c2(X)=42,deg⁡D∣C=13,γC=2,deg⁡D∣E=1,γE=1.


## 5. Tautological (Orbi)bundles

### 5.1. The Classical Story

LetΣ=G/P be a flag variety. A representation V of the parabolic subgroup P gives rise to a vector bundle ℰ on Σ as follows: (5.1)E=G×PV↓Σ=G/P.
In other words, the total space of ℰ consists of pairs (g,e)∈G×V modulo the equivalence (5.2)(gp,e)∼(g,pe),forp∈P.
The fiber of ℰ over each point Σ is isomorphic to the vector space underlying V.Example 5.1.
The simplest example isΣ=ℙn-1, a homogeneous variety G/P with G=GL(n) and P the parabolic subgroup consisting of matrices of the form
(5.3)A=(α*⋯*0⋮B0).
We obtain a one-dimensional representation of P by mapping A to α. The associated line bundle is just the tautological line bundle on ℙn-1, the dual of the hyperplane bundle 𝒪ℙn-1(1).Example 5.2.
More generally, considerΣ=Gr(k,n), the Grassmannian of k-planes in ℂn. Then G=GL(n) and the corresponding parabolic is the subgroup of matrices of the form
(5.4)A=(B1*0B2),
with B1,B2 of size k×k and (n-k)×(n-k), respectively. The representations of P defined by A↦B1, A↦B2, respectively, give the standard tautological sub- and quotient bundles 𝒮 and 𝒬 on the Grassmannian Gr(k,n), fitting into the exact sequence
(5.5)0⟶S⟶OGr(k,n)⊕n⟶Q⟶0.Example 5.3.
Finally consider theG2-variety Σ=G/P studied in Section 3. The smallest representations of the corresponding P have dimensions 2 and 5. The corresponding tautological bundles are easiest to describe using an embedding Σ↪Gr(2,7), mapping the G2 flag variety into the Grassmannian of 2-planes in a 7-dimensional vector space, the space Im𝕆 of imaginary octonions. Then the tautological bundles on the G2-variety Σ are the restrictions of the tautological sub- and quotient bundle from Gr(2,7).


### 5.2. Orbibundles on Calabi-Yau Sections

Recall that weighted flag varieties are constructed by first considering theℂ*-covering Σ̃∖{0}→Σ and then dividing Σ̃∖{0} by a different ℂ*-action given by the weights. A tautological vector bundle ℰ on Σ pulls back to a vector bundle ℰ̃ on Σ̃∖{0}. This can then be pushed forward to a weighted flag variety wΣ along the quotient map Σ̃∖{0}→wΣ. Because of the finite stabilizers that exist under this second action, the resulting object wℰ is not a vector bundle, but an orbibundle [9, Section  4.2], which trivializes on local orbifold covers with compatible transition maps. If X is a Calabi-Yau threefold inside wΣ, then we can define an orbi-bundle on X by restricting wℰ to X.In the constructions of Sections3 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 Y→X, 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].


## 5.1. The Classical Story

LetΣ=G/P be a flag variety. A representation V of the parabolic subgroup P gives rise to a vector bundle ℰ on Σ as follows: (5.1)E=G×PV↓Σ=G/P.
In other words, the total space of ℰ consists of pairs (g,e)∈G×V modulo the equivalence (5.2)(gp,e)∼(g,pe),forp∈P.
The fiber of ℰ over each point Σ is isomorphic to the vector space underlying V.Example 5.1.
The simplest example isΣ=ℙn-1, a homogeneous variety G/P with G=GL(n) and P the parabolic subgroup consisting of matrices of the form
(5.3)A=(α*⋯*0⋮B0).
We obtain a one-dimensional representation of P by mapping A to α. The associated line bundle is just the tautological line bundle on ℙn-1, the dual of the hyperplane bundle 𝒪ℙn-1(1).Example 5.2.
More generally, considerΣ=Gr(k,n), the Grassmannian of k-planes in ℂn. Then G=GL(n) and the corresponding parabolic is the subgroup of matrices of the form
(5.4)A=(B1*0B2),
with B1,B2 of size k×k and (n-k)×(n-k), respectively. The representations of P defined by A↦B1, A↦B2, respectively, give the standard tautological sub- and quotient bundles 𝒮 and 𝒬 on the Grassmannian Gr(k,n), fitting into the exact sequence
(5.5)0⟶S⟶OGr(k,n)⊕n⟶Q⟶0.Example 5.3.
Finally consider theG2-variety Σ=G/P studied in Section 3. The smallest representations of the corresponding P have dimensions 2 and 5. The corresponding tautological bundles are easiest to describe using an embedding Σ↪Gr(2,7), mapping the G2 flag variety into the Grassmannian of 2-planes in a 7-dimensional vector space, the space Im𝕆 of imaginary octonions. Then the tautological bundles on the G2-variety Σ are the restrictions of the tautological sub- and quotient bundle from Gr(2,7).


## 5.2. Orbibundles on Calabi-Yau Sections

Recall that weighted flag varieties are constructed by first considering theℂ*-covering Σ̃∖{0}→Σ and then dividing Σ̃∖{0} by a different ℂ*-action given by the weights. A tautological vector bundle ℰ on Σ pulls back to a vector bundle ℰ̃ on Σ̃∖{0}. This can then be pushed forward to a weighted flag variety wΣ along the quotient map Σ̃∖{0}→wΣ. Because of the finite stabilizers that exist under this second action, the resulting object wℰ is not a vector bundle, but an orbibundle [9, Section  4.2], which trivializes on local orbifold covers with compatible transition maps. If X is a Calabi-Yau threefold inside wΣ, then we can define an orbi-bundle on X by restricting wℰ to X.In the constructions of Sections3 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 Y→X, 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].



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*Source: 547317-2012-01-12.xml*