Calabi-Yau Threefolds in Weighted Flag Varieties

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.

1. Introduction

The classical flag varieties Σ=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 ℙ^𝑛 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the analogues of weighted projective space in this more general context, were defined by Corti and Reid [1] following unpublished work of Grojnowski.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 𝑤Σ⊂ℙ𝑤₀,…,𝑤_𝑛 (1.1) 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]: 𝑋⊂𝑤Σ⊂ℙ𝑤₀,…,𝑤_𝑛. (1.2) 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 𝑤_𝑖. 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 variety 𝑤Σ 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 models 𝑌 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 variety 𝑤Σ 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∈Λ_𝑊 in the weight lattice of 𝐺, the lattice of characters of the maximal torus 𝑇 of 𝐺. Then we have a corresponding parabolic subgroup 𝑃_𝜆, well defined up to conjugation. The quotient Σ=𝐺/𝑃_𝜆 is a homogeneous variety called a (generalized) flag variety, a projective subvariety of ℙ𝑉_𝜆, where 𝑉_𝜆 is the irreducible representation of 𝐺 with highest weight 𝜆.

Let Λ^∗_𝑊 denote the lattice of one-parameter subgroups of 𝑇, dual to the weight lattice Λ_𝑊. Choose 𝜇∈Λ^∗_𝑊 and an integer 𝑢∈ℤ such that 𝑤𝜆,𝜇+𝑢>0 (2.1) for all elements 𝑤 of the Weyl group of the Lie group 𝐺, where ⟨,⟩ denotes the perfect pairing between Λ_𝑊 and Λ^∗_𝑊.

Consider the affine cone ∑⊂𝑉_𝜆 of the embedding Σ↪ℙ𝑉_𝜆. There is a ℂ^∗-action on 𝑉_𝜆⧵{0} given by 𝜀∈ℂ^∗⟼𝑣⟼𝜀^𝑢𝜇(𝜀)∘𝑣 (2.2) which induces an action on Σ. Inequality (2.1) ensures that all the ℂ^∗-weights on 𝑉_𝜆 are positive, leading to a well-defined quotient 𝑤ℙ𝑉_𝜆=𝑉_𝜆⧵{0}/ℂ^∗. (2.3) This weighted projective space has weights 𝛼,𝜇+𝑢∣𝛼∈∇𝑉_𝜆, (2.4) where ∇(𝑉_𝜆) denotes the set of weights (understood with multiplicities) appearing in the weight space decomposition of the representation 𝑉_𝜆. Inside this weighted projective space, we consider the projective quotient 𝑤Σ=Σ⧵{0}/ℂ^∗⊂𝑤ℙ𝑉_𝜆. (2.5) We call 𝑤Σ a weighted flag variety. By definition, 𝑤Σ quasismooth, that is, its affine cone Σ is nonsingular outside its vertex 0. Hence it only has finite quotient singularities.

The weighted flag variety 𝑤Σ is called well formed [6], if no (𝑛−1) of weights 𝑤_𝑖 have a common factor, and moreover 𝑤Σ does not contain any codimension 𝑐+1 singular stratum of 𝑤ℙ𝑉_𝜆, where 𝑐 is the codimension of 𝑤Σ.

2.2. The Hilbert Series of a Weighted Flag Variety

Consider the embedding 𝑤Σ⊂𝑤ℙ𝑉_𝜆. The restriction of the line (orbi)bundle of degree one Weil divisors 𝒪_{𝑤ℙ𝑉𝜆}(1) gives a polarization 𝒪_𝑤Σ(1) on 𝑤Σ, a ℚ-ample line orbibundle some tensor power of which is a very ample line bundle. Powers of 𝒪_𝑤Σ(1) have well-defined spaces of sections 𝐻⁰(𝑤Σ,𝒪_𝑤Σ(𝑚)). The Hilbert series of the pair (𝑤Σ,𝒪_𝑤Σ(1)) is the power series given by 𝑃_𝑤Σ(𝑡)=_𝑚≥0dim𝐻⁰𝑤Σ,𝒪_𝑤Σ(𝑚)𝑡^𝑚. (2.6)

Theorem 2.1 (see [2, Theorem  3.1]). The Hilbert series 𝑃_𝑤Σ(𝑡) has the closed form 𝑃_𝑤Σ(𝑡)=∑_𝑤∈𝑊(−1)^𝑤𝑡^{⟨𝑤𝜌,𝜇⟩}/1−𝑡^{⟨𝑤𝜆,𝜇⟩+𝑢}∑_𝑤∈𝑊(−1)^𝑤𝑡^{⟨𝑤𝜌,𝜇⟩}. (2.7) 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 𝑃_𝑤Σ(𝑡)=𝑁(𝑡)∏_{𝛼∈∇(𝑉𝜆)}1−𝑡^{⟨𝛼,𝜇⟩+𝑢}, (2.8) where, as before, ∇(𝑉_𝜆) denotes the set of weights of the representation 𝑉_𝜆. 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 is projectively Gorenstein, which means that (i)𝐻^𝑖(𝑤Σ,𝒪_𝑤Σ(𝑚))=0 for all 𝑚and0<𝑖<dim(𝑤Σ);(ii)the Hilbert numerator 𝑁(𝑡) is a palindromic symmetric polynomial of degree 𝑞, called the adjunction number of 𝑤Σ; (iii)the canonical divisor of 𝑤Σ is given by 𝐾_𝑤Σ∼𝒪_𝑤Σ𝑞−𝑤_𝑖, (2.9) where, as above, the 𝑤_𝑖 are the weights of the projective space 𝑤ℙ𝑉_𝜆; the integer 𝑘=𝑞−∑𝑤_𝑖 is called the canonical weight.

2.3. Equations of Flag Varieties

The flag variety Σ=𝐺/𝑃↪ℙ𝑉_𝜆 is defined by an ideal 𝐼=⟨𝑄⟩ of quadratic equations generating a linear subspace 𝑄⊂𝑍=𝑆²𝑉^∗_𝜆 of the second symmetric power of the contragradient representation 𝑉^∗_𝜆. The 𝐺-representation 𝑍 has a decomposition 𝑍=𝑉_2𝜈⊕𝑉₁⊕⋯⊕𝑉_𝑛 (2.10) into irreducible direct summands, with 𝜈 being the highest weight of the representation 𝑉^∗_𝜆. As discussed in [7, 2.1], the subspace 𝑄 in fact consists of all the summands except 𝑉_2𝜈. 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 group 𝐺 and a 𝐺-representation 𝑉_𝜆 of dimension 𝑛 with highest weight 𝜆. We get a straight flag variety Σ=𝐺/𝑃_𝜆↪ℙ𝑉_𝜆 of computable dimension 𝑑 and codimension 𝑐=𝑛−1−𝑑. We choose 𝜇∈Λ^∗_𝑊 and 𝑢∈ℤ to get an embedding 𝑤Σ↪𝑤ℙ𝑉_𝜆=ℙ^𝑛−1[⟨𝛼_𝑖,𝜇⟩+𝑢], with 𝛼_𝑖∈∇(𝑉_𝜆) being the weights of the representation 𝑉_𝜆. The equations, the Hilbert series, and the canonical class of 𝑤Σ⊂𝑤ℙ can be found as described above.

(2) Take Threefold Calabi-Yau Section of 𝑤Σ
We take a quasilinear complete intersection 𝑋=𝑤Σ∩𝑤_𝑖1∩⋯∩𝑤_𝑖𝑙 (2.11) of 𝑙 generic hypersurfaces of degrees equal to some of the weights 𝑤_𝑖. We choose values so that dim(𝑋)=𝑑−𝑙=3 and 𝑘+∑^𝑙_𝑗=1𝑤_𝑖𝑗=0, thus 𝐾_𝑋∼𝒪_𝑋. After relabelling the weights, this gives an embedding 𝑋↪ℙ^𝑠[𝑤₀,…,𝑤_𝑠], with 𝑠=𝑛−𝑙−1, of codimension 𝑐, polarized by the ample ℚ-Cartier divisor 𝐷 with 𝒪_𝑋(𝐷)=𝒪_𝑤Σ(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 of 𝑤Σ all of whose singularities are induced by the weights of ℙ^𝑠[𝑤_𝑖]. Singular strata 𝑆 of ℙ^𝑠[𝑤_𝑖] correspond to sets of weights 𝑤_𝑖0,…,𝑤_𝑖𝑝 with gcd𝑤_𝑖0,…,𝑤_𝑖𝑝=𝑟 (2.12) 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 𝐴_𝑟−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 𝛾_𝐶 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 type 𝐺₂ 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 type 𝐺₂. Denote by 𝛼₁,𝛼₂∈Λ_𝑊 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⊂Λ_𝑊. Let {𝛽₁,𝛽₂} be the basis of the lattice Λ^∗_𝑊 dual to {𝛼₁,𝛼₂}.

We consider the 𝐺₂-representation with highest weight 𝜆=𝜔₂=3𝛼₁+2𝛼₂. The dimension of 𝑉_𝜆 is 14 [8, Chapter 22]. The homogeneous variety Σ⊂ℙ𝑉_𝜆 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∈Λ^∗_𝑊 and 𝑢∈ℤ.

Proposition 3.1. The Hilbert series of the codimension eight weighted 𝐺₂ flag variety is given by 𝑃_𝑤Σ(𝑡)=1−4+2∑_{𝛼∈∇𝑠}𝑡^{⟨𝛼,𝜇⟩}+∑_{𝛼∈∇𝑠}𝑡^{2⟨𝛼,𝜇⟩}+∑_{𝛼∈∇𝑙}𝑡^{⟨𝛼,𝜇⟩}𝑡^2𝑢+⋯+𝑡^11𝑢(1−𝑡^𝑢)²∏_𝛼∈∇1−𝑡^{⟨𝛼,𝜇⟩+𝑢}. (3.1) Moreover, if 𝑤Σ is well-formed, then the canonical bundle is 𝐾_𝑤Σ∼𝒪_𝑤Σ(−3𝑢).

The Hilbert series of the straight flag variety Σ↪ℙ¹³ can be computed to be 𝑃_Σ(𝑡)=1−28𝑡²+105𝑡³−⋯+105𝑡⁸−28𝑡⁹−𝑡¹¹(1−𝑡)¹⁴. (3.2) 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: 𝐾_𝑤Σ∼𝒪_𝑤Σ(33−∑_𝑖𝑤_𝑖)=𝒪(−9), as 𝑤Σ is well formed. (v)Variables on weighted projective space together with their weights𝑥_𝑖: Variables𝑥₁𝑥₂𝑥₃𝑥₄𝑥₅𝑥₆𝑥₇𝑥₈𝑥₉𝑥₁₀𝑥₁₁𝑥₁₂𝑥₁₃𝑥₁₄Weights24321242345433 (3.3) 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 𝑋=𝑤Σ∩𝑓₄𝑥_𝑖=0∩𝑔₅𝑥_𝑖=0⊂ℙ¹¹1,2⁴,3⁴,4³, (3.4) where the intersection is taken with general forms 𝑓₄,𝑔₅ of degrees four and five, respectively. The canonical divisor class of 𝑋 is 𝐾_𝑋∼𝒪_𝑋(−9+(5+4))=𝒪_𝑋. (3.5) 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 1𝑆=9𝑥₇𝑥₁₀+𝑥₂𝑥₁₂−1=03𝑥²₁₀+𝑥₇𝑥₁₂1=03𝑥²₇+𝑥₂𝑥₁₀=0⊂ℙ³_{𝑥2,𝑥7,𝑥10,𝑥12}. (3.6) 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 𝐷³=98,𝐷⋅𝑐₂(𝑋)=21,deg𝐷∣_𝐶=94,𝛾_𝐶=1. (3.7)

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 𝑤Σ. Therefore we get the embedding 𝒞𝑤Σ⊂ℙ¹⁴1²,2⁴,3⁴,4⁴,5. (3.8) The canonical divisor class of 𝒞𝑤Σ is 𝐾_𝒞𝑤Σ∼𝒪_𝒞𝑤Σ(−10). Consider the threefold quasilinear section 𝑋=𝒞𝑤Σ∩(5)∩(3)∩(2)⊂ℙ¹¹1²,2³,3³,4⁴,5 (3.9) with 𝐾_𝑋∼𝒪_𝑋; brackets (𝑤_𝑖) denote a general hypersurface of degree 𝑤_𝑖. (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 𝐷³=2716,𝐷⋅𝑐₂(𝑋)=21,deg𝐷∣_𝐶=34,𝛾_𝐶=2,deg𝐷∣_𝐸=34,𝛾_𝐸=1. (3.10)

Example 3.4. The next example is obtained by a slight generalization of the method described so far. The computation of the canonical class 𝐾_𝑤Σ, as the basic line bundle 𝒪_𝑤Σ(1) raised to the power equal to the difference of the adjunction number and the sum of the weights on 𝑤ℙ^𝑛, 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 𝒞𝒞𝑤Σ⊂ℙ¹⁵1²,2¹⁴ (3.11) with canonical class 𝐾_{𝒞𝒞𝑤Σ}∼𝒪_{𝒞𝒞𝑤Σ}(−8).
Consider the threefold quasilinear section 𝑋=𝒞𝒞𝑤Σ∩(2)⁴⊂ℙ¹¹1²,2¹⁰. (3.12) The canonical class 𝐾_𝑋 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: 𝐷³=92,𝐷⋅𝑐₂(𝑋)=42,deg𝐷∣_𝐶=9,𝛾_𝐶=1. (3.13)

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: 𝐾_𝑤Σ∼𝒪_𝑤Σ(−15), as 𝑤Σ is well formed.
We take a projective cone over 𝑤Σ to get the embedding 𝒞𝑤Σ⊂ℙ¹⁴1,3,4⁴,5⁴,6⁴,7 (3.14) with 𝐾_𝒞𝑤Σ∼𝒪_𝒞𝑤Σ(−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 𝑋=𝒞𝑤Σ∩(7)∩(5)∩(4)↪ℙ¹¹1,3,4³,5³,6⁴, (3.15) 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 𝐷³=524,𝐷⋅𝑐₂(𝑋)=17,deg𝐷∣_𝐶=54,𝛾_𝐶=9. (3.16)

4. The Codimension 6 Weighted Grassmannian Variety

4.1. The Weighted Flag Variety

We take 𝐺 to be the reductive Lie group of type GL(6,ℂ). The five simple roots are 𝛼_𝑖=𝑒_𝑖−𝑒_𝑖+1∈Λ_𝑊, the weight lattice with basis 𝑒₁,…,𝑒₆. The Weyl vector can be taken to be 𝜌=5𝑒₁+4𝑒₂+3𝑒₃+2𝑒₄+𝑒₅. (4.1) Consider the irreducible 𝐺-representation 𝑉_𝜆, with 𝜆=𝑒₁+𝑒₂. Then 𝑉_𝜆 is 15-dimensional, and all of the weights appear with multiplicity one. The highest weight orbit space Σ=𝐺/𝑃_𝜆⊂ℙ𝑉_𝜆=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 Σ⁸=Gr(2,6)↪ℙ𝑉_𝜆=ℙ¹⁴. (4.2)

Let {𝑓_𝑖,1≤𝑖≤6} be the dual basis of the dual lattice Λ^∗_𝑊. We choose 𝜇=⁶_𝑖=1𝑎_𝑖𝑓_𝑖∈Λ^∗_𝑊, (4.3)𝑢∈ℤ, to get the weighted version of Gr(2,6), 𝑤Σ𝜇,𝑢=𝑤Gr(2,6)_(𝜇,𝑢)↪𝑤ℙ¹⁴. (4.4) The set of weights on our projective space is {⟨𝜆_𝑖,𝜇⟩+𝑢}, where 𝜆_𝑖 are weights appearing in the 𝐺-representation 𝑉_𝜆. 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)∶𝑃_𝑤Gr(2,6)(𝑡)=1−𝑄₁(𝑡)𝑡^2𝑢+𝑄₂(𝑡)𝑡^3𝑢−𝑄₃(𝑡)𝑡^4𝑢−𝑄₄(𝑡)𝑡^5𝑢+𝑄₅(𝑡)𝑡^6𝑢−𝑄₆(𝑡)𝑡^7𝑢+𝑡^3𝑠+9𝑢∏_{1≤𝑖<𝑗≤6}1−𝑡^{𝑎𝑖+𝑎𝑗+𝑢}. (4.5) Here 𝑄₁(𝑡)=_{1≤𝑖<𝑗≤6}𝑡^{𝑠−(𝑎𝑖+𝑎𝑗)},𝑄₂(𝑡)=_{1≤𝑖,𝑗≤6}𝑡^{𝑠+(𝑎𝑖−𝑎𝑗)}−𝑡^𝑠,𝑄₃(𝑡)=_{1≤𝑖≤𝑗≤6}𝑡^{𝑠+(𝑎𝑖+𝑎𝑗)},𝑄₄(𝑡)=_{1≤𝑖≤𝑗≤6}𝑡^{2𝑠−(𝑎𝑖+𝑎𝑗)},𝑄₅(𝑡)=_{1≤𝑖,𝑗≤6}𝑡^{2𝑠+(𝑎𝑖−𝑎𝑗)}−𝑡^2𝑠,𝑄₆(𝑡)=_{1≤𝑖≤𝑗≤6}𝑡^{2𝑠+(𝑎𝑖+𝑎𝑗)}. (4.6) In particular, if 𝑤Gr(2,6)↪ℙ¹⁴[⟨𝑤_𝑖,𝜇⟩+𝑢] is well formed, then its canonical bundle is 𝐾_𝑤Gr(2,6)∼𝒪_𝑤Gr(2,6)(−2𝑠−6𝑢), with 𝑠=∑⁶_𝑖=1𝑎_𝑖.

The defining equations for Gr(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 𝐴=0𝑥₁𝑥₂𝑥₃𝑥₄𝑥₅0𝑥₆𝑥₇𝑥₈𝑥₉0𝑥₁₀𝑥₁₁𝑥₁₂0𝑥₁₃𝑥₁₄0𝑥₁₅0. (4.7)

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: 𝐾_𝑤Gr(2,6)∼𝒪_𝑤Gr(2,6)(−24).
Consider the threefold quasilinear section 𝑋=𝑤Gr(2,6)∩(7)∩(6)∩(5)∩(4)∩(2)⊂ℙ⁹1,2,3³,4²,5²,6. (4.8) Then 𝐾_𝑋 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 𝐷³=1130,𝐷⋅𝑐₂(𝑋)=685,deg𝐷∣_𝐶=13,𝛾_𝐶=−152,deg𝐷∣_𝐸=12,𝛾_𝐸=1. (4.9)

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: 𝐾_𝑤Gr(2,6)∼𝒪_𝑤Gr(2,6)(−12), as 𝑤Σ is well formed.
Consider the quasilinear section 𝑋=𝑤Gr(2,6)∩(3)²∩(2)³⊂ℙ⁹1⁴,2⁴,3², (4.10) then 𝐾_𝑋=𝒪_𝑋(−12+(2×3+3×2))=𝒪_𝑋. (4.11) 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 𝐷³=9718,𝐷⋅𝑐₂(𝑋)=42,deg𝐷∣_𝐶=13,𝛾_𝐶=2,deg𝐷∣_𝐸=1,𝛾_𝐸=1. (4.12)

5. Tautological (Orbi)bundles

5.1. The Classical Story

Let Σ=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: ℰ=𝐺×_𝑃𝑉↓Σ=𝐺/𝑃. (5.1) In other words, the total space of ℰ consists of pairs (𝑔,𝑒)∈𝐺×𝑉 modulo the equivalence 𝑔𝑝,𝑒∼𝑔,𝑝𝑒,for𝑝∈𝑃. (5.2) The fiber of ℰ over each point Σ is isomorphic to the vector space underlying 𝑉.

Example 5.1. The simplest example is Σ=ℙ^𝑛−1, a homogeneous variety 𝐺/𝑃 with 𝐺=GL(𝑛) and 𝑃 the parabolic subgroup consisting of matrices of the form 00𝐴=𝛼∗⋯∗⋮𝐵. (5.3) We obtain a one-dimensional representation of 𝑃 by mapping 𝐴 to 𝛼. The associated line bundle is just the tautological line bundle on ℙ^𝑛−1, the dual of the hyperplane bundle 𝒪_ℙ𝑛−1(1).

Example 5.2. More generally, consider Σ=Gr(𝑘,𝑛), the Grassmannian of 𝑘-planes in ℂ^𝑛. Then 𝐺=GL(𝑛) and the corresponding parabolic is the subgroup of matrices of the form 𝐵𝐴=¹∗0𝐵₂, (5.4) 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 0⟶𝒮⟶𝒪^⊕𝑛_{Gr()𝑘,𝑛}⟶𝒬⟶0. (5.5)

Example 5.3. Finally consider the 𝐺₂-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 the ℂ^∗-covering Σ⧵{0}→Σ and then dividing Σ⧵{0} by a different ℂ^∗-action given by the weights. A tautological vector bundle ℰ on Σ pulls back to a vector bundle ℰ on Σ⧵{0}. This can then be pushed forward to a weighted flag variety 𝑤Σ along the quotient map Σ⧵{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 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 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].

Acknowledgment

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