Calabi-Yau threefolds in weighted flag varieties

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 orbi-bundles on these varieties, which may be of interest in heterotic model building.


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). Weighted flag varieties wΣ, which are the analogues of weighted projective space in this more general context, were defined by Grojnowski and Corti-Reid [4]. They admit a Plücker-style embedding wΣ ⊂ P[w 0 , · · · , w n ] 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 [4,10,11]: X ⊂ wΣ ⊂ P[w 0 , · · · , w n ].
To be more precise, our examples are going to be quasi-linear 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 [10].
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 know 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 [3], 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 quasi-linear sections X and finally their resolutions Y . See for example [1] 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 quasi-linear section X carries natural orbi-bundles; 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.

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

The main definition
We start by recalling the notion of weighted flag variety due to Grojnowski and Corti-Reid [4]. Fix a reductive Lie group G and a highest weight λ ∈ Λ W , where Λ W is the weight lattice or lattice of characters 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 (generalized) flag variety.
Let Λ * W denote the lattice of one parameter subgroups, dual to the weight lattice Λ W . Choose µ ∈ Λ * W and an integer u ∈ Z such that 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 Σ ⊂ V λ of the embedding Σ ֒→ PV λ . There is a C * -action on V λ \ {0} given by which induces an action on Σ. The inequality (2.1) ensures that all the C * -weights on V λ are positive, leading to a well-defined quotient a weighted projective space, and inside it the projective quotient We call wΣ a weighted flag variety. By definition, wΣ quasismooth, i.e. 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.

The Hilbert series of a weighted flag variety
Here ρ is the Weyl vector, half the sum of the positive roots of G, and (−1) w = 1 or − 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 (2.2) can be converted into a form Here ∇(V λ ) denotes the set of weights (understood with multiplicities) appearing in the weight space decomposition of the representation V λ ; thus the set of weights w i =< α i , µ > +u in the denominator agrees with the set of weights of the weighted projective space wPV λ . The numerator is a polynomial N(t), the Hilbert numerator. Since (2.2) 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 i. H i (wΣ, O wΣ (m)) = 0 for all m and 0 < 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 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.

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 = S 2 V * λ of the second symmetric power of the contragradient representation V * λ . The G-representation Z has a decomposition into irreducible direct summands, with ν being the highest weight of the representation V * λ . As discussed in [8, 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 [10].

Constructing Calabi-Yau threefolds
We recall the different steps in the construction of Calabi-Yau threefolds as quasi-linear 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 ) 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 quasi-linear complete intersection 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=1 w i j = 0, thus K X ∼ O X . After re-labelling the weights, this gives an embedding X ֒→ P s [w 0 , · · · , w s ], with More generally, as in [4], 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 quasi-smooth 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 i 0 , · · · , w ip with gcd(w i 0 , · · · , w ip ) = r non-trivial. If the intersection X ∩ S is non-empty, 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 neighbourhood of points of P respectively 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 [3, Section 3] determines the Hilbert series of a polarized Calabi-Yau threefold (X, D) with quotient singularities in terms of the projective invariants D 3 and D.c 2 (X), as well as for each curve, the degree deg D| 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 h 0 (nD), and verify that the same Hilbert series can be recovered.

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 G 2 and A 5 , leading to Calabi-Yau families of codimension 8, respectively 6. Further examples for the Lie groups of type C 3 and A 3 , in codimensions 7 and 9, will be discussed in the forthcoming [11].
3 The codimension eight weighted flag variety 3

.1 Generalities
Consider the simple Lie group of type G 2 . Denote by α 1 , α 2 ∈ Λ W a pair of simple roots of the root system ∇ of G 2 , taking α 1 to be the short simple root and α 2 the long one. The fundamental weights are ω 1 = 2α 1 + α 2 and ω 2 = 3α 1 + 2α 2 . The sum of the fundamental weights, which is equal to half the sum of the positive roots, is ρ = 5α 1 +3α 2 . We partition the set of roots into long and short roots as ∇ = ∇ l ∪ ∇ s ⊂ Λ W . Let {β 1 , β 2 } be the basis of the lattice Λ * W dual to {α 1 , α 2 }. We consider the G 2 -representation with highest weight λ = ω 2 = 3α 1 + 2α 2 . The dimension of V λ is 14 [7,Chapter 22]. The homogeneous variety Σ ⊂ PV λ is five dimensional, so we have an embedding Σ 5 ֒→ P 13 of codimension 8. To work out the weighted version in this case, take µ = aβ 1 + bβ 2 ∈ Λ * W and u ∈ Z.
Proposition 3.2 The Hilbert series of the codimension eight weighted G 2 flag variety is given by .
The Hilbert series of the straight flag variety Σ ֒→ P 13 can be computed to be The image is defined by 28 quadratic equations, listed in the Appendix of [10].

Examples
Example 3.4 Consider the following initial data.
• Canonical divisor: , as wΣ is well-formed. 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 [10,Appendix].
Consider the threefold quasilinear section where the intersection is taken with general forms f 4 , g 5 of degrees four and five respectively. The canonical divisor class of X is 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. In this case, it is easy to see by hand (or certainly using Macaulay) that S ⊂ P 3 is in fact a twisted cubic curve isomorphic to P 1 . We then need to intersect this with the general X; the quintic equation will not give anything new, since x 2 , x 7 , x 10 , x 12 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).
• 1/3 singularities: the general X does not intersect this singular stratum, the equations from [10,Appendix] in the degree three variables give the empty locus; this is easiest to check by Macaulay. , γ C = 1.
Example 3.5 In this example, we consider the same initial data as in Example 3.4. To construct a new family of Calabi-Yau threefolds, we take a projective cone over wΣ. Therefore we get the embedding The canonical divisor class of CwΣ is K CwΣ ∼ O CwΣ (−10). Consider the threefold quasilinear section with K X ∼ O X ; brackets (w i ) denote a general hypersurface of degree w i .
• 1/4 singularities: since there is no quartic equation this time, the whole twisted cubic curve C ⊂ P 3 [x 2 , x 7 , x 10 , x 12 ], found above, is contained in the general X, and is a rational curve of singularities of type 1 4 (1, 3).
• 1/3 singularities: the general X does not intersect this singular stratum.
• 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 Example 3.6 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 wellformed variety by taking projective cones over it. We finally take a quasilinear section to construct a Calabi-Yau threefold (X, D).
We take a double projective cone over wΣ, by introducing two new variables x 15 and x 16 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 Consider the threefold quasilinear section 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 P 15 [1 2 , 2 14 ], the singular locus defined by weight two variables defines a curve in P 11 [1 2 , 2 10 ].
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.
Example 3.7 Our final initial data in this section consists of the following.
We take a projective cone over wΣ to get the embedding with K CwΣ ∼ O CwΣ (−16). We take a complete intersection inside CwΣ, with three general forms of degree seven, five and four in wP 14 . Therefore we get a threefold X = CwΣ ∩ (7) ∩ (5) ∩ (4) ֒→ P 11 [1, 3, 4 3 , 5 3 , 6 4 ], 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 4 The codimension 6 weighted Grassmannian variety

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 e 1 , . . . , e 6 . The Weyl vector can be taken to be ρ = 5e 1 + 4e 2 + 3e 3 + 2e 4 + e 5 .
Consider the irreducible G-representation V λ , with λ = e 1 +e 2 . 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 = P 14 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 Σ 8 = Gr(2, 6) ֒→ PV λ = P 14 .
Let {f i , 1 ≤ i ≤ 6} be the dual basis of the dual lattice Λ * W . We choose and u ∈ Z, to get the weighted version of Gr (2,6), 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, i.e. µ = (a 1 , a 2 , · · · , a 6 ). We expand the formula (2.2) for the given values of λ, µ to get the following formula for the Hilbert series of w Gr (2,6). . Here In particular, if w Gr(2, 6) ֒→ P 14 [< w i , µ > +u] is well-formed, then its canonical bundle The defining equations for Gr(2, 6) ⊂ P 14 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 (4.1)

Examples
Example 4.3 Consider the following data.
Then K X is trivial, and X is a Calabi-Yau 3-fold with a singular point of type Example 4. 4 We take the following.
Consider the quasilinear section 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 5 Tautological (orbi)bundles

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: In other words, the total space of E consists of pairs (g, e) ∈ G×V modulo the equivalence (gp, e) ∼ (g, pe) for p ∈ P.
The fiber of E over each point Σ is isomorphic to the vector space underlying V .
Example 5.2 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 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 P n−1 (1).

Example 5.3
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 with B 1 , B 2 of size k × k and (n − k) × (n − k) respectively. The representations of P defined by A → B 1 , respectively A → B 2 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.
Example 5.4 Finally consider the G 2 -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 G 2 flag variety into the Grassmannian of 2-planes in a 7-dimensional vector space, the space Im O of imaginary octonions. Then the tautological bundles on the G 2 -variety Σ are the restrictions of the tautological sub-and quotient-bundle from Gr(2, 7).

Orbi-bundles 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 can 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 [2, 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 3-4, the Calabi-Yau sections therefore carry possibly interesting orbi-bundles of ranks 2 and 5, respectively 4. We have not investigated the question whether these orbi-bundles 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 [9,5].