The Griffiths bundle is generated by groups
Introduction
First the Griffiths line bundle of a Q -VHS V is generalized to a Griffiths character grif (G , μ , r) associated to any triple (G , μ , r) , where G is a connected reductive group over an arbitrary field F, μ ∈ X ∗ (G) is a cocharacter (over F ¯ ) and r : G → G L (V) is an F-representation; the classical bundle studied by Griffiths is recovered by taking F = Q , G the Mumford–Tate group of V , r : G → G L (V) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to grif (G , μ , r) . The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of G -Zips. When G is F-simple, we show that, up to positive multiples, the Griffiths character grif (G , μ , r) (and thus also the Griffiths line bundle) is essentially independent of r with central kernel, and up to some identifications is given explicitly by - μ . As an application, we show that the Griffiths line bundle of a projective G - Zip μ -scheme is nef.
We are motivated by the general problem of understanding which geometric objects are generated by groups. We understand an object to be generated by groups if it is constructible and/or describable in terms of groups and associated data, such as subgroups, homogeneous spaces, representations, characters, cocharacters etc. In particular, in the case of reductive groups—the focus of this paper—we deem any geometric object which is describable in terms of root data of reductive groups as generated by groups.
As a more precise example of the general problem, we begin this paper by stating some questions about the group-generation of invariants of objects in a neutral Tannakian category. The paper is then concerned with showing that these questions have a particularly simple, explicit and positive answer when the invariant is the Griffiths line bundle of a variation of Hodge structure, or more generally the Griffiths character associated to a connected, reductive group G over an arbitrary field, a cocharacter and a representation r of G.
Tannakian generation
There are many neutral Tannakian categories whose objects have been studied in algebraic geometry independently of Tannakian categories. A key example in this paper will be the category of variations of -Hodge structure over a smooth, projective -scheme S.
Let be a Tannakian category over a field k which is neutralized by a fiber functor . Let G be the Tannaka group of , i.e., the affine k-group scheme which represents the functor of automorphisms (see [[10], 2.11]).
Question 1.1
Given an invariant i(X) associated to every object , is i(X) generated by G and additional group-theoretic data attached to G?
The prototypical type of additional group-theoretic data which we have in mind is a cocharacter . If G is reductive, then more generally any data deduced from a root datum of G would qualify.
A somewhat more local variant of Question 1.1 is the following: For every , let G(X) be the Tannaka group of the Tannakian sub-category generated by X. Then is i(X) generated by G(X) and group-theoretic data associated to G(X)? In this setting, one can even hope for more:
Question 1.2
Assume some invariant i is generated by G(X) and some additional data associated to G. Is i(X) essentially independent of X (and dependent only on G(X) and the additional data)?
A key component of Question 1.2 is of course to make precise the meaning of "essentially" in specific examples. The main result of this paper implies that Question 1.2 has a positive answer when X is a -VHS, i(X) is its Griffiths line bundle (Sect. 1.2) and G(X) is its Mumford–Tate group, provided the adjoint group is -simple, see Theorem 3.1. In this case, the additional group-theoretic data is the Hodge cocharacter and "essentially" means that the positive ray spanned by the Griffiths line bundle in the Picard group of the base is independent of i(X) and dependent only on the pair .
The Griffiths bundle of a variation of Hodge structure
The Griffiths line bundle arose historically in Hodge theory, where it was used by Griffiths to study the algebraicity of the period map of a variation of Hodge structure [[15]]. Suppose S is a connected, smooth, finite-type -scheme and is a (pure) polarized variation of Hodge structure on S with monodromy group and period domain D. Let be the (descending) Hodge filtration on ; for the sake of exposition suppose that . Griffiths (loc. cit., (7.13)) associated to the line bundle
1.2.1
Graph
We call the Griffiths line bundle[1] of . Griffiths also associated to a period map
1.2.2
Graph
By studying the positivity properties of the line bundle , Griffiths concluded that the image of the period map is projective algebraic when is discrete in and the base S is assumed projective (loc. cit., (9.7); see also [[4], 13.1.9]).[2]
Remark 1.3
It is clear from the definition (1.2.1) that to define requires much less than a polarized -VHS.
- The definition of only depends on the associated graded .
- The transversality, polarization and -structure are not used: The same definition applies to any filtered or graded vector bundle on S. (By contrast the former data is crucial for defining the period map (1.2.2).)
Remark 1.4
At least when the monodromy is discrete, it follows by pullback along the period map that the line bundle is trivial on S, see Sect. 3.2. Therefore one can replace the index set in the sums (1.2.1) with or ; the resulting line bundle will be unchanged.
Remark 1.5
For recent work generalizing that of Griffiths to the case that the base S is only assumed quasi-projective, see the preprints [[14]] and [[1]].[3]
Summary of the paper
Let F be an arbitrary field. Consider triples , where is a connected reductive F-group, is a cocharacter of and is a morphism of F-groups. In the vein of Remark 1.3, we explain in Sect. 3.1 how to generalize the Griffiths line bundle to a character of the Levi subgroup of . In Sect. 3.2, we describe how the Griffiths character gives rise to a Griffiths line bundle in two (a priori) different contexts: We recover the bundle associated to a VHS via Deligne's theory of pairs and we obtain a Griffiths line bundle on stacks of -Zips in characteristic .
The main result is stated in Sect. 3.3, see Theorem 3.1. Roughly speaking, it states that is, up to positive multiples and modulo the center, independent of r and given explicitly by . To make this precise requires some technical assumptions and identifications concerning a root datum of . For this purpose, the necessary structure theory associated to triples is given in Sect. 2. The sign change between and reflects the change in positivity/curvature between a Mumford–Tate domain and its compact dual (Remark 3.3).
By combining our result with our forthcoming joint work with Brunebarbe et al. [[3]] on the positivity of automorphic bundles, we obtain the following application: Assume X is a projective scheme in positive characteristic and is a morphism to the stack of -Zips associated to by Pink-Wedhorn-Ziegler [[20]]. As long as p is not too small relative (orbitally p-close to be precise, Sect. 2.4.4), then the Griffiths line bundle of X is nef (Corollary 3.7).
In Sect. 3.4, we give two examples of the main result: The first concerns the Hodge character and the Hodge line bundle. In the classical theory this amounts to the case that the VHS is polarized of weight one. Here we recover the results of our joint work with Koskivirta [[12]]. The second example provides explicit formulas for when is the adjoint representation, essentially in terms of the Coxeter number of the underlying root system.
The proof of Theorem 3.1 is given in Sect. 4. Some preliminary, general lemmas on roots and weights are given in Sect. 4.1; the proof proper occupies Sect. 4.2. The key is to translate the main result into a statement about weight pairings with coroots (Lemma 4.5). The simply-laced case of Theorem 3.1 then results from a simple change of variables in the root pairing expression (Lemma 4.7). The general case is reduced to the simply-laced one by the theory of root strings (Lemma 4.8). The application to nefness (Corollary 3.7) is proved in Sect. 4.3.
When we sent P. Deligne a draft of this paper he quickly replied with a considerable simplification of the proof of the main result Theorem 3.1(d). We are very grateful to Deligne for allowing us to include his simplification in Appendix A.
Notation and structure theory
Notation
Let F be a field and fix an algebraic closure of F. A subscript etc. will always denote base change to respectively. Thus denotes the multiplicative group scheme over , and if N is a -module, then is the associated -vector space.
If H is an algebraic F-group, then (resp. ) denotes the group of characters (resp. cocharacters) of . Write for the -conjugacy class of a cocharacter .
Similarly, denotes the category of representations of H over ; write for the representations over F.
The unipotent radical of H is denoted .
Root data
We follow [[12], Sect. 2.1], which in turn is based on [[16], Part II, Sect. 1], esp. 1.18.
Root datum of G
Let be a maximal torus[4] of . The root datum of is the quadruple
2.2.1
Graph
together with the -valued perfect pairing
2.2.2
Graph
and the bijection , , where (resp. ) is the set of roots (resp. coroots) of in .
Weyl group
For every , let be the corresponding root reflection. Let be the Weyl group of in . Recall that provides a canonical identification of W with the dual Weyl group of the dual root datum generated by the .
Weights
If V is an F-vector space and is a morphism of F-groups, write for the set of -weights in (not counting multiplicities). Given a weight , let denote its multiplicity (the dimension of the corresponding weight space).
Based root datum of G
Let be a basis of simple roots. Then is the corresponding basis of simple coroots and
2.2.3
Graph
is the based root datum of .
Derived subgroup, adjoint quotient and simply-connected covering
Let (resp. , ) be the derived subgroup of (resp. its adjoint quotient, the simply-connected covering of in the sense of root data[5]). Let be the natural quasi-section of the projection . The root datum (2.2.1) and the based root datum (2.2.3) naturally induce ones of , and as follows:
Root datum of Gder
Let
2.3.1
Graph
and
2.3.2
Graph
Then is a maximal torus in with character group and
2.3.3
Graph
is the root datum of , where the roots are restricted to . Similarly, by restriction we identify with a basis of simple roots for .
Root data of Gad and G~
Let (resp. ) denote the preimage of in (resp. the image of in ). Then and are maximal tori in and respectively; the roots (resp. simple roots, coroots, simple coroots) of the three pairs are identified via the central isogenies
2.3.4
Graph
Associated root system
The central isogenies (2.3.4) induce canonical identifications
2.3.5
Graph
In turn, the Weyl groups of the three pairs (2.3.4) are all canonically identified with W. Choose a positive definite, symmetric, -invariant bilinear form
2.3.6
Graph
When is F-simple, the form (,) is the unique one up to positive scalar satisfying the properties above (one reduces to the well-known fact that there is a unique W-invariant, nondegenerate, symmetric form up to scaling when is simple; the latter follows from Schur's Lemma, because the natural representation is then irreducible).
The triple is a root system associated to ; its isomorphism type is independent of the choice of (,).
Fundamental weights
Given , write (resp. ) for the corresponding fundamental weight (resp. fundamental coweight) of in (resp. ) defined by for all .
Cocharacter data and associated subgroups
Cocharacter data
Throughout much of this paper, we work with a pair , where . Given such a pair, we choose a maximal torus , a basis and a representative compatibly as follows: Choose over F and over such that ; this is always possible because all maximal tori of are conjugate. In the presence of , we always choose so that is -dominant.
Associated Levi subgroup
Given , let be the Levi subgroup of given as the centralizer . Let . Then is a basis of simple roots for in .
Parabolic subgroups and their flag varieties
Let . We define the standard parabolic subgroup of of type I to be the subgroup generated by and the root groups for . In particular, the standard Borel subgroup is generated by and the root groups of negative roots. Let be the flag variety of parabolics of of type I.
Orbitally p-close cocharacters
Let and let p be a prime number. Recall from [[13], Sect. N.5.3] (see also [[12], Sect. 5.1]) that is called if, for every and every root satisfying , one has
2.4.1
Graph
A cocharacter which is orbitally p-close for all p is called quasi-constant. The condition "orbitally p-close" is a weakening of certain p-smallness conditions, while "quasi-constant" generalizes "minuscule". See [[12]] for more on quasi-constant (co)characters.
The Griffiths character, the Griffiths bundle and the main result
The Griffiths character
Notation for real Hodge structures
Let . Recall that an -Hodge structure consists of a pair (V, h), where V is a finite-dimensional -vector space and is a morphism of -groups. Denote the (descending) Hodge filtration of (V, h) on by . Let be the associated cocharacter of . By Deligne's convention, acts on the graded piece by . Let (resp. ) be the largest (resp. smallest) -weight in . Then
3.1.1
Graph
and (resp. ) is characterized as the largest (resp. smallest) integer satisfying (3.1.1) (in other words, the Hodge filtration descends from to ).
The Griffiths character for Deligne pairs
Fix a pair , where G is a connected, reductive -group and is the -conjugacy class of a morphism of -groups . The reinterpretation of much of Griffiths' work in terms of such pairs was introduced by Deligne in his Bourbaki talk [[7]].
Given , redefine as the associated cocharacter of . Let be a morphism of -groups (later we will want to assume that G, r both arise by base change from objects over ). Then is an -Hodge structure. Define the Griffiths module of (G, h, r) by
3.1.2
Graph
and the Griffiths character of (G, h, r) by
3.1.3
Graph
Let L be the -group . Then is an L-module and is a character of L. The isomorphism class of the L-module and the conjugacy class do not depend on the choice of .
Central kernel assumption
We shall always assume that has central kernel; otherwise the component of the Griffiths character corresponding to some -simple factor of will be trivial. For example trivial should clearly be avoided, for then in .
Generalization to arbitrary fields
Since the Hodge filtration on and the cocharacter uniquely determine each other, we can generalize the Griffiths module and character to the setting of arbitrary cocharacter data over arbitrary fields by working with instead of h. Thus let F be a field and fix an algebraic closure (no restriction is imposed on the characteristic of F). Let be a connected, reductive F-group and let . For every -vector space V and every morphism of -groups, we have the cocharacter of GL(V) and the corresponding descending filtration on V given by
3.1.4
Graph
where is the b-weight space of acting on V.
Given r with central kernel (Sect. 3.1.3), define the Griffiths module as in (3.1.2) and set the Griffiths character to be its determinant: .
As before, let . Then is an -module and for every triple .
Griffiths line bundles associated to Griffiths characters
We explain how the Griffiths character gives rise to a line bundle by an associated bundle construction in two different settings: First we explain how to recover the Griffiths line bundle in the classical setting recalled in Sect. 1.2. Then we describe the Griffiths line bundle in the context of G-Zips in positive characteristic.
Associated sheaves
In both cases, one uses the following basic construction: Suppose X is an -scheme with an action of an algebraic -group H. Then there is an exact tensor functor from to the category of vector bundles on the quotient stack ([[13], N.4.1], see also [[16], Part I, 5.8] in the case H acts freely on X). An equivalent variant is: The quotient stack is equipped with a tautological H-torsor ; given , the pushout of via V gives a GL(V)-torsor on , which is the torsor of bases of the vector bundle on associated to V.
Associated line bundle I: F=Q. The Borel embedding, d'après Deligne [7, Sects. 5.6–5.10]
Return to the setting of Sections 3.1.1 and 3.1.2: G is an -group and , are morphisms of -groups; is the -conjugacy class of h. Assume r has central kernel and that is a pure -HS.
Then is a parabolic subgroup of which is independent of (V, r). Let be the type of P(h). Then I is independent of . The compact dual of is the flag variety .
The Borel embedding is the injection given by ; it identifies with an open subset of , [[7], Lemma 5.8].
Applying Sect. 3.2.1 to the tautological P(h)-torsor on , combined with the functor given by extending trivially on the unipotent radical , one gets a diagram of exact tensor functors:
3.2.1
Graph
Applying (3.2.1) to the Griffiths character gives a -equivariant line bundle on .
The line bundle is the line bundle associated to the tautological family of Hodge structures on (with fiber at ) via Sect. 1.2. However, this family of Hodge structures is rarely a VHS (i.e., it rarely satisfies transversality): assuming that is of weight 0, the tautological family of HS over X is a VHS if and only if is of type , [[8], Prop. 1.1.14]. Finally, one obtains the line bundle of a (Sect. 1.2) by pullback along the period map (1.2.2): For every period domain D (more generally for every Mumford–Tate domain D), there exists a Deligne pair such that (resp. ) is a classical topology connected component of (resp. ).
Associated line bundle II: F=Fp. G-Zipμ-schemes, d'après Pink-Wedhorn-Ziegler [20, 21]
Let and . Up to possibly conjugating , we assume fixed a compatible choice of as in Sect. 2.4.1. Let be the associated stack of -Zips of type . Let P be the standard parabolic of type (Sect. 2.4.3), its opposite relative and put . Recall that a -Zip of type on an -scheme S is a quadruple , where I is a -torsor on S, (resp. ) is a P-structure (resp. Q-structure) on I and is an isomorphism of -torsors (Sect. 2.1).
Since part of the datum of a -Zip of type is a P-torsor, every representation of yields a vector bundle on via Sect. 3.2.1. We define the Griffiths line bundle of to be the line bundle associated to the Griffiths character . If X is an -scheme and is a morphism, we define the Griffiths line bundle of by pullback: .
G-Zipμ-schemes from de Rham cohomolgy, d'après Moonen-Wedhorn [18] and Pink-Wedhorn-Ziegler [...
In order for Sect. 3.2.3 to be useful, one needs an interesting supply of morphisms . In analogy with Sect. 1.2, we recall how maps arise from de Rham cohomology in characteristic p, see also the introduction to [[11]]. Suppose is a proper smooth morphism of schemes in characteristic p, that the Hodge-de Rham spectral sequence of degenerates at and that both the Hodge and de Rham cohomology sheaves of are locally free. Let and consider the conjugacy class of cocharacters of GL(n) whose -weight space has dimension . Then is a GL(n)-Zip of type ; thus it determines a morphism .
The analogy between and the period map (Sect. 1.2) was first suggested by Moonen-Wedhorn in the introduction to [[18]]. We thank Y. Brunebarbe for suggesting to pursue this analogy further. It is an interesting open problem to understand what should be the right analogue of the Mumford–Tate group for . Still, if the Hodge filtration is compatible with certain tensors, then will factor through a stack of -Zips, where is the subgroup stabilizing those tensors. For example, as already observed in [[18]], when Y / X is a family of polarized abelian schemes (resp. K3 surfaces) then factors through a stack of -Zips, where is a symplectic similitude group (resp. ).
Main result
We continue to use the notation for root data introduced in Sect. 2; we always choose compatibly as in Sect. 2.4.1 and the pairing (,) on is always chosen -invariant and positive definite (Sect. 2.3.3).
The Griffiths ray
Let
3.3.1
Graph
be the positive ray generated by in . We call the Griffiths ray of . More generally, if v is a vector of a -vector space, write for the ray which it generates.
Write for the projection of onto . Via the identifications (2.3.5), one has .
Weight pairing sums
Given a root and a representation , recall Sect. 2.2.3 and let
3.3.2
Graph
Since , each summand in (3.3.2) is invariant under .
Theorem 3.1
Let be a connected, reductive F-group. Assume is F-simple, and is a representation of over F with central kernel. Then:
- For all , one has
- 3.3.3
Graph
- The value is independent of .
- Under the identification afforded by (,), for every one has
- 3.3.4
Graph
- In particular,
- 3.3.5
Graph
Remark 3.2
- Both of the terms and , when viewed in , depend on (,). Further, the dependence among the two is inverse proportional, so the right-hand side of (3.3.4) is independent of (,)
- For fixed r, the value depends on , not just on the isomorphism class of the root system .
Remark 3.3
The sign difference between the cocharacter and the Griffiths character is a reflection of the change in curvature/positivity between a period domain (or more generally a Mumford–Tate domain, or a Griffiths–Schmid manifold) and its compact dual . For example, the Hodge line bundle is ample on a Hodge type Shimura variety (Sect. 3.4.1), but over it arises via the construction 3.2.1 from a line bundle on which is anti-ample on .
Under the same hypotheses, two immediate corollaries of Theorem 3.1 are:
Corollary 3.4
(Independence)
- Given a cocharacter datum , the Griffiths ray is independent of r (always assumed with central kernel).
- Given with central kernel, the positive scalar such that is independent of .
Corollary 3.5
In addition to the hypotheses of Theorem 3.1, assume that all roots satisfying have the same length.[6] Then, without reference to (,), one has
3.3.6
Graph
Remark 3.6
The assumption that is F-simple and the need to consider associated rays (i.e., to allow positive scalar multiples) are both already essential in the setting of the Hodge line bundle, see [[12], Sect. 4.5] and [[13], Sect. 2.1.6, Footnote 7] for respective examples.
As an application of our joint work with Brunebarbe, Koskivirta and Stroh on positivity of automorphic bundles [[3]], one obtains the nefness of the Griffiths bundle on a proper -scheme.
Corollary 3.7
(Nefness) Assume F has characteristic p, that X is a proper -scheme of finite type and that is a morphism. If is orbitally p-close (Sect. 2.4.4), then the pullback of the Griffiths line bundle to X is a nef line bundle on X.
Remark 3.8
We emphasize that [[3]] contains stronger positivity results and that our sole contribution here is to show that is -negative, see Corollary 4.6.
Remark 3.9
Unlike the property "ampleness", the property "nef" is not always open on the base, cf. [[19]]. So we do not know if one can reprove Griffiths' result that the Griffiths line bundle of a polarized -VHS over a projective base is nef via Corollary 3.7.
Examples
The Hodge character and line bundle I: F=Q
Consider the special case of triples where
-
(i.e., is a -group),
-
for some and is a Shimura datum, where ,
-
is of type .
Then the Shimura datum is of Hodge type and r is a symplectic embedding [[8], Lemma 1.3.3]. In this case, the Griffiths character is the Hodge character giving rise to the Hodge line bundle of the Shimura variety . Since is minuscule in this example, the condition of Corollary 3.5 holds. Then Corollary 3.5 recovers Theorem 1.4.4 and Corollary 1.4.5 of our joint work with Koskivirta [[12]], which state that the Hodge character is quasi-constant (Sect. 2.4.4) and that the Hodge ray it determines is independent of r and given by (3.3.6).
For applications of these results to the "tautological" ring of Hodge-type Shimura varieties and the cycle classes of Ekedahl-Oort strata, see the recent preprint of Wedhorn-Ziegler [[23]].
The Hodge character and line bundle II: G-Zips
Let . Let be a symplectic space over of dimension g and the corresponding symplectic similitude group. Let be a non-central, minuscule cocharacter of . The Hodge character is defined for any symplectic embedding of cocharacter data
3.4.1
Graph
and the associated line bundle is the Hodge line bundle of , [[13], Sect. 1.3]. Theorem 3.1 extends the results of [[12]], recalled in Sect. 3.4.1, to symplectic embeddings 3.4.1 which need not arise from an embedding of Shimura data. In particular, the Hodge character is quasi-constant even if it does not arise from a Shimura variety setting by reduction mod p.
The adjoint representation via the Coxeter number
When is the adjoint representation, the sums (Sect. 3.3.2) are sums of squares of root pairings, and their values are computed explicitly using the computations by Bourbaki of the "canonical bilinear form" and -invariant of a root system ([[2], Chap. 6, Sect. 1.12] and exercise 5 of Chap. 6, Sect. 1 in loc. cit.; see also Remark A.2.1). In this way, given the root system associated to (Sect. 2.3.3), one obtains the proportionality constant c of Corollary 3.4(b) for .
When is irreducible and simply-laced, one has for all , where h is the Coxeter number of . When is irreducible and multi-laced one has two invariants of the root system: for short and long respectively. These are recorded in Table 1, together with the Coxeter number and ; the latter are taken from the Planches in loc. cit. When is simply-laced, one has .
The root pairing sums S(α∨,Ad)
Type of | Coxeter Number h |
| simply-laced | short | long |
---|
| n |
| 4n | | |
| 2n |
| |
|
|
| 2n |
| |
|
|
|
|
|
| | |
| 6 | 48 | | 16 | 48 |
| 12 | 162 | | 36 | 72 |
| 12 | 144 | 48 | | |
| 18 | 324 | 72 | | |
| 30 | 900 | 120 | | |
Proof of the main result and its application
General lemmas on roots and weights
Let k be an algebraically closed field.
Lemma 4.1
Let T be a k-torus. Assume
Graph
is a finite-dimensional representation with finite kernel. Then the -span of the T-weights of is all of .
Proof
Let . Then is a torus, and the induced map is faithful. Moreover, since is finite. Thus we reduce to the case that is faithful.
Since T is a torus, the category is semisimple and Tannakian, neutralized by the forgetful functor . Since is faithful, every is a factor of some . In other words, is a T-weight of some . The T-weights of are -linear combinations of the T-weights of V. So lies in the -span of the T-weights of V.
Recall the notation for the T-weights of V (Sect. 2.2.3).
Lemma 4.2
Assume that is a representation with central kernel. Then, for every simple root , there exists a weight such that .
Proof
Let be the inclusion. If is a T-weight of V, then , where we identify the coroots of in with those of T in G as in Sect. 2.3.1. Thus we reduce to the case that is semisimple. Then r has finite kernel, hence so does its restriction to . Now the result follows from Lemma 4.1, for otherwise the -span of the weights would lie in a root hyperplane.
Corollary 4.3
Assume that is a representation with central kernel and that is not central in G. Then there exists a weight such that .
Proof
Since is not central, there exists such that . By Lemma 4.2 applied to , there exists such that .
Since , we conclude that either or . Since the set of weights is W-stable, the corollary is proved.
Return to the setting that F is an arbitrary field with algebraic closure .
Lemma 4.4
Assume is an adjoint, simple F-group. Then two roots have the same length if and only if they are conjugate under .
Proof
The action of preserves length (recall from Sect. 2.3.3 that the bilinear form (,) is chosen -invariant).
Conversely, assume are two roots of equal length. Since is simple over F, the -simple factors of are permuted transitively by . Thus there exist -simple factors and of , with respective maximal tori , contained in , such that the root systems are naturally irreducible components of the root system and , . Let map to . In a reduced and irreducible root system, two roots are conjugate under the Weyl group if and only if they have the same length; apply this to the system and the roots .
Root-theoretic analysis of the Griffiths module
Consider the setting of Theorem 3.1: Let be a connected, reductive F-group, V an F-vector space and a morphism F-groups with central kernel. Let .
The expression of in the basis of fundamental coweights (Sect. 2.3.4) in is
Graph
Since is identified with in via (,), is identified with the linear combination
Graph
in the basis of fundamental weights in . Therefore, in Theorem 3.1,(a) and (b) jointly imply (c). In turn, (c) trivially implies (d). Thus it suffices to prove (a) and (b). Part (a) is proved in Lemma 4.5. Part (b) is shown first for roots with the same length in Lemma 4.7. The general case is reduced to that of equal length in Lemma 4.8.
Lemma 4.5
For all , one has
4.2.1
Graph
Proof
By definition of the Griffiths module (3.1.2), the set of -weights of r is contained in that of (not counting multiplicities): (Sect. 2.2.3). To understand , we must determine the multiplicity of a weight in in terms of its multiplicity in (V, r). Let . Choose a maximal torus in containing . Let be the identity. By construction, we have an equality of -representations:
Graph
The -weights of all have multiplicity one ( is minuscule). The multiplicity of a -weight in is given by the distance to the top of the filtration, namely
4.2.2
Graph
Let . Then . Since there are precisely different -weights which pull back to , the multiplicity of in is
4.2.3
Graph
Since is defined (3.1.3) as the determinant of ,
4.2.4
Graph
If , then , so contributes zero in the sum (4.2.4). We group the remaining terms in (4.2.4) in pairs . Recalling that and that the multiplicities are W-invariant gives
4.2.5
Graph
Substituting the definition into (4.2.5) yields (4.2.1).
Recall that is normalized to be -dominant.
Corollary 4.6
The Griffiths character is -anti-dominant. Furthermore,
4.2.6
Graph
if and only if .
Proof
The anti-dominance is clear from (3.3.2) and (4.2.1). Moreover, if , then since , so again by (4.2.1).
Finally, assume . Since is assumed -dominant, one has . By Lemma 4.2, one of the terms in the sum (3.3.2) is nonzero, so .
Lemma 4.7
Assume is a simple F-group. If have the same length, then
4.2.7
Graph
Proof
Assume have the same length. By Lemma 4.4, there exists such that . One has
4.2.8
Graph
Since r is a morphism of F-groups and is an F-torus, the set of weights is stable under and for all . Applying the orthogonality of W relative , together with the substitution gives (4.2.7).
We pause to note that we have shown the simply-laced case of the main result:
Proof of Corollary 3.5
Combine Lemma 4.5, Corollary 4.6 and Lemma 4.7.
Root strings
To treat the multi-laced case, given two roots , we will use what Knapp calls the " -root-string containing " (see the paragraph preceding Prop. 2.48 in [[17]]) and what Bourbaki call the " -chaîne de racines définie par ", [[2], Chapter 6, No. 1.3]. The preceding synonymous terminology refers to the set .
Assume is a pair of adjacent simple roots. Recall from [[2], Chapter 6, No. 1.3, Prop. 9]) that the length of the -root string containing is then given as follows: If is strictly longer than , then the root string is . Otherwise, is at least as short as and the length of the root string is
4.2.9
Graph
Note that the right-hand side of (4.2.9) is the number of edges connecting and in the Dynkin diagram of , as defined for instance in [[22], Sect. 9.5.1]. Moreover, if is strictly shorter than , then all of the members of the root string are short, except for the two endpoints and which are both long.[7]
Lemma 4.8
Assume is a simple F-group and . Then
4.2.10
Graph
Proof
By Lemma 4.7 and its proof, since is simple over F we reduce to the case that
-
is multi-laced,
-
-
are adjacent (in particular they belong to the same -simple factor of ), and
-
is strictly longer than .
Given , let
Graph
Since are adjacent, is a coroot.[8] Since is strictly longer than , is strictly shorter than .
We now apply Sect. 4.2.1, but to the root system of coroots. Thus, is a short coroot and is a coroot. The coroot is long when is doubly-laced (types ) but short when is triply-laced (type .
One has
4.2.11a
Graph
4.2.11b
Graph
On the other hand, it follows from Lemma 4.7 that , since both and are short. Also, or according to whether is doubly or triply-laced respectively. Solving the two equations in each of the two cases for the ratio , one finds (resp. ) when is doubly (resp. triply) laced.
The proof of the main result is now complete:
Proof of Theorem 3.1
Combine Lemma 4.5, Corollary 4.6 and Lemma 4.8.
Nefness of the Griffiths bundle of G-Zipμ-schemes
Proof of Corollary 3.7
Let as in Corollary 3.7. Write for the Griffiths line bundle on associated to some choice of r with central kernel (Sect. 3.4.2) and put . In view of Theorem 3.1 nothing in the argument below will depend on the auxiliary choice of r.
Assume that is orbitally p-close (Sect. 2.4.4). By Theorem 3.1, the Griffiths ray is also orbitally p-close (by definition, one element of a ray is orbitally p-close if and only if all are). Thus the Griffiths character is a Hasse generator of by [[13], Th. 3.2.3], i.e., for every stratum with Zariski closure there exists and a section such that the nonvanishing locus of is precisely .
In [[3]], it is shown that any line bundle corresponding to any Hasse generator is nef on a projective -scheme X. Indeed, suppose is a closed, integral subscheme. Then we have a nonzero section in as follows: Since Z is irreducible, there exists a unique stratum in such that the intersection is dense in Z. Then take .
Acknowledgements
Open access funding provided by Stockholm University. We are grateful to the GS Magnusons Fund, the Knut & Alice Wallenberg Foundation and Stockholm University for their support. This paper was completed during a visit to the University of Zurich. I thank the Institute of Mathematics for its hospitality and the opportunity to speak about this work. We thank the referee for his/her work on the paper. I want to thank my coauthors Y. Brunebarbe, J.-S. Koskivirta and B. Stroh for hours and hours of stimulating discussions on several topics related to this paper; in particular the idea for this work was born when Brunebarbe taught me about the Griffiths bundle and suggested to think of maps as analogues of period maps. I am grateful to Koskivirta and J. Ayoub for comments on an earlier version of this work, which led to improvements in the statement of the main result. I thank B. Moonen and T. Wedhorn for discussions about G-Zips and the link with classical Hodge theory. I am grateful to B. Klingler and J. Daniel for explaining to me the connection between the Griffiths bundle and Daniel's work on loop Hodge structures [[6]]; while we do not study this connection here, it would be interesting to understand the relationship between this paper and Daniel's work in the future. Finally, it should be clear to the reader how much this paper owes to the works of Griffiths and Deligne. In addition to reshaping Hodge theory, we also thank them both for inspiring correspondence and discussions. In particular, I thank Griffiths for correspondence on positivity of Hodge bundles. I thank Deligne, first for sending me a two-line proof of the independence of from r when , in response to my question about the dependence of on r, and second for his simplification of the main result given in Appendix A.
Appendix A. Deligne's Simplification
We present Deligne's elegant simplification of the proof of Theorem 3.1(d) following a letter from him [[9]]. In particular, Deligne's argument does not use root strings and it does not differentiate according to the lacing of the Dynkin diagram of .
Deligne's argument works in the setting of semisimple , so we first explain in §A.1 how to reduce Theorem 3.1(d) to that case. The reduction is a formality, so the reader who believes it may happily jump to Deligne's argument proper in §A.2.
A.1. Reduction to the semisimple case
The proof of Lemma 4.5, up to (4.2.3), gives the equality of characters
A.1.1
Graph
Note that the next step (4.2.4) in the proof of Lemma 4.5 is the result of pairing both sides of (A.1.1) with but we now avoid doing this. As -representations, one has
A.1.2
Graph
Let be the inclusion. Since is semisimple, the pullback of to is trivial. Therefore the image of (A.1.2) in is zero. Pulling back (A.1.1) to , the first term on the right pulls back to zero, so
A.1.3
Graph
It follows from (A.1.3) that the validity of Theorem 3.1(d) is invariant under replacing with a (strictly) positive scalar multiple. Hence we may assume that arises from a cocharacter of , which we denote the same way. For such , the cocharacter of is central in .
For any cocharacter which is central in , the sum
A.1.4
Graph
is a character of . One can see this in two ways: Since is central, by Schur's Lemma it acts on each -irreducible constituent of by a character of ; the action of on is then given by (A.1.4). Alternatively, one checks that the pairing of (A.1.4) with every coroot is zero by the method of proof in Lemma 4.5, i.e., grouping the summands with in pairs . As before, the pullback of (A.1.4) to is zero.
Applying the above with and using (A.1.3), we conclude that
A.1.5
Graph
where is the restriction of r to . It follows that Theorem 3.1(d) for implies it for .
A.2. Deligne's argument
By §A.1, we may henceforth assume that is semisimple. We may also assume that , as otherwise both sides of Theorem 3.1(d) are trivial. Define a pairing on by setting, for all ,
A.2.1
Graph
Since the -representation V is defined over F, the pairing is -invariant. It is positive definite by Corollary 4.3 and thus induces an isomorphism . Under this isomorphism, maps to by (A.1.3). Since is F-simple, any other positive definite, -invariant pairing (, ) on is a positive scalar multiple of (Sect. 2.3.3). This proves Theorem 3.1(d).
Remark A.2.1
When is the adjoint representation, the pairing (A.2.1) is the inverse of the "canonical bilinear form" of [[2], Chapter 6, Sect. 1.12].
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Footnotes
In loc. cit., was called the canonical bundle of ; in [[14]] and [[1]] it was called the Hodge (line) bundle.
The condition " discrete in " is already necessary to ensure that is an analytic space.
It appears that the authors of [[14]] are working to fix an error in their preprint.
The two possible interpretations of 'maximal torus' here are equivalent [[5], Appendix A].
One of several equivalent definitions is that the -span of the coroots is the whole cocharacter group.
In particular this holds if is simply-laced.
The last claim is checked by expanding and plugging in the value of .
Caution: While is a root and is a coroot, one has whenever are adjacent of different lengths.
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By Wushi Goldring
Reported by Author