The Griffiths bundle is generated by groups

First the Griffiths line bundle of a $\mathbf Q$-VHS $\mathscr V$ is generalized to a Griffiths character ${\rm grif}(\mathbf G, \mu,r)$ associated to any triple $(\mathbf G, \mu, r)$, where $\mathbf G$ is a connected reductive group over an arbitrary field $F$, $\mu \in X_*(\mathbf G)$ is a cocharacter (over $\overline{F}$) and $r:\mathbf G \to GL(V)$ is an $F$-representation; the classical bundle studied by Griffiths is recovered by taking $F=\mathbf Q$, $\mathbf G$ the Mumford-Tate group of $\mathscr V$, $r:\mathbf G \to GL(V)$ the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to ${\rm grif}(\mathbf G, \mu, r)$. 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 $\mathbf G$-Zips. When $\mathbf G$ is $F$-simple, we show that, up to positive multiples, the Griffiths character ${\rm grif}(\mathbf G,\mu,r)$ (and thus also the Griffiths line bundle) is essentially independent of $r$ with central kernel, and up to some identifications is given explicitly by $-\mu$. As an application, we show that the Griffiths line bundle of a projective $\mathbf G{\rm -Zip}^{\mu}$-scheme is nef.


Introduction
We are motivated by the general problem of understanding which geometric objects are generated by groups. As an 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 µ ∈ X * (G) and a representation r of G.
1.1. 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 Q-VHS S of variations of Q-Hodge structure over a smooth, projective C-scheme S.
Let C be a Tannakian category over a field k which is neutralized by a fiber functor ω : C → Vec k . Let G be the Tannaka group of (C , ω), i.e., the affine group scheme which represents the functor of automorphisms Aut ⊗ (ω). Question 1.1.1. Given an invariant i(X) associated to every object X ∈ C , 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 µ ∈ X * (G). 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 1.1.1 is the following: For every X ∈ C , let G(X) be the Tannaka group of the Tannakian sub-category X ⊗ 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 1.1.2. 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 1.1.2 is of course to make precise the meaning of "essentially" in specific examples. The main result of this paper implies that Question 1.1.2 has a positive answer when X is a Q-VHS, i(X) is its Griffiths line bundle ( §1.2) and G(X) is its Mumford-Tate group, provided the adjoint group G(X) ad is Q-simple, see Theorem 3.3.5. In this case, the additional group-theoretic data is the Hodge cocharacter µ ∈ X * (G(X)) 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 (G(X), µ).
1.2. 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 [12]. Suppose S is a connected, smooth projective C-scheme and V is a polarized variation of Hodge structure on S with monodromy group Γ and period domain D. Let Fil • V be the (descending) Hodge filtration on V ; for the sake of exposition suppose that Fil 0 V = V . Griffiths (loc. cit., (7.13)) associated to V the line bundle We call grif(V ) the Griffiths line bundle 1 of V . Griffiths also associated to V a period map By studying the positivity properties of the line bundle grif(V ), Griffiths concluded that the image of the period map Φ(S) is projective algebraic when Γ is discrete in Aut(D) (loc. cit., (9.7), see also [3, 13. 1.3. Summary of the paper. Let F be an arbitrary field. Consider triples (G, µ, r), where G is a connected reductive F -group, µ ∈ X * (G) is a cocharacter of G F and r : G → GL(V ) is a morphism of F -groups. In the vein of Remark 1.2.3, we explain in §3.1 how to generalize the Griffiths line bundle to a character grif(G, µ, r) of the Levi subgroup L := Cent G F (µ) of G F . In §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 grif(V ) associated to a VHS via Deligne's theory of pairs (G, X) and we obtain a Griffiths line bundle on stacks of G-Zips in characteristic p > 0.
The main result is stated in §3.3, see Theorem 3.3.5. Roughly speaking, it states that grif(G, µ, r) is, up to positive multiples and modulo the center, independent of r and given explicitly by −µ. To make this precise requires some technical assumptions and identifications concerning a root datum of G. For this purpose, the necessary structure theory associated to triples (G, µ, r) is given in §2. The sign change between µ and grif(G, µ, r) reflects the change in positivity/curvature between a Mumford-Tate domain and its compact dual (Remark 3.3.10).
By combining our result with our forthcoming joint work with Y. Brunebarbe, J.-S. Koskivirta and B. Stroh [2] on the positivity of automorphic bundles, we obtain the following application: Assume X is a projective scheme in positive characteristic p > 0 and ζ : X → G-Zip µ is a morphism to the stack of G-Zips associated to (G, µ) by Pink-Wedhorn-Ziegler [18,17]. As long as p is not too small relative µ (orbitally p-close to be precise, §2.4.4), then the Griffiths line bundle of X is nef (Corollary 3.3.15).
In §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 V is polarized of weight one. Here we recover the results of our joint work with Koskivirta [11]. The second example provides explicit formulas for grif(G, µ, r) when r = Ad is the adjoint representation, essentially in terms of the Coxeter number of the underlying root system.
The proof of Theorem 3.3.5 is given in §4. Some preliminary, general lemmas on roots and weights are given in §4.1; the proof proper occupies §4.2. The key is to translate the main result into a statement about weight pairings with coroots (Lemma 4.2.1). The simply-laced case of Theorem 3.3.5 then results from a simple change of variables in the root pairing expression (Lemma 4.2.9). The general case is reduced to the simply-laced one by the theory of root strings (Lemma 4.2.14). The application to nefness (Corollary 3.3.15) is proved in §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 3.3.5(d). We are very grateful to Deligne for allowing us to include his simplification in Appendix A.

Acknowledgements
First, 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 ζ : X → G-Zip µ 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 [5]; 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. 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.
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 grif(G, µ, r) from r when G = GL(n), in response to my question about the dependence of grif(G, µ, r) on r, and second for his simplification of the main result given in Appendix A.
2. Notation and structure theory 2.1. Notation. Let F be a field and fix an algebraic closure F of F . A subscript F , Q, R, C etc. will always denote base change to F , Q, R, C respectively. Thus G m,F denotes the multiplicative group scheme over F , and if N is a Z-module, then N Q := N ⊗ Z Q is the associated Q-vector space.
If H is an algebraic F -group, then X * (H) = Hom(H F , G m,F ) (resp. X * (H) = Hom(G m,F , H F )) denotes the group of characters (resp. cocharacters) of H F . Write [µ] for the H(F )-conjugacy class of a cocharacter µ ∈ X * (H).
Similarly 2.2.1. Root datum of G. Let T be a maximal torus 3 of G. The root datum of (G, T) is the quadruple together with the Z-valued perfect pairing is the set of roots (resp. coroots) of T F in G F .

Weyl group.
For every α ∈ Φ, let s α be the corresponding root reflection. Let W := W (G, T) := s α |α ∈ Φ be the Weyl group of T F in G F . Recall that s α ↔ s α ∨ provides a canonical identification of W with the dual Weyl group of the dual root datum generated by the s α ∨ .
. Given a weight χ ∈ Φ(V, T), let m V (χ) denote its multiplicity (the dimension of the corresponding weight space).
2.2.6. Based root datum of G. Let ∆ ⊂ Φ be a basis of simple roots. Then ∆ ∨ := {α ∨ | α ∈ ∆} is the corresponding basis of simple coroots and is the based root datum of (G, T, ∆).

2.3.
Derived subgroup, adjoint quotient and simply-connected covering. Let G der (resp. G ad ,G) be the derived subgroup of G (resp. its adjoint quotient, the simply-connected covering G der in the sense of root data 4 ). Let s :G → G be the natural quasi-section of the projection pr : G → G ad . The root datum (2.2.2) and the based root datum (2.2.7) naturally induce ones of G der , G ad andG as follows: Then T der is a maximal torus in G der with character group X * (T der ) = X * (T)/X * 0 (T) and is the root datum of (G der , T der ), where the roots Φ are restricted to T der . Similarly, by restriction we identify ∆ with a basis of simple roots for (G der , T der ).

2.3.5.
Root data of G ad andG. LetT (resp. T ad ) denote the preimage of T der inG (resp. the image of T der in G ad ). ThenT and T ad are maximal tori inG and G ad respectively; the roots (resp. simple roots, coroots, simple coroots) of the three pairs (G,T), (G der , T der ), (G ad , T ad ) are identified via the central isogenies In turn, the Weyl groups of the three pairs (2.3.6) are all canonically identified with W . Choose a positive definite, symmetric, When G ad 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 G ad F is simple; the latter follows from Schur's Lemma, because the natural representation W → GL(X * (T) Q ) is then irreducible).
The triple (X * (T) Q , Φ, (, )) is a root system associated to (G, T); its isomorphism type is independent of the choice of (, ).

2.4.1.
Cocharacter data. Throughout much of this paper, we work with a pair (G, [µ]), where µ ∈ X * (G). Given such a pair, we choose a maximal torus T, a basis ∆ ⊂ Φ and a representative µ ′ ∈ [µ] compatibly as follows: Choose T over F and µ ′ over F such that Im(µ ′ ) ⊂ T F ; this is always possible because all maximal tori of G F are conjugate. In the presence of µ ∈ X * (G), we always choose ∆ so that µ is ∆-dominant.

Associated Levi subgroup.
Given µ ∈ X * (G), let L be the Levi subgroup of G F given as the centralizer Then ∆ L is a basis of simple roots for T F in L.

2.4.3.
Parabolic subgroups and their flag varieties. Let I ⊂ ∆. We define the standard parabolic subgroup of G F of type I to be the subgroup generated by T F and the root groups U α for α ∈ −∆ ∪ I. In particular, the standard Borel subgroup is generated by T F and the root groups of negative roots. Let Par I be the flag variety of parabolics of G F of type I.
3. The Griffiths character, the Griffiths bundle and the main result . By Deligne's convention, µ acts on the graded piece Gr a V C := Fil a V C /Fil a+1 V C by z −a . Let µ max (resp. µ min ) be the largest (resp. smallest) µ-weight in V C . Then and −µ max (resp. 1 − µ min ) is characterized as the largest (resp. smallest) integer satisfying (3.1.2) (in other words, the Hodge filtration descends from −µ max to −µ min ).
3.1.3. The Griffiths character for Deligne pairs. Fix a pair (G, X), where G is a connected, reductive R-group and X := Class G(R) (h) is the G(R)-conjugacy class of a morphism of R-groups h : S → G. The reinterpretation of much of Griffiths' work in terms of such pairs (G, X) was introduced by Deligne in his Bourbaki talk [6]. Given h ∈ X, redefine µ(z) := (h ⊗ C)(z, 1) ∈ X * (G) as the associated cocharacter of G C . Let r : G → GL(V ) be a morphism of R-groups (later we will want to assume that G, r both arise by base change from objects over Q). Then r • h is an R-Hodge structure. Define the Griffiths module of (G, h, r) by 3.1.6. Central kernel assumption. We shall always assume that r : G → GL(V ) has central kernel; otherwise the component of the Griffiths character corresponding to some C-simple factor ofG C will be trivial. For example r = 1 trivial should clearly be avoided, for then grif(G, h, 1) = 0 in X * (L).
3.1.7. Generalization to arbitrary fields. Since the Hodge filtration on V C and the cocharacter µ 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 µ instead of h. Thus let F be a field and fix an algebraic closure F (no restriction is imposed on the characteristic of F ). Let G be a connected, reductive F -group and let µ ∈ X * (G). For every F -vector space V and every morphism r : G F → GL(V ) of F -groups, we have the cocharacter r • µ of GL(V ) and the corresponding descending filtration Fil • V on V given by Given r with central kernel ( §3.1.6), define the Griffiths module Grif(G, µ, r) as in (3.1.4) and set the Griffiths character to be its determinant: grif(G, µ, r) := det Grif(G, µ, r).

3.2.4.
Associated line bundle II: F = F p . G-Zip µ -schemes, d'après Pink-Wedhorn-Ziegler [17,18]. Let F = F p and µ ∈ X * (G). Up to possibly conjugating µ, we assume fixed a compatible choice of µ, T, ∆ as in §2.4.1. Let G-Zip µ be the associated stack of G-Zips of type µ. Let P be the standard parabolic of type ∆ L ( §2.4.3), P opp its opposite relative L and put Q := (P opp ) (p) . Recall that a G-Zip of type µ on an F -scheme S is a quadruple (I, I P , I Q , ϕ), where I is a G-torsor on S, I P (resp. I Q ) is a P -structure (resp. Q-structure) on I and ϕ : ( Since part of the datum of a G-Zip of type µ is a P -torsor, every representation of L yields a vector bundle on G-Zip µ via §3.2.1. We define the Griffiths line bundle grif(G-Zip µ , r) of G-Zip µ to be the line bundle associated to the Griffiths character grif(G, µ, r). If X is an F -scheme and ζ : X → G-Zip µ is a morphism, we define the Griffiths line bundle of (X, ζ) by pullback: grif(X, ζ, r) := ζ * grif(G-Zip µ , r). [15] and Pink-Wedhorn-Ziegler [18,17]. In order for §3.2.4 to be useful, one needs an interesting supply of morphisms ζ : X → G-Zip µ . In analogy with §1.2, we recall how maps ζ : X → G-Zip µ arise from de Rham cohomology in characteristic p, see also the introduction to [10]. Suppose π : Y → X is a proper smooth morphism of schemes in characteristic p, that the Hodge-de Rham spectral sequence of π degenerates at E 1 and that both the Hodge and de Rham cohomology sheaves of π are locally free. Let n = rk H i dR (Y /X) and consider the conjugacy class [µ] of cocharacters of GL(n) whose −a-weight space has dimension rk R i−a π * Ω a Y /X . Then H i dR (Y /X) is a GL(n)-Zip of type µ; thus it determines a morphism ζ : X → GL(n)-Zip µ . The analogy between ζ and the period map Φ ( §1.2) was first suggested by Moonen-Wedhorn in the introduction to [15]. 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 ζ. Still, if the Hodge filtration is compatible with certain tensors, then ζ will factor through a stack of G-Zips, where G ⊂ GL(n) is the subgroup stabilizing those tensors. For example, when Y /X is a family of polarized abelian schemes (resp. K3 surfaces) then ζ factors through a stack of G-Zips, where G is a symplectic similitude group (resp. G = SO(21)).
(a) Both of the terms (α, α) and µ ad , when viewed in X * (T) Q , depend on (, ). Further, the dependence among the two is inverse proportional, so the right-hand side of (3.3.7) is independent of (, ) (b) For fixed r, the value (α, α)S(α ∨ , r) depends on Φ ⊂ X * (T) Q , not just on the isomorphism class of the root system (X * (T) Q , Φ, (, )). Under the same hypotheses, two immediate corollaries of Theorem 3.3.5 are: (a) Given a cocharacter datum (G, µ), the Griffiths ray grif(G, µ, r) is independent of r (always assumed with central kernel). (b) Given r ∈ Rep F (G) with central kernel, the positive scalar c ∈ Q > 0 such that s * grif(G, µ, r) = −cµ ad is independent of µ.
Corollary 3.3.12. In addition to the hypotheses of Theorem 3.3.5, assume that all roots α satisfying α, µ = 0 have the same length 5 . Then, without reference to (, ), one has Remark 3.3.14.
The assumption that G 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 [11, §4.5] and [9, §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 [2], one obtains the nefness of the Griffiths bundle on a proper G-Zip µ -scheme. Corollary 3.3.15. Assume X is a proper F -scheme of finite type and ζ : X → G-Zip µ is a morphism. If grif(G, µ, r) is orbitally p-close ( §2.4.4), then the pullback of the Griffiths line bundle to X is a nef line bundle on X.
Remark 3.3.16. We emphasize that [2] contains stronger positivity results and that our sole contribution here is to show that grif(G, µ, r) is ∆ \ ∆ L -negative, see Corollary 4.2.7.  [11], which state that the Hodge character is quasi-constant ( §2. 4.4) and that the Hodge ray it determines is independent of r and given by (3.3.13).
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 [20].

3.4.2.
The Hodge character and line bundle II: G-Zips. Let F = F p . Let (V, ψ) be a symplectic space over F p of dimension g and GSp(V, ψ) the corresponding symplectic similitude group. Let µ g be a non-central, minuscule cocharacter of GSp(V, ψ). The Hodge character is defined for any symplectic embedding of cocharacter data When ∆ is irreducible and simply-laced, one has S(α ∨ , Ad) = 4h for all α ∈ Φ, where h is the Coxeter number of ∆. When ∆ is irreducible and multi-laced one has two invariants of the root system: S(α ∨ , r) for α short and long respectively. These are recorded in Table 3.4.4, together with the Coxeter number and γ(∆); the latter are taken from the Planches in loc. cit. When ∆ is simply-laced, one has h 2 = γ(∆).  Proof. Let T ′ := T / ker ϕ. Then T ′ is a torus, and the induced map ϕ : Thus we reduce to the case that ϕ is faithful. Since T is a torus, the category Rep k (T ) is semisimple and Tannakian, neutralized by the forgetful functor Rep k (T ) → Vec k . Since ϕ is faithful, every χ ∈ X * (T ) is a factor of some T m,n (V ) := V ⊗m ⊗ (V ∨ ) ⊗n . In other words, χ is a T -weight of some T m,n (V ). The T -weights of T m,n (V ) are Z-linear combinations of the T -weights of V . So χ lies in the Q-span of the T -weights of V .  Proof. Let ι : T der → T be the inclusion. If χ is a T -weight of V , then χ, α ∨ = ι * χ, α ∨ , where we identify the coroots of T der in G der those of T in G as in §2.3.1. Thus we reduce to the case that G = G der is semisimple. Then r has finite kernel, hence so does its restriction to T ⊂ G. Now the result follows from Lemma 4.1.1, for otherwise the Q-span of the weights would lie in a root hyperplane. Proof. Since µ is not central, there exists α ∈ ∆ such that α, µ = 0. By Lemma 4.1.2 applied to α, there exists χ ∈ Φ(V, T ) such that χ, α ∨ = 0.
Return to the setting that F is an arbitrary field with algebraic closure F . Proof. The action of W ⋊ Gal(F /F ) preserves length (recall from §2.3.7 that the bilinear form (, ) is chosen W ⋊ Gal(F /F )-invariant).
Conversely, assume α, β ∈ Φ are two roots of equal length. Since G is simple over F , the F -simple factors of G F are permuted transitively by Gal(F /F ). Thus there exist F -simple factors G i and G j of G F , with respective maximal tori T i , T j contained in T F , such that the root systems (X * (T i ), Φ i , (, )), (X * (T j ), Φ j , (, )) are naturally irreducible components of the root system (X * (T) Q , Φ, (, )) and α ∈ Φ i , β ∈ Φ j . Let σ 0 ∈ Gal(F /F ) map Φ i to Φ j . 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 Φ j and the roots σ 0 α, β.

4.2.
Root-theoretic analysis of the Griffiths module. Consider the setting of Theorem 3.3.5: Let G be a connected, reductive F -group, V an F -vector space and r : G → GL(V ) a morphism F -groups with central kernel. Let µ ∈ X * (G).
The T V -weights of Id all have multiplicity one (Id is minuscule). The multiplicity of a T V -weightχ in Grif(GL(V ), r • µ, Id) is given by the distance to the top of the filtration, namely Let χ = r * χ . Then χ, r • µ = χ, µ . Since there are precisely m V (χ) different T V -weights which pull back to χ, the multiplicity of χ in Grif(G, µ, r) is Since grif(G, µ, r) is defined (3.1.5) as the determinant of Grif(G, µ, r), m Grif(G,µ,r) (χ) χ, α ∨ .