Über die Autorin bzw. den Autor

Mark Green is professor of mathematics at the University of California, Los Angeles and is Director Emeritus of the Institute for Pure and Applied Mathematics. Phillip A. Griffiths is Professor Emeritus of Mathematics and former director at the Institute for Advanced Study in Princeton. Matt Kerr is assistant professor of mathematics at Washington University in St. Louis.

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Mumford-Tate Groups and Domains

PRINCETON UNIVERSITY PRESS

Contents

Introduction..................................................................................................................1I Mumford-Tate Groups.........................................................................................................28I.A Hodge structures..........................................................................................................28I.B Mumford-Tate groups.......................................................................................................32I.C Mixed Hodge structures and their Mumford-Tate groups......................................................................38II Period Domains and Mumford-Tate Domains....................................................................................45II.A Period domains and their compact duals...................................................................................45II.B Mumford-Tate domains and their compact duals.............................................................................55II.C Noether-Lefschetz loci in period domains.................................................................................61III The Mumford-Tate Group of a Variation of Hodge Structure..................................................................67III.A The structure theorem for variations of Hodge structures................................................................69III.B An application of Mumford-Tate groups...................................................................................78III.C Noether-Lefschetz loci and variations of Hodge structure................................................................81IV Hodge Representations and Hodge Domains....................................................................................85IV.A Part I: Hodge representations............................................................................................86IV.B The adjoint representation and characterization of which weights give faithful Hodge representations.....................109IV.C Examples: The classical groups...........................................................................................117IV.D Examples: The exceptional groups.........................................................................................126IV.E Characterization of Mumford-Tate groups..................................................................................132IV.F Hodge domains............................................................................................................149IV.G Mumford-Tate domains as particular homogeneous complex manifolds.........................................................168Appendix: Notation from the structure theory of semi-simple Lie algebras......................................................179V Hodge Structures with Complex Multiplication................................................................................187V.A Oriented number fields....................................................................................................189V.B Hodge structures with special endomorphisms...............................................................................193V.C A categorical equivalence.................................................................................................196V.D Polarization and Mumford-Tate groups......................................................................................198V.E An extended example.......................................................................................................202V.F Proofs of Propositions V.D.4 and V.D.5 in the Galois case.................................................................209VI Arithmetic Aspects of Mumford-Tate Domains.................................................................................213VI.A Groups stabilizing subsets of D..........................................................................................215VI.B Decomposition of Noether-Lefschetz into Hodge orientations...............................................................219VI.C Weyl groups and permutations of Hodge orientations.......................................................................231VI.D Galois groups and fields of definition...................................................................................234Appendix: CM points in unitary Mumford-Tate domains...........................................................................239VII Classification of Mumford-Tate Subdomains.................................................................................240VII.A A general algorithm.....................................................................................................240VII.B Classification of some CM-Hodge structures..............................................................................243VII.C Determination of sub-Hodge-Lie-algebras.................................................................................246VII.D Existence of domains of type IV(f)......................................................................................251VII.E Characterization of domains of type IV(a) and IV(f).....................................................................253VII.F Completion of the classification for weight 3...........................................................................256VII.G The weight 1 case.......................................................................................................260VII.H Algebro-geometric examples for the Noether-Lefschetzlocus types.........................................................265VIII Arithmetic of Period Maps of Geometric Origin............................................................................269VIII.A Behavior of fields of definition under the period map — image and preimage.......................................270VIII.B Existence and density of CM points in motivic VHS......................................................................275Bibliography..................................................................................................................277Index.........................................................................................................................287

Chapter One

I.A HODGE STRUCTURES

Let V be a Q-vector space. For k = R or C we set V_k = V [cross product] k and GL(V)_k = GL(V_k). There are three definitions of a Hodge structure of weight n, here given in historical order. In the first two definitions, we assume that n is positive and the p, q's in the definitions are non-negative. In the third definition, which will be the one used primarily in this monograph, n and p, q are arbitrary integers.

Definition (i): A Hodge structure of weight n is given by a Hodge decomposition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition (ii): A Hodge structure of weight n is given by a Hodge filtration

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These are equivalent by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the third definition we shall use the Deligne torus

S =: Res_C/R G_m,C

where "Res" denotes the restriction of scalars à la Weil. A concrete, and in some ways more useful, description of S will be given below. It is an R-algebraic group whose real points are

S(R) [congruent to] C^*.

Definition (iii): A Hodge structure of weight n is given by a homomorphism of R-algebraic groups

(I.A.1) [??] : S(R) → GL(V)(R)

such that for r [member of] R^* [subset] S(R)

[??](r) = rⁿ id_V.

This is equivalent to definition (i) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In other words, the action of [??](z) on V_C decomposes into eigenspaces V^p,q as above, and this action is defined over R and is therefore given by a homomorphism (I.A.1). The Weil operator C is defined by

ITLITL = [??](i)

and thus C = i^p-q on V^p,q.

The alternate way one may formulate definition (iii) is the following: For k = Q, R or C set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and U(R) [congruent to] S¹ is the maximal compact subgroup of S(R). We set

φ = [??]|_U(R).

Then (I.A.1) gives

(I.A.2) φ : U(R) → SL(V)(R)

where

φ(z)v = z^p-qv; v [member of] V^p,q.

This formulation, which is most useful when studying polarized Hodge structures, determines the Hodge decomposition but not the weight, for which one needs the scaling factor.

The Tate structure Q(1) is defined by V = 2πiQ [subset] C and

[??](z) = z^-1[bar.z]^-1.

Thus V is of pure Hodge type (-1, -1). It is also sometimes referred to as the Hodge-Tate structure. We denote by Q(p) the p^th tensor power of Q(1) and note that Q(-1) [congruent to] Q(1).

In this paper we shall use both the (I.A.1) and (I.A.2) versions of definition (iii), and therefore shall denote a Hodge structure by (V, [??]) and (V, φ) or, when no confusion is likely, simply by V_[??] and V_φ.

Remark: In the literature there are two sign conventions concerning the weight. Ours is this: For a complex torus X = C^g/[conjunction] where C^g has coordinates z¹, ..., z^g and [conjunction] [congruent to] = Z^2g is a lattice, a differential form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defines a class in Hⁿ_DR(X) [congruent to] Hⁿ(X,C). Under a homothety zⁱ → λzⁱ, λ [member of] R^*, we have

Ψ → λⁿΨ.

Thus, we will say that the Hodge structure on Hⁿ(X,Q) given by specifying that Hⁿ(X,C)^(p,q) is represented by Ψ as above where |I| = p, |J| = q has weight n. In [Mo1] the weight, as defined there and following Deligne's convention [DMOS], is -n.

For later use it will be convenient to extend the action of φ to S(C) [congruent to] C^* × C^* as given above where

[??](z,w) = z^pw^q on V^p,q.

This agrees with the above via S(R) [subset] S(C) given by z → (z, [bar.z]). Note that U(C) [congruent to] C^*, and the circle U(R) [congruent to] S¹ in particular maps into S(C) by z → (z, z^-1). The diagonal inclusion G_m [subset] S maps C^* [congruent to] G_m(C) (and R^* [congruent to] G_m(R)) into C^* × C^* by α [??] (α,α).

Hodge structures of weight n are sometimes called pure Hodge structures, and the term Hodge structure then refers to a direct sum of pure Hodge structures.

Definition: A Hodge structure is given by a Q-vector space and a representation

[??] : S(R) → GL(V_R)

with the following property: the restriction of [??] to G_m should be a morphism of Q-algebraic groups, yielding a homomorphism of rational points

[??] : Q^* → GL(V).

Here, G_m [congruent to] Q^* is the subgroup [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of S(Q). Over Q we have the weight space decomposition

V = [direct sum]V⁽ⁿ⁾

where [??](r) = rⁿ on V⁽ⁿ⁾, and over C we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where φ(z) = z^p[bar.z]^q, p + q = n, on V_C^(n)p,q

Hodge structures admit the standard operations of linear algebra, in particular [cross product] and Hom. We denote by [??] the dual vector space of V and denote by [??]_φ the Hodge structure of weight -n induced on [??] by a weight n Hodge structure V_φ.

A sub-Hodge structure is given by a Q-vector subspace U [subset]_V such that

[??](S(R)) : U(R) → U(R);

i.e., the action of [??](S(R)) leaves U(R) invariant. When one decomposes V into a direct sum of pure Hodge structures, there will be a corresponding induced decomposition of U as a direct sum of pure sub-Hodge structures.

Polarizations are most conveniently defined using definition (ii). Let

Q : V [cross product] V → Q

be a non-degenerate form with

Q(u, v) = (-1)ⁿQ(v, u) u, v [member of] V.

We shall denote by G = Aut(V,Q) the Q-algebraic group associated to (V,Q), and by G(R),G(C) the real and complex points of G.

Definition: A polarized Hodge structure of weight n is given by (V, Q, φ) where φ is as in (I.A.2) and where the Hodge-Riemann bilinear relations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are satisfied.

The bilinear relations are equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this work we shall mainly but not exclusively be concerned with polarized Hodge structures, which is why we shall frequently work with φ instead of [??].

When working with Hodge structures arising from geometry there will usually be given a lattice H_Z [subset] H_R such that H = H_Z [cross product] Q.

I.B MUMFORD-TATE GROUPS

Mumford-Tate groups, sometimes abbreviated MT-groups, are the natural symmetry groups that encode information about the Q-structure and the Hodge structure.

Definitions: (i) The Mumford-Tate group M_[??] associated to a Hodge structure (V, [??]) is the Q-algebraic closure of

[??] : S(R) → GL(V_R).

(ii) The Mumford-Tate group M_φ associated to a Hodge structure (V, φ) of weight n is the Q-algebraic closure of

φ : U(R) → SL(V_R).

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and in each case the Mumford-Tate groups are the intersection of the Q-algebraic subgroups of GL(V) whose real points contain [??](S(R)), respectively φ(U(R)). In the literature, M_φ is usually called the special Mumford-Tate group or "Hodge group." Because of the centrality of M_φ in this monograph, and because the term "Hodge group" will be used in Chapter IV in another context, we will simply refer to both M_[??] and M_φ as Mumford-Tate groups and let the subscripts e' and ' designate which one we are referring to.

We may define M_[??] and M_φ for a general Hodge structure [??] : S(R) → GL(V_R) (i.e., a Q-direct sum of pure weight Hodge structures) in exactly the same way. As long as (V, [??]) is not pure of weight zero, the Q-closure definition implies that

M_[??] is the semi-direct product of its subgroups M_φ and G_m,Q.

If (V, Q, φ) is a weight n polarized Hodge structure, then φ preserves Q [member of] [??] [cross product] [??] and thus M_φ [subset] G = Aut(V,Q).

We shall now formulate and prove the basic property for M_φ in the pure case; subsequently we shall do the same for M_[??]. In each case the basic property is an answer to the question:

What are the defining equations for the Q-algebraic group M_φ, respectively M_[??]?

For this we let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

be the tensor spaces and tensor algebra of V. Recall that for a pure Hodge structure of even weight n = 2p, the Hodge classes are defined by

Hg(V_φ) = V [intersection] V^p,p.

Setting Hg^k,l_φ = Hg(T^k,l_φ), we then have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

consisting of the Hodge classes in T^k,l and the algebra of Hodge tensors in T^[??],[??].

(I.B.1) BASIC PROPERTY (I): M_φ is the subgroup of G fixing Hg^[??],[??]_φ.

This result provides our answer to the previous question. The proof will be given in several steps.

Step one: If t [member of] HgITL^[k,l]ITL_φ, then M_φ fixes t.

Proof. Since t is rational, fixing it defines a Q-algebraic subgroup G(t) of GL(V). If t is of Hodge type (p, p), where n(k-l) = 2p, then φ(z)t = z^p-pt. It follows that

φ(U(R)) [subset or equal to] G(t)(R),

and by the minimality of M_φ we conclude that M_φ [subset] G(t).

Step two: If M_φ stabilizes the line Qt spanned by t [member of] T^k,l, then t [member of] Hg^k,l_φ is a Hodge tensor.

Proof. Since φ(U(C)) [subset] M_φ(C), if M_φ stabilizes Qt, then t will be an eigenvector for φ(z) for a general z [member of] U(R). Hence t is of pure Hodge type, and since it is rational it must be a Hodge tensor.

We then have

Chevalley's theorem: Let M be a closed Q-algebraic subgroup of GL(V). Then M is the stabilizer of a line L in a finite direct sum [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A proof of this result will be given below.

Completion of the proof of (I.B.1). Denoting by Fix(*) the Q-algebraic subgroup of GL(V) defined by fixing pointwise a set * of tensors in T^[??],[??], by step one we have

M_φ [subset or equal to] Fix(Hg^[??],[??]_φ).

For the reverse inclusion, by Chevalley's theorem there exists

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that M_φ [subset] GL(V) is defined by stabilizing the line Qτ. In particular, M_φ stabilizes each line [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and by step two each t_i is Hodge. Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of chevalley's theorem. From the open embedding GL(V) [??] End(V) of algebraic varieties comes the injection of coordinate rings

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The action of GL(V) on itself by conjugation extends to the adjoint action on End(V), and these induce compatible actions on the coordinate rings that, moreover, are compatible with the action of GL(V) on [direct sum]T^k,k. Write S := Q[End(V)]. If we choose a basis for V, we may think of S as polynomials P(Xⁱ_j) in the matrix entries of X [member of] End(V).

The stabilizer of the ideal

I(M) [subset or equal to] Q[GL(V)]

of M, viewed as a subvariety of GL(V), is the largest algebraic subgroup contained in the Zariski closure of M. But since M is Zariski closed, and a subgroup, the stabilizer is just M. Note that S and

I := I(M) [intersection] S

inherit a nonnegative GL(V)-invariant grading from the above injection of coordinate rings, and M is also the subgroup of GL(V) stabilizing I in S. This follows from the above because Q[GL(V)] = Q[End(V)][1/det] and det is non-vanishing on M.

Since M is an algebraic variety in GL(V), I(M) is finitely generated; the same goes for I, the ideal of the Zariski closure of M in End(V). Let {P_i} [subset] I^≤k be a generating set for I, and set d := dim(I^≤k). Since I^≤k generates I and the action is compatible with products, M is the stabilizer of I^≤k in S^≤k, and this is, by linear algebra, the same as the stabilizer of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark: The basic idea behind this argument is very simple:

(i) If M is a Q-algebraic group acting on a finite dimensional Q-vector space W, then stabilizing a subspace U [subset] W is the same as stabilizing the line [conjunction]^dU [subset] [conjunction]^dW where dim U = d;

(ii) In the coordinate ring Q[Xⁱ_j], M is exactly the stabilizer of the ideal that defines M;

(iii) By finite generation, (ii) will hold on a finite dimensional subspace of Q[Xⁱ_j].

(Continues...)

Excerpted from Mumford-Tate Groups and Domainsby Mark Green Phillip Griffiths Matt Kerr Copyright © 2012 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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