Table Of ContentLecture Notes in Mathematics 900
Editors:
A. Dold, Heidelberg
B. Eckmann,Zurich
Pierre Deligne James S. Milne
Arthur Ogus Kuang-yen Shih
Hodge Cycles, Motives,
and Shimura Varieties
Springer-Verlag
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Authors
PierreDeligne
InstituteforAdvancedStudy
School ofMathematics
Princeton, NJ08540, USA
James S. Milne
MathematicsDepartment, UniversityofMichigan
AnnArbor,MI48109, USA
ArthurOgus
MathematicsDepartment, UniversityofCalifornia
Berkeley, CA94720, USA
Kuang-yen Shih
3511 W. 229thStreet
Torrance, CA90505, USA
1stEdition 1982
2nd, CorrectedPrinting 1989
ISBN3-540-11174-3 Springer-Verlag Berlin Heidelberg NewYork
ISBN3-540-11174-3 Springer-Verlag NewYork Berlin Heidelberg
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PREFACE
This volume collects six related articles.
The first is the notes (written by J.S. Milne) of a major
part of the seminar "Periodes des Int~gralesAbeliennes" given
by P. Deligne at I'.B.E.S., 1978-79.
The second article was written for this volume (by P.
Deligne and J.S. Milne) and is largely based on: N Saavedra
Rivano, Categories tannakiennes, Lecture Notes in Math. 265,
Springer, Heidelberg 1972.
The third article is a slight expansion of part of:
J.S. Milne and Kuang-yen Shih, Sh~ura varieties: conjugates
and the action of complex conjugation 154 pp. (Unpublished
manuscript, October 1979).
The fourth article is based on a letter from P. De1igne
to R. Langlands, dated 10th April, 1979, and was revised and
completed (by De1igne) in July, 1981.
The fifth article is a slight revision of another section
of the manuscript of Milne and Shih referred to above.
The sixth article, by A. Ogus, dates from July, 1980.
P. Deligne
J.S. Milne
A. Ogus
Kuanq-yen Shih
Note to the second printing 1989
For this printing an Addendum and Erratum were added (pp. 415,
(16).
CONTENTS
Introduction • . • • • • • • • • • • . • • • • • • • • • •
Notations and conventions 8
I P. Deligne(Notes by J. Milne): Hodge cycles on abelian
varieties • • • • · • • • • . . . • 9
II P. Deligne, J. Milne: Tannakian categories. 101
III J. Milne, K.-y. Shih: Langlands' construction of the
Taniyama group • • • • • • • • • • • • • • • 229
IV P. Deligne: Motifs et groupe de Taniyama 261
V J. Milne and K.-y. Shih: Conjugates of Sh~ura varieties. 280
VI A. Ogus: Hodge cycles and crystalline cohomology • • • •• 357
CONTENTS
Introduction • . • • • • • • • • • • . • • • • • • • • • •
Notations and conventions 8
I P. Deligne(Notes by J. Milne): Hodge cycles on abelian
varieties • • • • · • • • • . . . • 9
II P. Deligne, J. Milne: Tannakian categories. 101
III J. Milne, K.-y. Shih: Langlands' construction of the
Taniyama group • • • • • • • • • • • • • • • 229
IV P. Deligne: Motifs et groupe de Taniyama 261
V J. Milne and K.-y. Shih: Conjugates of Sh~ura varieties. 280
VI A. Ogus: Hodge cycles and crystalline cohomology • • • •• 357
General Introduction
Let X be a smooth projective variety over ~. Hodge
co~jectured that certain cohomology classes on X are algebraic.
The work of Deligne that is described in the first article of
this volume shows that, when X is an abelian variety, the
classes considered by Hodge have many of the properties of
algebraic classes.
In more detail, let xan be the complex analytic manifold
associated with X, and consider the singular cohomology groups
Hn{Xan,m> • The variety Xan being of Kahler type (any p~ojective
embedding defines a Kahler structure>, its cohomology groups
Hn{Xan,~) =Hn(Xan,;) ~ ~ have a canonical decomposition
The cohomology class c~(Z) e H2p(Xan,~) of an algebraic
subvariety Z of codimension p in X is rational (it lies
in H2p(Xan,W» and of bidegree (p,p) (it lies in HP,P) .
The Hodge conjecture states that, conversely, any element of
H2p(X,;}A HP,P is a linear combination over m of the classes
of algebraic subvarieties. Since the conjecture is unproven,
it is convenient to call these rational (p,p)-classes
Hodge cycles on X.
Now consider a smooth projective variety X over a field
k that is of characteristic zero, algebraically closed, and
small enough to be embeddable in ~. The algebraic de Rham
2
n
cohomology groups H (X/k) have the property that, for any
DR
embedding a: k ~ ~ , there is a canonical isomorphism
Hn (X/k) ~ ~ ~ Hn (Xan) = Hn(Xan ~) It is natural to
DR k,a DR ' •
say that t e H~~(X/k) is a Hodge cycle on X relative to a
if its image in H2p(Xan,~) is (2~i)P times a Hodge cycle
on X 8 k,a ~. Deligne's results show that, if X is an
2p
abelian variety, then an element of H (X/k) that is a
DR
Hodge cycle on X relative to one embedding of k in ~ is
a Hodge cycle relative to all embeddingsi further, for any
t~es H2p(Xan,~)
embedding, (2ni)P a Hodge cycle in always
2p
lies in the image of H (X/k) Thus the notion of a Hodge
DR
cycle on an abelian variety is intrinsic to the variety: it
is a purely algebraic notion. In the case that k = ~ the
theorem shows that the image of a Hodge cycle under an automorphism
of ~ is again a Hodge cycle; equivalently, the notion of a
Hodge cycle on an abelian variety X over ~ does not depend
on the map X ~ spec ~. Of course, all of this would be
obvious if only one knew the Hodge conjecture.
In fact, in the first article a stronger result is proved
in which a Hodge cycle is defined to be an element of
H~R(X) x ~ Hn (X ,m ) • As the title of the original seminar
et 1
suggests, the stronger result has consequences for the algebraicity
of the periods of abelian integrals: briefly, Deligne's result
allows one to prove all arithmetic properties of abelian
periods that would follow from knowing the Hodge conjecture
for abelian varieties.
3
The second article is mainly expository. Since Tannakian
categories are used in several articles, we thought it useful
to include an account of the essential features of the theory.
The exposition largely follows that of Saavedra [lJ except at
three places: in §3 we point out an error in Saavedra's results
concerning a non-neutral Tannakian category; in §4 we eliminate
an unnecessary connectedness assumption in the theory of
polarized Tannakian categories; and in §6 we discuss motives
relative to absolute Hodge cycles rather than algebraic cycles.
A neutralized Tannakian category is a k-linear category
C with an operation 8: C x C ~ C and a functor to finite
dimensional vector spaces over k satisfying certain conditions
sufficient to ensure that C is equivalent to the category of
finite-dimensional representations of an affine group scheme
Gover k. The importance of this notion is that properties
of an abstract category C will be faithfully reflected in
properties of the associated group G.
The category of polarizable ~-rational Hodge structures
is Tannakian. The group associated to its Tannakian subcategory
of those Hodge structures of eM-type is called the connected
Serre group So.
It would follow from Grothendieck's standard conjectures
that the category of motives, arising from the category of
projective smooth varieties over a field k , is Tannakian.
If, in the definition of motive, "algebraic cycle" is replaced
by "Hodge cycle" and the initial category is taken to consist
of abelian varieties over a field k of characteristic zero,
4
then the resulting category of Hodge motives is Tannakian.
(Of course, for this to make sense, one needs Deligne's Theorem.)
If the initial category is taken to be all abelian varieties
over ~ of CM-type, then the group associated with the
category of motives is again SO i if the initial category
is taken to be all abelian varieties over m that become of
CM-type over ~, then the associated group is called the
Serre group S. The identity component of S is SO , and
S is an extension of Gal(~/m) by So. There is a canonical
f ~
continuous splitting S(lA ) Ga1(m/(D) over the ring of
f
finite ade1es lA •
The third article is again largely expository: it
describes Langlands's construction of his Taniyama group.
Langlands's study of the zeta functions of Shimura varieties
led him to make a conjecture concerning the conjugates of a
Shimura variety (Langlands [1, p 417]). In the belief that
this conjecture was too imprecise to be proved by the methods
usually applied to Shimura varieties, he then made a second,
stronger conjecture (Langlands [2, p 232-33]). This second
conjecture is stated in terms of the Taniyama group T which,
like the Serre group, is an extension of Gal(m/m) by SO
together with a continuous splitting over ~f. In the
following two articles, this conjecture is proved for most
Shimura varieties, viz. for those of abelian type (see
article V.I for a definition of this term).
In the first of the two articles, Deligne proves that
the Serre group, together with its structure as an extension
