The space of Hodge cycles of the general Prym variety is proved to be
generated by its Neron-Severi group.Comment: LaTeX-fil

Biswas, Indranil

Paranjape, Kapil H.

English

arXiv

The space of Hodge cycles of the general Prym variety is proved to be generated by its Neron-Severi group

Publications of the IAS Fellows

ar
X
iv
:m
at
h/
00
07
19
2v
1 
 [m
ath
.A
G]
  3
1 J
ul 
20
00
THE HODGE CONJECTURE FOR GENERAL PRYM
VARIETIES
INDRANIL BISWAS AND KAPIL H. PARANJAPE
Introduction
We work over C, the field of complex numbers.
The Prym variety of a double cover C → D of a smooth connected
projective curve D by a smooth connected curve C is defined (see [7])
as the identity component of the kernel of the norm homomorphism
N : J(C) → J(D) between the Jacobians of the curves. This is an
abelian variety polarised by the restriction of the canonical polarisation
on J(C); we denote this variety by P (C → D) or simply P when there
is no possibility of ambiguity.
A Hodge class on a variety X is an integral singular cohomology class
on the complex manifold X(C) which is represented by a closed differ-
ential form of type (p, p). The Hodge conjecture (see [3]) asserts that
some multiple of such a class is the cohomology class of an algebraic
cycle on X.
Let A be an abelian variety. The Ku¨nneth decomposition implies
that the rational singular cohomology of A × · · · × A is a direct sum
of subquotients of tensor products of H1(A(C),Q). Hence we have an
action of a linear automorphism of this vector space on these cohomol-
ogy groups. The Mumford-Tate group H(A) of A can thus be defined
(see [2]) as the group of all linear automorphisms of H1(A(C),Q) which
stabilise all Hodge cycles on the varieties A× · · · ×A.
The aim of this note is to show that the Mumford-Tate group H(P )
of a general Prym variety P (C → D) is isomorphic to the full sym-
plectic group Sp(2g); where the class in
2
∧H1(P (C),Q) = H2(P (C),Q)
which is stabilised by this group is the first Chern class of the natural
polarisation on the Prym variety. Invariant theory (see [9] or [5] and
[6]) then implies that the only Hodge cycles on P are powers (under
cup-product) of this polarisation class. In particular, we obtain the
Hodge conjecture for P as a consequence of this result.
As a particular case the Ne´ron-Severi group of a general Prym variety
is Z. This was proved earlier by Pirola (see [8]). We do not give a new
proof of that result and use it in an essential way to prove our result.
1
2 I. BISWAS AND K. H. PARANJAPE
The outline of the paper is as follows. In section 1 we set out some
standard arguments about Mumford-Tate groups in families. In sec-
tion 2 we use an extension (due to Beauville [1]) of the definition of
Prym varieties to the case where C and D are singular curves. The
results on Mumford-Tate groups are applied to this larger family of
Prym varieties in section three. In addition we use the semi-simplicity
of the Mumford-Tate group (see [2]) and the result of Pirola (see [8])
to reduce the problem to an elementary lemma on subgroups of the
symplectic group.
1. Mumford-Tate groups in families
Let f : X → S be a family of smooth projective varieties parametrised
by a smooth connected variety S. For some positive integer k let
V = Rif∗QX denote the variation of pure Hodge structures of weight
k on S. More generally we can consider any variation V of Hodge
structures of weight k on S.
Let V a,b = V ⊗a ⊗ V ∗⊗b be the associated tensor variations of pure
Hodge structures of weight (a−b)k. For every (a, b) such that (a−b)k =
2p is even, we have the nested sequence of analytic subvarieties
Ha,b := V a,b
Z
∩ F pV a,b ⊂ V
a,b
Z
⊂ V
a,b
C
of the complex vector bundle V a,b
C
over S associated with V a,b. The
analytic variety Ha,b parametrises pairs (s, c), where s is a point of S
and c an integral class of type (p, p) in V a,bs ; i. e. c is a Hodge cycle.
If W is an irreducible component of Ha,b such that the natural map
W → S is open at some point, thenW contains an open subset of V a,b
Z
;
hence W is a connected component of V a,b
Z
. Let Aa,b be the the union
of all such components. The map Aa,b → S makes each component of
the former a covering space of S.
Now, if W is an irreducible component of Ha,b for which the map
W → S is not open at any point then its image in S is a set of measure
zero by Sard’s theorem. Let B be the (countable) union of these images
as we vary over all the components of Ha,b and as we vary a and b.
If s is any point of S which is not in B, then by the above reasoning,
the only points of Ha,b that lie over it are in Aa,b. Let t be any other
point of S and γ be a path in S connecting s and t. We can use γ
to identify V a,b
Z,s with V
a,b
Z,t ; this then gives an identification of A
a,b
s with
A
a,b
t . Hence, under this identification, the collection of Hodge cycles in
V
a,b
Z,s is contained in the collection of Hodge cycles in V
a,b
Z,t . Thus the
Mumford-Tate group Gt of Vt is identified by γ with a subgroup of the
Mumford-Tate group Gs of Vs. In other words, we have
THE HODGE CONJECTURE FOR GENERAL PRYM VARIETIES 3
The Mumford-Tate group at a general point contains (a con-
jugate of) the Mumford-Tate group at a special point in a
variation of Hodge structures over a smooth connected vari-
ety.
2. Degenerate covers
A connected projective curve which has at worst ordinary double
points as its singularities is called a semi-stable curve. The dual graph
of such a curve has as its vertices the irreducible components; each
singular point gives an edge incident on the two vertices corresponding
to the components that contain it. We will be interested in semi-stable
curves whose dual graph is contractible and hence a tree; such curves
are called tree-like. In this case, the first cohomology of the curve is a
direct sum of the first cohomology of its components with the induced
(pure) Hodge structure. In particular, the Jacobian of a tree-like semi-
stable curve is the product of the Jacobians of its components.
A finite morphism C → D of semi-stable curves is called a semi-
stable cover (or an admissible cover) if
1. This is a topological cover of constant degree of D outside a finite
set of points which includes the singular locus of D.
2. The inverse image of a singular point of D consists of singular
points of C.
3. The order of ramification on the two branches at a singular point
of C must be equal.
This notion was first defined by Beauville [1] for the case of degree two
covers (which are the case of interest) and later generalised (see [4]).
In these papers, it is shown that the deformations of such a semi-stable
cover of tree-like curves are unobstructed. In other words, there is a
smooth (open) curve S, a flat morphism p : D → S and a finite flat
morphism f : C → D. There is a point o of S over which the f restricts
to the given semi-stable cover C → D. Moreover, the general fibre is a
double cover C ′ → D′ of a smooth curve D′ by a smooth curve C ′.
We are interested in the case of degree two covers C → D; here the
singular points of C are either unramified on each branch or ramified
of order two on each branch. Let us further assume that C and D are
tree-like. For each component Di of D there are two possibilities:
1. There is exactly one component Ci of C that lies over it. The
map Ci → Di is a double cover in the usual sense.
2. There are two components C ′i and C
′′
i of C that lie over Di and
the given map is an isomorphism between these components and
Di.
4 I. BISWAS AND K. H. PARANJAPE
The Prym variety can be defined as before as the identity component of
the kernel of the natural norm homomorphism between the Jacobians
J(C) → J(D). It follows that the Prym variety is the product of the
Prym varieties of the covers Ci → Di corresponding to the first case
and the Jacobians of the curves Di corresponding to the second case.
In particular, the product of these components gives an abelian variety.
Hence we have
The family of Prym varieties can be extended to include the
Prym varieties of degenerate tree-like covers. In particular,
the Mumford-Tate group of a general Prym variety contains
(a conjugate of) the Mumford-Tate group of the Prym vari-
ety of any degenerate tree-like cover.
In the special case when D has two exactly components (call them
D1 and D2), such a cover can be constructed in one of two ways:
I Let C1 → D1 be a double cover that is not branched at the com-
mon point p = D1 ∩D2. Then, C is obtained by attaching to C1
two copies of D2, one at each point lying over p.
II Let C1 → D1 and C2 → D2 be double covers that are both
branched at the common point p. We obtain C by attaching
the curves C1 and C2 along their respective ramification points
lying over p.
The specific covers that we are interested in are the following.
1. A covering of type (II) which is the degeneration of a double cover
C → D where D is rational and C is of genus g. The curves Di
are smooth rational curves. The curve C1 is an elliptic curve and
the curve C2 is any hyperelliptic curve of genus g − 1.
2. A covering of type (I) which is the degeneration of a double cover
C → D where D has genus at least 2 and the cover is e´tale. The
curve D1 is any elliptic curve, C1 → D1 is an e´tale double cover
and D2 is any curve of genus one less than that of D.
3. A covering of type (I) which is the degeneration of a double cover
C → D where D has genus at least 1 and the cover is ramified
at some point. The curve D1 is any curve of genus one less than
that of D and C1 → D1 is a double cover ramified at the same
number of points as the cover C → D; D2 is any elliptic curve.
As a result we have
Lemma 1. We have containments of Mumford-Tate groups as enu-
merated below.
1. The Mumford-Tate group of a general hyperelliptic curve of genus
g contains a conjugate of the product of the Mumford-Tate group
THE HODGE CONJECTURE FOR GENERAL PRYM VARIETIES 5
of any elliptic curve with the Mumford-Tate group of any hyper-
elliptic curve of genus g − 1.
2. The Mumford-Tate group of the Prym variety of a general e´tale
cover of a curve of genus g ≥ 2 contains a conjugate of the
Mumford-Tate group of any curve of genus g − 1.
3. The Mumford-Tate group of the Prym variety of a general cover of
a curve of genus ≥ 1 ramified at r ≥ 1 points contains a conjugate
of the product of the Mumford-Tate group of any elliptic curve
with the Mumford-Tate group of the Prym variety of any cover of
a curve of genus g − 1 which is ramified at r points.
Proof. The first cohomology group of the product of two abelian vari-
eties is the direct sum of the first cohomology groups of the individual
abelian varieties. Moreover, the Hodge cycles on the individual vari-
eties pull-back to give Hodge cycles on the product. Thus it follows
that the Mumford-Tate group of the product contains the product of
the Mumford-Tate groups. The result now follows from the above con-
structions.
3. The Main result
To prove the main result we need the following three lemmas.
Lemma 2 (Pirola). The Ne´ron-Severi group of a general Prym variety
is the free group on 1 generator.
This lemma is proved in [8]. We note that this case includes the case
of a general hyperelliptic curve.
Lemma 3. Let G be a connected semi-simple subgroup of the symplec-
tic group Sp(2n) which contains (a conjugate of) the product Sp(2a)×
Sp(2n−2a), then G is either this product or it is the the full symplectic
group
Proof. Let V be the standard representation of Sp(2n). Let ⊕i∈IWi
be its decomposition into isotypical components as a representation of
G. Let V = V1 ⊕ V2 be the decomposition of V as a representation of
Sp(2a)×Sp(2n− 2a). Then each Wi is either V1 or V2 or V = V1⊕V2.
The result follows by dimension counting.
The lemma also follows from the fact that the quotient
sp(2n)
sp(2a)× sp(2n− 2a)
of the Lie algebras is an irreducible module over Sp(2a) × Sp(2n −
2a).
6 I. BISWAS AND K. H. PARANJAPE
Lemma 4. Let A be an abelian variety of dimension n whose Mumford-
Tate group is the product Sp(2a) × Sp(2n − 2a) in Sp(2n). Then the
Ne´ron-Severi group of A is of rank at least 2.
Proof. The first cohomology group of A decomposes as a direct sum of
two (polarised) sub-Hodge structures. It follows that A is the product
of two abelian subvarieties. Hence we have the result.
Theorem 5. The Mumford-Tate group of a general Prym variety is
the full symplectic group.
Proof. We begin with the case where the base curve has genus zero.
In this case the Prym varieties are the Jacobians of the corresponding
hyperelliptic double cover. The result is classical for elliptic curves
which can be considered as the Prym varieties associated with double
covers of smooth rational curves branched at 4 points. By induction, let
us assume that the result is known for hyperelliptic Jacobians of genus
less than g ≥ 2. The lemma 1 then shows that the Mumford-Tate group
of a general hyperelliptic curve of genus g contains Sp(2)×Sp(2g− 2).
By the above results we see that thus Mumford-Tate group must be
either Sp(2g) or Sp(2)×Sp(2g−2). In the latter case, the Ne´ron-Severi
group of the curve would have rank at least two but by Pirola’s result
we know that this is not true for the general hyperelliptic curve. Hence
we see that the Mumford-Tate group of a general hyperelliptic curve
must be Sp(2g) where g is the genus of the curve.
Now let us consider the case where the cover is unramified. Then
we may assume that the base curve of the double cover has genus g at
least 2 (else the Prym variety is just a point). In this case the Prym
variety has dimension n = g − 1. By the lemma 1 we know that the
Mumford-Tate group contains the Mumford-Tate group of any curve of
genus g − 1. In particular, it contains the Mumford-Tate group of any
hyperelliptic curve of genus g−1 and hence by the previous paragraph
it contains (and is thus equal to) Sp(2n).
Now assume that the base curve of the double cover has genus g at
least 1 and the cover is ramified. We argue by induction on the genus
of the base curve. We can begin the induction since we already know
the result for the hyperelliptic curves. Let us assume that the result is
known for base curves of genus less than g. By the lemma 1 we know
that the Mumford-Tate group contains the product of the Mumford-
Tate group of an elliptic curve with the Mumford-Tate group of the
Prym variety of a the double of a curve of genus g − 1; in other words
it contains Sp(2) × Sp(2n − 2) by induction. Now as argued above,
THE HODGE CONJECTURE FOR GENERAL PRYM VARIETIES 7
the three lemmas above imply that the Mumford-Tate group must be
Sp(2n).
References
[1] Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41
(1977), no. 2, 149–196.
[2] Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cy-
cles, motives, and Shimura varieties, Lecture Notes in Mathematics, no. 900,
Springer-Verlag, Berlin, 1982.
[3] A. Grothendieck, Hodge’s general conjecture is false for trivial reasons, Topology
8 (1969), 299–303.
[4] Joe Harris and David Mumford, On the Kodaira dimension of the moduli space
of curves, Invent. Math. 67 (1982), no. 1, 23–88, With an appendix by William
Fulton.
[5] Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc.
313 (1989), no. 2, 539–570.
[6] , Erratum to: “Remarks on classical invariant theory”, Trans. Amer.
Math. Soc. 318 (1990), no. 2, 823.
[7] David Mumford, Prym varieties. I, (1974), 325–350.
[8] Gian Pietro Pirola, Base number theorem for abelian varieties. An infinitesimal
approach, Math. Ann. 282 (1988), no. 3, 361–368.
[9] Hermann Weyl, The Classical Groups. Their Invariants and Representations,
Princeton University Press, Princeton, N.J., 1939.
School of Maths, TIFR, Homi Bhabha Road, Mumbai 400 005, India.
E-mail address : indranil@math.tifr.res.in
IMSc, CIT Campus, Tharamani, Chennai 600 113, India.
E-mail address : kapil@imsc.ernet.in


The Hodge Conjecture for general Prym varieties

Abstract

Similar works

Full text

Available Versions

Publications of the IAS Fellows