Lagrangian subvarieties of abelian fourfolds

28 Pages·2002·0.27 MB·English

LAGRANGIAN SUBVARIETIES OF ABELIAN
FOURFOLDS
by
Fedor Bogomolov and Yuri Tschinkel
2 FEDOR BOGOMOLOV and YURI TSCHINKEL
Dedicated to Professor K. Kodaira
1. Introduction
Let (W,ω) be a smooth projective algebraic variety of dimension 2n
over C together with a holomorphic (2,0)-form of maximal rank 2n. A
subvariety X ⊂ W is called weakly lagrangian if dimX ≤ n and if the
restriction of ω to X is trivial (notice that X can be singular). An n-
dimensional subvariety X ⊂ W with this property is called lagrangian.
For example, any curve C contained in a K3 or abelian surface S is
lagrangian. Further examples of lagrangian subvarieties are obtained by
taking a curve C ⊂ S and by considering the corresponding symmetric
products. Alternatively, one could look at a product of different curves
(of genus > 1) inside a product of abelian varieties. We will say that a
variety X ⊂ W is fibered if it admits a dominant map onto a curve of
genus > 1. In this note we construct examples of nonfibered lagrangian
surfaces in abelian varieties. Our motivation comes from the following
Problem 1.1. — Findexamplesofprojectivesurfaceswithanontrivial
fundamental group. In particular, find examples where the fundamental
group has a nontrivial nilpotent tower.
If X is fibered over a curve C of genus > 1 then the fundamental group
π (X) surjects onto a subgroup of finite index in π (C) and consequently
1 1
both π (X) and its nilpotent tower are big. Therefore, we are interested
1
in examples of surfaces where the nontriviality of π (X) is not induced
1
from curves. Consider the map to the Albanese variety alb : X →
Alb(X). One is interested in situations where the natural map
H2(Alb(X),C) → H2(X,C)
has a nontrivial kernel – the triviality of the kernel implies the triviality
ofthenilpotenttower(tensorQ). SuchexamplesweregivenbyCampana
([3], Cor. 1.2) and Sommese-Van de Ven ([7]). However, there the kernel
LAGRANGIAN SUBVARIETIES OF ABELIAN FOURFOLDS 3
was found in the map
Pic(Alb(X)) → Pic(X).
In their construction the fundamental group of the variety is a central
extension of an abelian group (and the lower central series has only two
steps). Thelagrangianpropertyofalb(X) ⊂ Alb(X)impliesthatthereis
a nontrivial kernel in H2,0 (rather than on the level of the Picard groups).
We produce an infinite series of surfaces X of different topological types
which are contained in abelian varieties and are lagrangian (or weakly
lagrangian) with respect to a nondegenerate (2,0)-form. The weakly la-
grangian property for a subvariety X of an abelian variety A is related to
the nontriviality of the fundamental group π (X) as follows: the number
1
of linearly independent generators of the second quotient of the central
series of π (X) is bounded from below by the dimension of the rational
1
envelope of the space of those (2,0)-forms on A which restrict trivially
to X.
Our construction uses dominant maps between K3 surfaces. Let S
be a K3 surface and g ,g dominant rational maps of S to Kummer
1 2
K3 surfaces S ,S (blowups of quotients of abelian surfaces A ,A by
1 2 1 2
standard involutions). Then the (birational) preimage X of (g ,g )(S) ⊂
1 2
S × S in A × A is a lagrangian surface (in general, singular). For
1 2 1 2
special choices of g ,g we can compute some basic invariants of X and,
1 2
inparticular,showthatX isnonfibered. Forexample,letAbeanabelian
surface which is not isogenous to a product of elliptic curves and S the
associated Kummer surface. Assume that g : S → S is an isomorphism
1
and g : S → S is not an isomorphism. Then X ⊂ A × A is not
2
fibered, Alb(X) is isogenous to A×A and X is lagrangian with respect
to exactly one 2-form on A×A. We analyze other constructions, with S
an elliptic Kummer surface and g induced from the group law. We don’t
j
determine the actual structure of π (X) (it presumably depends on X),
1
but it seems quite plausible that for some X from our list π (X) has a
1
rather nontrivial nilpotent tower.
Acknowledgments. The first author was partially supported by the
NSF. The second author was partially supported by the NSA. The paper
was motivated by a question raised by F. Catanese. We would like to
4 FEDOR BOGOMOLOV and YURI TSCHINKEL
thank him for useful discussions in the early stages of this work. We
thank the referee for comments which helped to improve the exposition.
2. Preliminaries
Let V be a finite dimensional Q-vector space and V its complexifi-
Q C
cation. Let w ∈ V be a vector. We denote by L(w) ⊂ V its rational
C Q
envelope, i.e., the smallest linear subspace such that w ∈ L(w) . More
C
generally, if W ⊂ V is any set, we will denote by L(W) ⊂ V the small-
C Q
est linear subspace such that L(w) ⊂ L(W) for all w ∈ W. An element
w ∈ V will be called k-generic if the dimension of L(w) is dimV −k.
C Q
Remark 2.1. — The set of rationally defined subspaces in V is count-
C
able. Therefore, for any linear subspace W ⊂ V we have L(W) = L(w)
C
for all w ∈ W which are not contained in a countable number of linear
subspaces.
Proposition 2.2. — Let (A,ω) be an abelian variety of dimension 2n
together with a nondegenerate holomorphic (2,0)-form ω. Assume that
ω defines a k-generic element in H2,0(A,C) with k < 3. Then there are
no weakly lagrangian surfaces in A.
Proof. Assume that we have a weakly lagrangian surface i : X (cid:44)→ A.
Consider the induced Q-rational homomorphism
i∗ : H2(A,Q) → H2(X,Q).
Theclassofω iscontainedinthe(Q-rationallydefined)kernelofi∗. Since
ω is k-generic in H2,0(A,C) for k < 3 the image of H2(A,C) in H2(X,C)
has rank at most 2. If we had a holomorphic form w(cid:48) ∈ H2,0(A,C)
restricting nontrivially to X then the image of H2(A,C) in H2(X,C)
would contain at least three linearly independent forms w(cid:48),w¯(cid:48) and the
image of a polarization from Pic(A). Thus the triviality of ω on X
implies that all (2,0)-forms restrict trivially to X. Consequently, the
dimension of X is ≤ 1, contradiction. (cid:3)
We end this section with a simple description of maps of fibered sur-
faces onto curves (see also [4]).
LAGRANGIAN SUBVARIETIES OF ABELIAN FOURFOLDS 5
Proposition 2.3. — For every smooth projective algebraic surface V
there exists a universal algebraic variety U(V) such that every dominant
rational map V → C onto a smooth curve of genus ≥ 2 factors through
U(V).
Proof. First observe that the set of dominant rational maps of V onto
smoothcurveswithgenericallyirreduciblefibersiscountable. Indeed,the
class f∗(L) (where L is some polarization on the image-curve) defines f.
If two such classes [f] and [f(cid:48)] differ by an element in Pic0(V), then the
maps must be the same. Otherwise, some fiber of f would surject onto
the image of f(cid:48) and therefore, the degrees of the two classes [f] and [f(cid:48)]
on this fiber would be different.
Further, dominant maps onto smooth curves of genus ≥ 2 define a
linear subspace W ⊂ H0(V,Ω1) by the property that wedge-products of
linearly independent forms w,w(cid:48) ∈ W are trivial. The set of maximal
subspaces W in H0(V,Ω1) with this property is an algebraic variety.
Remark that for two forms ω,ω(cid:48) with ω ∧ ω(cid:48) = 0 their ratio ω/ω(cid:48) is
a nonconstant rational function, which is constant on the fibers of the
foliationsonV definedbyω, resp. ω(cid:48). Thus, anysuchsubspaceW defines
afoliationF onV withcompactnonintersectingfibers(locallytheform
W
defines a holomorphic map, its fibers are the fibers of F ). Therefore,
W
F defines a dominant rational map onto a curve C (which lifts to a
W W
map of V onto the normalization of C ). Different W define different
W
foliations and different morphisms (the sum of two spaces W and W(cid:48)
with the same foliations F = F has the same property, contradicting
W W(cid:48)
the maximality of W and W(cid:48)). Since the set of such spaces W is on the
one hand algebraic (a finite union of subvarieties of a Grassmannian) and
on the other hand countable, it must be finite. (cid:3)
Corollary 2.4. — For every smooth projective algebraic surface V
there exists a universal algebraic variety U(V) such that every dominant
rational map V → V(cid:48), where V(cid:48) is a product of smooth curves of genus
≥ 2, factors through U(V).
Proof. We have shown in Lemma 2.3 that there is a finite number of
dominant rational maps onto curves of genus ≥ 2. The product of these
curves and maps is U(V). Universality follows. (cid:3)
6 FEDOR BOGOMOLOV and YURI TSCHINKEL
3. Construction
Let A be an abelian surface and τ the involution τ(a) = −a (for
a ∈ A). This involution acts on A with 16 fixed points. We denote
by S˜ the (singular) quotient A/τ. Let A∗ be the blowup of A in the
16 points. The involution τ extends to a fixed point free action on A∗
and the quotient A∗/τ is a K3 (Kummer) surface S, a blowup of S˜. It
contains 16 exceptional curves and we will denote by ∆ = ∪16 ∆k their
k=1
union. Weshalldenotebyδ : A → S˜thedoublecover. EveryK3surface
has a unique (up to constants) nondegenerate holomorphic (2,0)-form.
We start with two simple abelian surfaces A and A (in particular, A
1 2 j
don’t contain elliptic curves - this assumption simplifies the discussion
in Section 4). We consider the corresponding Kummer surfaces S and
j
we assume that there exists another K3 surface S together with two
dominant rational maps g : S → S . We denote by g˜ the induced
j j j
maps S → S˜ . Now consider the map
j
ϕ˜ = (g˜ ,g˜ ) : S → S˜ ×S˜ .
1 2 1 2
We have two double covers
(δ ,1) : A ×S˜ → S˜ ×S˜
1 1 2 1 2
and
(1,δ ) : S˜ ×A → S˜ ×S˜ .
2 1 2 1 2
Denote by Y˜ = (δ ,1)−1(ϕ˜(S)) and by Y˜ = (1,δ )−1(ϕ˜(S)). Let
1 1 2 2
X˜ = X˜ := (δ ,δ )−1(ϕ˜(S))
ϕ˜ 1 2
be the preimage of S in A ×A . We see that the (2,2)-covering X˜ →
1 2
ϕ˜(S) is
X˜ = Y˜ × Y˜ ⊂ A ×A .
1 ϕ˜(S) 2 1 2
˜
The surface X is, in general, singular.
Lemma 3.1. — There exists a nondegenerate holomorphic 2-form ω(cid:48)
on A ×A such that X˜ is lagrangian with respect to ω(cid:48).
1 2
LAGRANGIAN SUBVARIETIES OF ABELIAN FOURFOLDS 7
˜
Proof. The singular surface S carries a nondegenerate holomorphic
j
2-form, which we again denote by ω . Evidently, g˜∗ω = λ ω for some
j j j j
nonzero numbers λ . Consider the form
j
ω(cid:48) = λ ω −λ ω
2 1 1 2
on S˜ ×S˜ . The form ω(cid:48) is identically zero on the image ϕ˜(S) (since it
1 2
is trivial on the open part of S where ϕ˜ is smooth). Since both forms ω
j
lift to nondegenerate forms on the abelian surfaces A the form ω(cid:48) lifts
j
to a nondegenerate form on A ×A . The restriction of the lift of ω(cid:48) to
1 2
X˜ is identically zero on the smooth points of X˜. (cid:3)
Example 3.2. — If A = A = A and the maps g˜ = id then the
1 2 j
˜
resulting surface X is the abelian surface A, embedded diagonally into
A×A.
Example 3.3. — Let S be aK3 surface, which is simultaneously Kum-
mer and a double cover of P2. Denote by θ the covering involution on
S. Put S = S = S, g˜ = id and g˜ = θ. Then the corresponding
1 2 1 2
˜
surface X is lagrangian. A special case of this construction is obtained
as follows: Let C be a curve of genus 2 and σ a hyperelliptic involution
on C. Consider C ×C, together with the involutions: σ interchanging
12
the factors and
σ : (c ,c ) → (σ(c ),σ(c )).
a 1 2 2 1
Denote by Y = C × C/σ and Y = C × C/σ . We see that both Y
1 12 2 a 1
and Y are isomorphic to a symmetric square of C, which is birational
2
˜
to the same abelian surface A. Denote by S the quotient of A by the
standard involution τ. Observe that S is realized as a double cover of
P2 = P1 ×P1/σ . Then
12
(τ,τ)−1((1,θ)(S)) = C ×C ⊂ A×A.
In the following sections we will show that we can arrange a situation
where X˜ is not contained in any abelian subvariety of A ×A and where
1 2
it does not admit any dominant morphisms onto a curve of genus ≥ 2.
8 FEDOR BOGOMOLOV and YURI TSCHINKEL
4. Details and proofs
˜
4.1. Desingularization. — To analyze the surface X constructed in
Section 3 we will need its sufficiently explicit partial desingularization.
It will be constructed in two steps.
Let Σ be the minimal finite set of points in S such that both maps
g ,g are well defined on the complement S0 = S \Σ. On S0 the maps
1 2
g and g are local isomorphisms. Consider the divisors ∆ ⊂ S . The
1 2 j j
preimages D0 of these divisors (under the maps g ) in S0 are smooth and
j j
thecomponentsofD0 don’tintersectinS0. However, thedivisorsD0 and
j 1
D0 do intersect in S0, and we denote by p ,...,p ∈ S0 their intersection
2 1 l
points. Let D be the closure of D0 in S. We fix a blowup S¯ of S
j j
with centers supported in Σ such that the preimage of the intersection
D ∩ D in the neighborhood of every point in Σ is a normal crossing
1 2
¯ ˜ ¯
divisor. Of course, S has a map to S and we denote by Y the fibered
j j
productS¯× A . The surface Y¯ is adouble cover ofS¯ andit hasatmost
S˜j j j
A1-singularities (double points), since D0 is smooth in S0. Consider the
j
fibered product X¯ = Y¯ × Y¯ . All singular points of X¯ which are not of
1 S¯ 2
type A lie over the intersection points of D0 and D0 in S0. The surface
1 1 2
X¯ has a natural action of Z/2 + Z/2. It admits equivariant surjective
maps δ¯ : X¯ → Y¯ , where Y¯ is the quotient of X¯ by the involution τ¯ .
j j j j
¯ ¯
We denote by Y the quotient of X by τ¯ = τ¯ τ¯ , it is still singular.
12 12 1 2
LetSˆbetheminimalblowupofS¯withsupportinthepointsp ,...,p ∈
1 l
S0 ⊂ S¯ such that proper transforms of the irreducible components D0
1
and D0 are disjoint in the preimage of S0 in Sˆ. Now we define Y as
2 j
ˆ
the induced (from the open part) double covers of S. Their ramification
is contained in the full transforms of the divisors D . Define X as the
j
fiberedproductY × Y . Wehavetheinducedinvolutions(againdenoted
1 Sˆ 2
byτ ,τ )onX andwedefineY asthequotientofX underτ (itadmits
j 12 12 12
ˆ
a map onto S). By construction, the surfaces X, Y and Y all have at
j 12
most A1-singularities. One has surjective regular maps X → X˜,Y → Y˜
j j
etc.
LAGRANGIAN SUBVARIETIES OF ABELIAN FOURFOLDS 9
(cid:119)(cid:119) (cid:119)(cid:119)
(cid:126)(cid:126)(cid:126)τ˜(cid:126)1(cid:126)(cid:126)(cid:126)(cid:126)(cid:126)(cid:126)X˜(cid:15)(cid:15) τ˜1(cid:64)2(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)τ˜(cid:64)2(cid:64)(cid:32)(cid:32) (cid:126)(cid:126)(cid:126)τ¯(cid:126)1(cid:126)(cid:126)(cid:126)(cid:126)(cid:126)(cid:126)X¯(cid:15)(cid:15) τ¯1(cid:64)2(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)τ¯(cid:64)2(cid:64)(cid:32)(cid:32) (cid:126)(cid:126)(cid:126)τ(cid:126)1(cid:126)(cid:126)(cid:126)(cid:126)(cid:126)(cid:126)X(cid:15)(cid:15) τ1(cid:64)2(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)τ(cid:64)2(cid:64)(cid:32)(cid:32)
Y˜ Y˜ Y˜ Y¯ Y¯ Y¯ Y Y Y
1 (cid:64)(cid:64)(cid:64)(cid:64) 12 (cid:126)(cid:126)(cid:126) 2 1 (cid:64)(cid:64)(cid:64)(cid:64) 12 (cid:126)(cid:126)(cid:126) 2 1 (cid:64)(cid:64)(cid:64)(cid:64) 12 (cid:126)(cid:126)(cid:126) 2
(cid:64) (cid:126) (cid:64) (cid:126) (cid:64) (cid:126)
(cid:64) (cid:126) (cid:64) (cid:126) (cid:64) (cid:126)
(cid:64) (cid:126) (cid:64) (cid:126) (cid:64) (cid:126)
(cid:64)(cid:32)(cid:32) (cid:15)(cid:15) (cid:126)(cid:126)(cid:126)(cid:126) (cid:64)(cid:32)(cid:32) (cid:15)(cid:15) (cid:126)(cid:126)(cid:126)(cid:126) (cid:64)(cid:32)(cid:32) (cid:15)(cid:15) (cid:126)(cid:126)(cid:126)(cid:126)
S˜ S¯ Sˆ
Lemma 4.2. — For all k = 1,...,l every irreducible component of the
preimage of p in Y is a rational curve.
k 12
Proof. Notice that D0 are smooth in S0 and that their components
j
don’t intersect in S0. Every point p is a point of intersection of an
k
irreducible component ofD0 with an irreducible component of D0. In the
1 2
neighborhood of p these two divisors have a canonical form: D is given
k 1
by x = 0 and D is given by y = xn, where n is the order of tangency.
2
There is a standard chain of blowups separating the proper transforms
of D and D over p (which intersect one of the ends of the chain in two
1 2 k
distinct points). The induced double cover on every component of the
preimage of p is ramified in at most three points (hence in fact, two)
k
and is therefore a rational curve. (cid:3)
Corollary 4.3. — The map alb : Y → Alb(Y ) factors through
12 12
¯
Y .
12
Proof. Indeed, the natural map Y → Y¯ contracts connected graphs
12 12
¯
of rational curves to distinct points in Y . Since these connected graphs
12
of rational curves map into points in Alb(Y ) our claim follows. (cid:3)
12
4.4. Elliptic fibrations. — LetE → P1 beaJacobianellipticfibration
and M ,M two irreducible horizontal divisors on E. Let M ⊂ E be
1 2 12
the divisor of pairwise differences: M ∩E is the set of all points of the
12 b
form p − p , where p ∈ M ∩ E . The divisor M may have several
1 2 j j b 12
irreducible components. We shall say that the divisor M is torsion if
12
every irreducible component of M ⊂ E consists of torsion points.
12
10 FEDOR BOGOMOLOV and YURI TSCHINKEL
Lemma 4.5. — Consider the restriction map to the generic fiber
η∗ : Pic(E) → Pic(E ).
η
If the divisor M is torsion then there exists a positive integer N such
12
that the class η∗([M ]) ∈ Pic(0)(E ) is annihilated by N.
12 η
Proof. For every irreducible component of M there exists a positive
12
integer N(cid:48) such that all points p in this component are annihilated by
N(cid:48). Since the divisor M has only a finite number of irreducible com-
12
ponents we can find an N annihilating all points in M . Consider the
12
class N[M ]. The corresponding divisor is trivial upon restriction to the
12
generic fiber E . (cid:3)
η
Corollary 4.6. — The kernel of the map η∗ : Pic(E) → Pic(E ) is
η
isomorphic to the subgroup of Pic(E) generated by the components of the
singular fibers.
Lemma 4.7. — Let E → P1 be a Jacobian elliptic fibration with sin-
gular fibers of simple multiplicative type (irreducible nodal curves). Let
M ,M ,M ⊂ S be three different irreducible horizontal divisors which
1 2 3
are linearly independent in Pic(E). Then at most one of the divisors M ,
ij
(i,j = 1,2,3, and i (cid:54)= j) is torsion.
Proof. Indeed if two of the above divisors, for example M and M ,
12 13
are torsion then there is a positive integer N which annihilates both
η∗([M ]) and η∗([M ]). Hence the kernel of η∗ has rank ≥ 2. By as-
12 13
sumption, the singular fibers of E generate a subgroup of rank 1. Con-
tradiction. (cid:3)
Let E → P1 be a nonisotrivial elliptic fibration and r the order of
E in the Tate-Shafarevich group of the corresponding Jacobian elliptic
fibration J(E). Recall that for each integer r(cid:48) we have a principal homo-
geneous fibration J(r(cid:48))(E) of (relative) zero cycles of degree r(cid:48), together
with natural maps
J(r(cid:48))(E)×J(r(cid:48)(cid:48))(E) → J(r(cid:48)+r(cid:48)(cid:48))(E)
and an identification of J(0)(E) and J(r)(E), depending on the choice of
a global section of the Jacobian fibration J(E) = J(0)(E). After fixing