IITAKA FIBRATION
JUNGKAI A. CHEN AND CHRISTOPHER D. HACON Abstract. We characterize the irregularity of the image of the Iitaka fibration in terms of the dimension of certain cohomological support loci.
1. Introduction
Let L be a Cartier divisor on a projective variety X with κ(X, L) ≥ 0. The Iitaka fibration associated to L is a birational model of the map induced by the linear series |mL| for m sufficiently big and divisible. We will denote it by fL : X0 → YL, where X0 is an appropriate birational
model of X. It is a fundamental tool in the birational classification of higher dimensional varieties. Therefore, it is important to understand the geometry of the image of the Iitaka fibration.
In this note, we characterize the irregularity of YL in terms of
di-mension of certain cohomological support loci. To be more precise, let
Vm(L) = Vm := {P ∈ Pic0(X)|h0(X, L⊗m⊗P ) 6= 0},
and V = ∪m≥1Vm. We show that the maximal irreducible component
of V passing through the origin, denoted by G, is in fact the subgroup
f∗Pic0(Y
L) ⊂ Pic0(X). In particular, q(YL) = dimG.
In general, the locus V need not be a group, it could have infinitely many irreducible components and there may exists P, Q ∈ Pic0(X) such that fL⊗P and fL⊗Q are not birational. Nevertheless, we show
that when one considers the case of the canonical divisor L = KX,
then the set V (KX) is a subgroup of Pic0(X) such that its components
consist of finitely many torsion translates of Pic0(Y ). Moreover, we show that, for all P ∈ V , one has that the Iitaka fibration fKX⊗P is
birational to fKX.
Finally, we study the case when X is of maximal Albanese dimension. In [CH2], the authors have shown that the translates through the origin of the components of V1(KX) generate Pic0(X). In §4 we show that the
locous T given by the intersection of all (translates through the origin of the components of V1(KX)) plays a fundamental role in understanding
the geometry of X. In particular we show that: Let X be of general type
and maximal Albanese dimension. If P1(X) = 1 then q(X) = dim(X).
Moreover, we construct examples of such varieties. This answers a question of Koll´ar (cf. [Kol2], Conjecture (17.9.2)).
Acknowledgement. The second author would like to thank R. Par-dini for valuable conversations.
Notation and conventions.
(1.1) Throughout this paper, we work over the field of complex num-bers C. X will always denote a projective variety.
(1.2) |D| will denote the linear series associated to the divisor D. We do not distinguish between line bundles, linear equivalence classes of divisors and invertible sheaves. Let |V | ⊂ |L| be a linear subsystem. A log resolution of |V | is a proper birational morphism µ : X0 −→ X
such that X0 is smooth, µ∗|V | = |W | + F , where |W | is base point free
and the union of support of F and the exceptional set of µ is a divisor with normal crossings support.
(1.3) For a real number a, let bac be the largest integer ≤ a and
dae be the smallest integer ≥ a. For a Q-divisor D = PaiDi, let
bDc =PbaicDi and dDe =
P
daieDi.
(1.4) We will denote by A(X) the Albanese variety of X, by albX :
X −→ A(X) the Albanese morphism. As usual Pic0(X) is the abelian variety dual to A(X) parameterizing all topologically trivial line bun-dles on X. Pic0(X)tors will denote the set of torsion elements in
Pic0(X). We will denote by m : A × A −→ A the group law of an
abelian variety A. Given subsets Z, W ⊂ A, we define Z + W :=
image (m|Z×W : Z × W −→ A).
(1.5) Let F be a coherent sheaf on X, then hi(X, F) denotes the
com-plex dimension of Hi(X, F). In particular, the plurigenera h0(X, ω⊗m X )
are denoted by Pm(X) and the irregularity h0(X, Ω1X) is denoted by
q(X).
(1.6) Let L be a Cartier divisor on X. If h0(X, L) > 0, then there is
a rational map φ|L| : X 99K P(H0(X, L)) defined by the sections of L.
The Iitaka dimension of a line bundle L is defined as
κ(L) := max{dim φ|mL|(X); m ∈ N}.
If |mL| = ∅ for all m > 0, we set κ(L) = −∞. A nonsingular rep-resentative of the Iitaka fibration of X is a morphism of smooth com-plex projective varieties f0
L = f0 : X0 −→ Y such that for all
suffi-ciently big and divisible integers m, f0 : X0 −→ Y is birational to
f|mL| : X 99K f|mL|(X). It is characterized up to birational equivalence
by the following properties:
i) f0 : X0 −→ Y is an algebraic fiber space (i.e. it is surjective with
connected fibers); ii) dim(Y ) = κ(L); iii) κ(X0
y, L|Xy0) = 0 (where Xy0 is a generic geometric fiber of f0).
There is a semi-group N(L) = N(X, L) := {m ∈ N; H0(X, L⊗m) 6=
0}. The exponent e(L) is the greatest common divisor of all elements of N(L). For all m À 0, one has that m ∈ N(L) iff e(L)|m.
2. cohomological support loci
Lemma 2.1. Let X be a closed subvariety (reduced and irreducible) of
an abelian variety A passing through the origin. If X is closed under the group law, then X is an abelian subvariety.
Proof. Let g be the dimension of X. Assume that g > 0. Since X is
closed under multiplication, we have an induced morphism
m : X × X → X.
Let Fx be the fiber over x ∈ X. It is clear that dimFx ≥ g. On the
other hand, one notices that Fx ∩ ({a} × X) consists of at most one
point for all a ∈ X. This, together with the fact that dimFx ≥ g, shows
that Fx∩ ({a} × X) has exactly one point for all a ∈ X. Let x = 0 then
it follows that X is a subgroup of A hence an abelian subvariety. ¤ Let X be a smooth projective variety with q(X) > 0, L a Cartier divisor on X with κ(L) ≥ 0 and exponent e := e(L). We define
Vm = Vm(X, L) := {P ∈ Pic0(X)|h0(X, L⊗m⊗P ) 6= 0}.
By semi-continuity, each component of Vmis a closed subset of Pic0(X).
Let V = ∪∞
m=1Vm. Then V ⊂ Pic0(X) is a semi-group. We define Gm
to be the union of all irreducible components of Vm passing through the
origin and let G = ∪m>0Gm. Note that Gm is non-empty only when
m ∈ N(L). Recall that for all m À 0, we have m ∈ N(L) iff e(L)|m.
Lemma 2.2. There exists an integer m > 0 such that G = Gm is an
abelian subvariety of Pic0(X).
Proof. Pick any maximal irreducible component W ⊂ G, that is, if W0 is an irreducible component of G containing W then W = W0.
(This is possible since Pic0(X) is of course Noetherian). Assume that
W ⊂ Gm0. We claim that W = G and W is closed under multiplication.
Then we are done by Lemma 2.1. To see the claim, observe that if Z is any irreducible component of Gm, then Z + W is irreducible since it
is the image of m : Z × W → Pic0(X). Moreover,
Z + W ⊂ Gm0 + Gm ⊂ Gm0+m⊂ G.
Let W0 be an irreducible component of G
m0+m containing Z + W . It
follows that W ⊂ Z + W ⊂ W0. By the maximality of W , one has
W = W0. In particular, Z ⊂ W and hence W = G.
It then suffices to check that W is closed under multiplication. But
W ⊂ W + W ⊂ G + G = G = W and hence W + W = W . ¤
Corollary 2.3. There exists an integer t0 > 0 such that Gte = G for
all t ≥ t0.
Question: Are the loci V (resp. Vm) union of translates of subgroups
Remark: It is easy to see that if k(L) = 0 and P ∈ V is a non torsion element, then P∨ is not in V . In particular, dimG = 0. The following
example, shows that dimVm could be positive even if κ(L) = 0.
Example 2.4. Consider A = E × E a product of elliptic cures and
let M = p∗
1H + p∗2P with H ample of degree 1 and P ∈ Pic0(E) −
Pic0(E)
tors. Let X = P(OA⊕ M), π : X −→ A and L := OX(1). Since
π : X −→ A is a projective bundle, it follows that π is the Albanese morphism of X. Since
π∗OX(m) = Sm(M ⊕ OA) = M⊗m⊕ M⊗m−1⊕ .... ⊕ M ⊕ OA,
one sees that
Vm = {OA} ∪ m [ i=1 (p∗ 1Pic0(E) + ip∗2P∨). We have (1) κ(L) = 0 and dimG(L) = 0. (2) For all m > 0, Q ∈ (p∗
1Pic0(E) + mp∗2P∨), κ(L⊗Q) = 1 and
dimVm = dimV = 1.
This gives an example in which κ(L) = 0 but κ(L⊗P ) > 0 for some P ∈ Pic0(X). Moreover, V = ∪V
m is a semi-group (with infinitely
many components) but not a group.
Consider the following diagram relating the Iitaka fibration fL and
the Albanese morphism albX
X albX −−−→ A(X) fL y ψL y YL albY −−−→ A(YL).
Notice that fL is not a morphism but simply a rational map. However
there exists a birational model µ : X0 −→ X such that the induced
map f0
L : X0 −→ YL is a morphism. Let πL : X0 −→ A(YL) be the
induced map.
Lemma 2.5. If P ∈ Vm then for all sufficiently big and divisible
in-tegers, s > 0 one has that sP + Pic0(Y
L⊗m⊗P) ⊂ V . In particular, if κ(L) ≥ 0, then Pic0(YL) ⊂ G(L)
Proof. Let N = L⊗m⊗P . Fix H an ample line bundle on A(Y
L⊗m⊗P).
For k À 0 sufficiently divisible, we have that h0(X0, µ∗N⊗k⊗π∗H∨) >
0. Clearly, µ∗OX0 = OX and Riµ∗OX0 = 0 for all i > 0. Since µ∗N⊗k⊗π∗
NH∨ = µ∗(N⊗k⊗alb∗Xψ∗NH∨), by the projection formula, we
have h0(X, N⊗k⊗alb∗
XψN∗H∨) > 0. It is easy to see that for any integer
j À 0, one has h0(Hj⊗Q) > 0 for any Q ∈ Pic0(Y
L⊗m⊗P). It follows
that h0(X, N⊗kj⊗Q⊗j) > 0, i.e. kjP + jQ ∈ V
mkj. The Lemma follows
In particular, take P = OX ∈ Vm, then OX ∈ Pic0(YL) ⊂ Vmkj and
hence Pic0(YL) ⊂ G. ¤
It follows that for any m > 0 and P ∈ Pic0(X), we have q(YL⊗P) ≤
dimG(L⊗P ) ≤ dimV (L). Recall though that it is possible dimV (L) >
G(L).
It also follows that if L is big, then there exists an integer m0 > 0
such that Vm = Pic0(X) for all m ≥ m0.
Lemma 2.6. i) ψ : A(X) −→ A(Y ) is surjective and has connected
fibers.
ii) Let K := ker(ψ). Then there is an exact sequence
0 −→ Pic0(Y ) −→ Pic0(X) −→ Pic0(K) −→ 0.
iii) Fix a general fiber i : Xy −→ X. Let
H := ker¡i∗ : Pic0(X) −→ Pic0(X y)
¢
. Then, H ⊃ Pic0(Y ) and H/Pic0(Y ) is a finite group. Proof. i) See [HP] Proposition 2.1.b.
ii) This is an immediate consequence of i) obtained by dualizing the exact sequence of abelian varieties
0 −→ K −→ A(X) −→ A(Y ) −→ 0.
iii) Let Jy be the abelian subvariety generated by the translates of
albX(Xy) through origin. Since there are at most countably many
sub-abelian varieties of a fixed abelian variety, we have that J = Jy
does not depend on y ∈ Y for general y. Let B := A(X)/J. Then
Y −→ A(Y ) factors through B, and hence there is a map of abelian
varieties A(Y ) −→ B inducing the map Y −→ B. The image of X in
A(X) generates A(X), hence its image in B generates B as well. It
follows that image of A(Y ) in B generates B. And hence, one sees that
B = A(Y ) i.e. K = J.
We have the following commutative diagram Pic0(X) −−−→ Pic0(K) i∗ y = y Pic0(X y) ←−−− Picα 0(J),
where i : Xy → X is the inclusion and α is induced from albX : Xy → J.
Note that, since A(Xy) → J is surjective, the homomorphism α has
finite kernel. The assertion now follows from ii). ¤ Theorem 2.7. G(L) = Pic0(YL).
Proof. By Lemma 2.5, one has G(L) ⊃ Pic0(YL).
If κ(L) = 0, it suffices to prove that dim G = 0. This is clear as otherwise one may consider m > 0 such that G(L) ⊂ Vm and the
morphisms
|mL + P | × |mL − P | −→ |2mL|
for P ∈ G. It is easy to see that if dim G > 0 then dim |2mL| > 0 a contradiction.
Assume now that κ(L) > 0. If P ∈ G(L), then for some m À 0,
h0(L⊗m⊗P ) 6= 0 in particular h0(X
y, L⊗m⊗P |Xy) 6= 0. It follows that
i∗G(X, L) ⊂ V
m(Xy, L|Xy). And hence i∗G(X, L) ⊂ G(Xy, L|Xy).
However, G(Xy, L|Xy) = {OXy} as we have seen in the preceding
para-graph. Thus we have
i∗G(X, L) = G(X
y, L|Xy) = {OXy}.
By Lemma 2.6, G(X, L) ⊂ H. Since G(X, L) is connected and contains the origin, it follows that G(X, L) ⊂ Pic0(Y
L). ¤
3. Iitaka fibrations of canonical divisors ¿From now on we will assume L = KX and κ(X) ≥ 0.
Proposition 3.1. Fix an integer m ≥ 2. For all Q ∈ Pic0(X)
tors and
all P ∈ Pic0(Y
KX) one has h0(ω⊗m
X ⊗P ⊗Q) = h0(ω⊗mX ⊗Q).
Proof. Just follow the proof of [HP] Proposition 2.12. ¤ Theorem 3.2. The loci Vm(KX) consist of a finite union of torsion
translates of abelian subvarieties of Pic0(X).
Proof. By a result of Simpson, the loci V1(KX) are torsion translates
of abelian subvarieties of Pic0(X). Fix m ≥ 2, and let P ∈ V
m(KX).
Let µ : X0 −→ X be a log resolution of the non empty linear series
|r((m−1)KX+P )|. Let D be a general member of µ∗|r((m−1)KX+P )|.
D has normal crossings support. Let N := OX0((m−1)KX0+P −bD
rc),
then the locus
V1(ωX0⊗N) := {Q ∈ Pic0(X)|h0(ωX0⊗N⊗Q) > 0}
consist of a finite union of torsion translates of subtori of Pic0(X)
(cf. [Sim] and [ClH] §7). Comparing base loci, it is easy to see that
h0(ω
X0⊗N) = h0(ωX⊗m0 ⊗P ) and for all Q ∈ TP we have h0(ω⊗m
X0 ⊗Q) = h0(ωX0(bD
r c)⊗N⊗Q⊗P
∨) ≥ h0(ω
X0⊗N⊗Q⊗P∨) > 0.
Hence for each P ∈ Vm(KX) there exists TP a torsion translate of a
subtorous such that P ∈ TP ⊂ Vm(KX) ⊂ Pic0(X). It follows that
Vm(KX) = ∪TP. Since Vm(KX) ⊂ Pic0(X) is closed, it follows that
Lemma 3.3. If κ(X) = 0 then there exists an integer c(X) = c > 0 and
an element P ∈ Pic0(X)tors such that for all m > 0, the set Vm(KX)
is non-empty if and only if c divides m. Moreover, if c divides m, then Vm = {P⊗
m
c}. In particular V is a subgroup of Pic0(X).
Proof. It is easy to see that since κ(X) = 0, each Vm can have at most
1 torsion point. Therefore Vm(KX) consists of at most one element say
Pm ∈ Pic0(X)tors. Let
c := min{m > 0|Vm 6= ∅}.
It is clear that Vm 6= ∅ for all m divisible by c. Let d be any positive
integer such that Vd6= ∅. We claim that c|d. We may write d = cq + r
with q > 0 and 0 ≤ r ≤ c − 1. Let Dc (resp. Dd) be the unique
divisor in the linear series |cKX + Pc| (resp. |dKX + Pd|). Since Vcd
has only one point, we have dPc = cPd and moreover cDd = dDc as
h0(cdK
X + dPc) = 1.
Consider now the divisor Dd−qDc. It is effective since c(Dd−qDc) =
rDc is effective. It follows that h0(rKX + Pd − qPc) 6= 0 and hence
Vr 6= ∅. One sees that r = 0. Therefore, Vm(KX) is non-empty iff c
divides m. The Lemma now follows easily. ¤
Corollary 3.4. Let eH := i∗−1(V (X
y)). For all m > 0 one has Vm ⊂
e
H. Moreover, V ⊂ eH is a subgroup of finite index containing G. Proof. For any Q ∈ Vm, if h0(ωX⊗m⊗Q) 6= 0, then h0(ωX⊗my ⊗Q) 6= 0.
Therefore, Vm ⊂ eH for all m > 0 and hence, V ⊂ eH.
Since eH/G is finite and G ⊂ V ⊂ eH, it suffices to show that the
semigroup V is in fact a group.
To this end, pick any P ∈ V , we may assume that P ∈ Vm for m ≥ 2
since V1 ⊂ V1+t for all sufficiently big and divisible t. By Proposition
3.1 and Theorem 3.2, one sees that P is in an irreducible component of the type G + P0 with P0 torsion. Let k be the order of P0. One sees
that
−P ⊂ G − P0 = (k − 1)(G + P0) ⊂ V.
It follows that V is a group. ¤
Corollary 3.5.
q(X) − q(Y ) ≤ q(Xy).
Proof. Since κ(Xy) = 0, by [Kaw], the map albXy : Xy −→ A(Xy) is
surjective. The map Xy −→ J is factored through A(Xy). It follows
that A(Xy) −→ J is surjective by the definition of J. Thus q(Xy) =
dimA(Xy) ≥ dimJ. But J = K and dimK = dimA(X) − dimA(Y ).
The inequality follows. ¤
Corollary 3.6. X is of maximal Albanese dimension if and only if Y
is of maximal Albanese dimension and
Proof. If X is of maximal Albanese dimension, then albX : X → A(X)
is generically finite. It follows that albX : Xy → A(X) is generically
finite. Therefore, Xy −→ J is an ´etale map of abelian varieties and
dimXy = dimJ. Moreover, since albX is generically finite, for a general
point z ∈ albY(Y ), the preimage in X consits of at most finite union
of fibers Xy. Thus albY is generically finite and hence Y is of maximal
Albanese dimension.
On the other hand, let Y → S → A(Y ) be the Stein factorization. And let T := S ×A(Y ) A(X). It suffices to check that X → T is
generically finite. Notice that κ(Xy) = 0 and q(Xy) ≥ dimXy. It
follows by [Kaw] that Xy is birational to an abelian variety and hence
that Xy → K is generically finite. Therefore Xy → T is generically
finite, and so is X → T . ¤
Corollary 3.7. For all P ∈ Pic0(Y ), the Iitaka fibration Y (KX + P )
is birational to the Iitaka fibration Y (KX).
Proof. Let P, Q ∈ Pic0(Y ) such that P = 2Q. From the morphism |mKX| × |m(KX + P )| −→ |2m(KX + Q)|,
one sees that Y (KX + Q) dominates Y (KX) and Y (KX + P ). By
Proposition 3.1,
dim Y (KX + Q) = dim Y (KX + P ) = dim Y (KX).
since the various Iitaka fibrations X −→ Y (...) have connected fibers, then Y (KX + Q) is birational to Y (KX) and to Y (KX + P ). ¤
Lemma 3.8. Let a : X → A(X) be the Albanese map. If q(Y ) = 0,
then a∗ωX is a homogeneous vector bundle. In particular, c1(a∗ωX) =
0.
Proof. Let a = albX. If q(Y ) = 0, then V1 is supported on finitely
many points. We recall that by definition
Vi(a
∗ωX) := {P ∈ Pic0(A(X))|hi(a∗ωX⊗P ) 6= 0}.
By [CH1], we have
V1 = V0 ⊇ V1 ⊇ ... ⊇ Vq,
where q = q(X). It follows that all the Vi are supported on finitely
many points.
We now follow [Laz] to show that a∗ωX is a homogeneous vector
bundle. Let pi be the projection of A(X) × Pic(X) = A × bA on to the
i-th factor and P be the normalized Poincare’ line bundle on A × bA.
Then RSa∗ωX = Rp2∗(p∗1a∗ωX⊗P) is the Fourier-Mukai transform of
a∗ωX. By [Muk], it is enough to show that RgSa∗ωX ∼= RSa∗ωX is a
that for an open affine U = Spec(B) ⊂ ˆA, there is a complex of finitely
generated free B-modules, denoted by E•, such that
RiSa
∗ωX ∼= Hi(E•),
and
Hi(X
y, a∗ωX⊗Py) ∼= Hi(E•⊗k(y)).
It suffices to show that Hi(E•) = 0 for all i < q and Hq(E•) is
sup-ported on finite point. We may assume that B is local since localization is flat. Let
Wi(E•) := coker(Ei−1→ Ei).
It easy to see that Wi(E•)⊗k(y) = Wi(E•⊗k(y)) since ⊗ is right exact.
One has exact sequence
0 → Hi(E•) → Wi(E•) → Ei+1.
Our hypothesis on the cohomologies of a∗ωX implies that Hi(E•) is
Artinian for all i. Thus by Mumford’s acyclic lemma ([Mum], page
127), we are done. ¤
4. Varieties of Maximal Albanese dimension
Throughout this section, we assume that X is of maximal Albanese dimension i.e. dim (albX(X)) = dimX.
In (17.9.3) of [Kol2], Koll´ar Conjectured that If X is of general type
and Maximal Albanese dimension, then χ(X, OX) > 0. A consequence
of this conjecture would be: i) V1(X, KX) = Pic0(X) and
ii) (cf. (17.9.2) of [Kol2]) that P1(X) ≥ 2.
In [EL], Ein and Lazarsfeld show that χ(X, OX) ≥ 0, and they produce
a counterexample to the above conjecture of Koll´ar ((17.9.3) of [Kol2]). In [CH2], the authors show that:
Theorem 4.1. If X is of maximal Albanese dimension then (the
trans-lates through the origin of) the irreducible components of V1(X, KX)
generate Pic0(Y ), and for all m ≥ 2 one has G
m(X, KX) = Pic0(Y ).
In particular if X is of general type, they generate Pic0(X) (see i)
above).
Let X be of maximal Albanese dimension. If χ(OX) = 0, let Si0 be
the translates of the components of V1(X, KX) through the origin and
T := ∩S0 i.
Theorem 4.2. Let X be a variety of general type, maximal Albanese
dimension with χ(ωX) = 0. Let A(X) −→ T∨ be the corresponding
map of abelian varieties, and π : X −→ T∨ the induced morphism. For
general w in W := π(X), one has that χ(ωXw) = 0 and albX(Xw) is a translate of the abelian subvariety ker (A(X) −→ T∨). In particular,
Proof. Recall that by [EL] Theorem 3, the Albanese image Z := albX(X)
⊂ Alb(X) is fibered by tori. In fact [EL] shows the following: Let S be any positive dimensional component of V0(ω
X) = V1(X, KX).
Con-sider the quotient map π : A := Alb(X) −→ C dual to the inclu-sion S ,→ Pic0(A). Let g : X −→ C be the induced morphism, then dim g(X) = dim(X) − (dim(A) − dim(C)), i.e. Z −→ g(X) is fibered
by tori. By [CH2] Theorem 2.3, the translates through the origin of
the components of V1(ωX) generate Pic0(X). X is of general type and
hence not ruled by tori. Assume that the intersection of the translates through the origin of the components of V1(ωX) is just {OX}. Let Si
be any component of V1(ωX) and Bi be the kernel of the
correspond-ing projection of abelian varieties πi : A −→ Ci := Si∨. Then the
union of the Bi generates A. As observed above, by the result of [EL],
Z + Bi = Z and hence Z = A, i.e. q(X) = dimX.
Suppose now that dimT > 0 and consider the following dual exact sequences of abelian varieties
0 −→ T −→ Pic0(X)−→ Picφ 0(X)/T −→ 0,
0 −→ F −→ A(X) −→ T∨ −→ 0.
Let Si := Si/T ⊂ Pic0(F ) = Pic0(X)/T and π : X −→ T∨ the induced
morphism. For any P ∈ Pic0(X)
torssuch that P /∈ V1(X, KX), one has
that
φ−1(φ(P )) = P + T ∩ V
1(X, KX) = ∅.
So for all Q ∈ T one has h0(X, ω
X⊗P ⊗π∗Q) = 0 and therefore also
hi(X, ω
X⊗P ⊗π∗Q) = 0 for all i ≥ 0. By a result of Koll´ar (cf. [Kol1],
Corollary 3.3) one sees that for all Q ∈ T , the group Hi(π
∗(ωX⊗P )⊗Q)
is a direct summand of the group Hi(X, ω
X⊗P ⊗π∗Q) and hence also
vanishes. By a result of Mukai (cf. [Muk]) it follows that π∗(ωX⊗P ) =
0. So for general w ∈ W = π(X), one has h0(X
w, ωXw⊗P ) = 0.
Therefore, χ(ωXw) = 0 and the locous V1(Xw, KXw, Pic
0(F )) := {R ∈ Pic0(F )|h0(ω
Xw⊗R) > 0}
(which is determined by its torsion points) is a subset of V1(X, KX)/T .
It follows that the intersection of the translates through the origin of the components of V1(Xw, KXw, Pic
0(F )) is just {O
F}. Therefore,
arguing as in the preceeding paragraph, one sees that albX(Xw) = F
is an abelian subvariety of A(X). ¤
Corollary 4.3. Let X be a 3-fold of general type and maximal Albanese
dimension. If χ(ωX) = 0 then q(X) = dimX.
Proof. We may assume that dimX > dimT > 0 and hence Xw is of
general type and has dimension 1 or 2. By the classification theory of curves and surfaces χ(ωXw) > 0. This is the required contradiction. ¤
Corollary 4.4. Let X be a variety of general type, maximal Albanese
Proof. If P1(X) = 1 then χ(ωX) = 0. Let ˜W −→ W be an appropriate
desingularization (of W = π(X) as above). Since X is of maximal Al-banese dimension, one has P1( ˜W ) = 1 and hence since W is a subvariety
of the abelian variety T∨, it follows that W = T∨. Since alb
X(Xw) = F
one sees that albX(X) = A(X) and hence q(X) = dim(X). ¤
Corollary 4.5. Let X be a variety of general type, maximal Albanese
dimension with χ(ωX) = 0. If dimT = 0, then h1(X, P ) = 0 for all
P ∈ Pic0(X) − {OX}.
Proof. Assume that h1(X, P ) > 0 for some P ∈ Pic0(X) − {O X}.
By [Sim], we may assume that P is torsion. Consider the ´etale cover
Y −→ X corresponding to the subgroup < P >⊂ Pic0(X). Clearly Y
is of general type, maximal Albanese dimension and χ(ωY) = 0 so by
the Theorem 4.2
dim(Y ) = q(Y ) = h1(O Y) =
X
h1(X, P⊗i).
In particular h1(X, P⊗i) = 0 for i > 0. ¤
The following example shows that Koll´ar’s conjecture (17.9.2) of [Kol2] also fails:
Example. Let d : D −→ E be the Z2
2 cover of an elliptic curve defined
by d∗OD = OE⊕L∨⊕P ⊕L∨⊗P where deg(L) = 1 and P is a non-zero
2-torsion element of Pic0(E). Then g(D) = 3 and we will denote by l, p, lp the elements of Z2
2 whose eigensheves with eigenvalue 1 are OE
and L∨, P, L∨⊗P respectively. It follows that
δ : D × D × D −→ E × E × E
is a Z6
2 cover. Let X := D × D × D/G where G ∼= Z42 is the subgroup
of Z6
2 generated by {(id, p, l), (p, l, id), (l, id, p), (p, p, p)}. By a direct
check, one can see that
(δ∗OD×D×D)G ∼=
OE£ OE£ OE⊕ L∨£ L∨⊗P £ P ⊕ P £ L∨£ L∨⊗P ⊕ L∨⊗P £ P £ L∨.
The singularities of X are of type 1
2(1, 1, 1) i.e. locally isomorphic to
C3/ < (−1, −1, −1) >. In particular all singularities are rational. Let
˜
X −→ X be a resolution of the singularities of X and ν : ˜X −→ E × E × E be the induced morphism. One can compute that
ν∗(ωX˜) = OE£OE£OE⊕L£L⊗P £P ⊕P £L£L⊗P ⊕L⊗P £P £L.
Therefore, ˜X is a threefold of maximal Albanese dimension with P1(X) = 1 and q(X) = 3.
We remark that this example easily generalizes to dim X ≥ 3. The product X × C with C a curve of genus 2, is a variety of general type and maximal Albanese dimension with χ(ωX×C) = 0 and 5 = q(X) >
χ( ˜X) = 0. One can then construct (infinitely many) examples where P1( ˜X) = P1(X), q( ˜X) = q(X).
References
[CH1] J. A. Chen, C. D. Hacon, On algebraic fiber spaces over varieties of maximal
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