Annales de l’institut Fourier (AIF)

Tonini, Fabio

Yasuda, Takehiko

Moduli of formal torsors II

Campana, Frédéric

Université Grenoble Alpes
ANNALES DE
L’INSTITUT FOURIER
Frédéric Campana
Albanese map of special manifolds: a correction
Tome 73, n
o
5 (2023), p. 1903-1908.
https://doi.org/10.5802/aif.3563
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Creative Commons attribution – pas de modification 3.0 France
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Ann. Inst. Fourier, Grenoble
73, 5 (2023) 1903-1908
ALBANESE MAP OF SPECIAL MANIFOLDS: A
CORRECTION
by Frédéric CAMPANA
Abstract. — We show that any fibration of a “special” compact Kähler man-
ifold X onto an Abelian variety has no multiple fibre in codimension one. This
statement strengthens and extends previous results of Kawamata and Viehweg
when κ(X) = 0. This also corrects the proof given in the present journal in 2004,
which was incomplete.
Résumé. — On montre qu’une fibration d’une variété compacte Kählérienne X
sur une variété Abélienne n’a pas de fibre multiple en codimension 1. Ce résultat
renforce et généralise des résultats précédents de Kawamata et Viehweg lorsque
κ(X) = 0, et corrige la preuve incomplète donnée dans ce journal en 2004.
1. Introduction
The following statement is given in [3, Proposition 5.3]:
Theorem 1.1. — Let aX : X → AX is the Albanese map of X, assumed
to be special. Then aX is onto, has connected fibres, and no multiple fibre
in codimension one.
This statement is used neither in [3], nor in [4].
Recall ([3, Definition 2.1 and Theorem 2.27]) that X is special if κ(X,L) <
p for any p > 0, and any rank-one coherent subsheaf of ΩpX . This implies
that X has no surjective meromorphic g : X 99K Z map onto a manifold Z
of general type and positive dimension p, and more generally exactly means
that the “orbifold base” (Z, Dg) of such a g, constructed out of its multiple
fibres, is never of “general type”. See Section 2 below for some more details.
While the proofs given there for the first two properties (which gener-
alise earlier results by Y. Kawamata and Kawamata–Viehweg in the case
Keywords: Albanese map, special variety, orbifold base.
2020 Mathematics Subject Classification: 14K99, 14K12, 14D99, 14D06, 14J40.
1904 Frédéric CAMPANA
when κ(X) = 0) are complete, the proof of the third property, even in
the projective case, is not (as pointed out to me by K. Yamanoi and E.
Rousseau(1) ). The aim of this note is to correct it in the projective case
(but only partially in the compact Kähler case) by using the main result
of [4], itself based on [1] (or, alternatively, [5]).
Theorem 1.2. — Let f : X → A be a holomorphic map from the
connected, compact, Kähler manifold to a compact complex torus A. If X
is special, then:
(1) f is surjective,
(2) If f is the Albanese map of X, then f has connected fibres,
(3) f has no multiple fibre(2) in codimension one if the fibres of f are
connected(3) , and if A is an Abelian variety (equivalently: Df = 0
if (A, Df ) is the “orbifold base” of f).
Remarks 1.3.
(1) Although the assumption that A is an Abelian variety in (3) is cer-
tainly not necessary, our proof rests on [1] (or [5]) for A projective.
Although the Kähler version is known, by [9], we cannot treat the
Kähler case in general with the present arguments, because we use
Poincaré reducibility.
(2) An easy case, not covered by Theorem 1.2 in which (3) can be
proved, is when the algebraic dimension a(A) of X vanishes, be-
cause then A does not contain any effective divisor. The argument
of the second step of the proof applies, more generally, to show that
the conclusion of Theorem 1.2(3) still holds true when A = T × V
is a compact complex torus product of a subtorus T of algebraic di-
mension zero, with V , an Abelian variety, because every irreducible
divisor on A is a product T × D, where D is a divisor on V .
(3) (1) and (2), and their proofs, still apply when X is “weakly special”,
which means that no finite étale cover X ′ of X has a surjective mero-
morphic fibration onto a positive-dimensional projective manifold of
(1) The problem comes from the potential multiplicity-one exceptional divisors of aX
which are no longer f -exceptional if g = q ◦ f , where q : A → B is a non-trivial torus
quotient sending Df to an ample divisor of B. We overcome this difficulty in the second
step of the proof, by cutting the fibration by means of Poincaré reducibility. The first
step is the same as in [3].
(2) We use here the “inf” multiplicities, not the more classical “gcd” version, which would
give a weaker result.
(3) This can be realised, using the Stein factorisation of f . Indeed, since f factorises
through the Albanese map aX of X, which has connected fibres the Stein factorisation
of f coincides with the one of aX , which maps Alb(X) to a torus, hence finite étale
over A.
ANNALES DE L’INSTITUT FOURIER
ALBANESE MAP OF SPECIAL MANIFOLDS: A CORRECTION 1905
general type. Weak-specialness coincides with specialness for curves
and surfaces ([3, Proposition 9.29]), but differs from dimension 3 on,
by examples of Bogomolov–Tschinkel ([2], [3, Theorem 9.30]).
2. Multiple fibres of maps to complex tori
If f : X → Y is a fibration, that is: a holomorphic map with con-
nected fibres onto a (smooth) compact complex manifold Y , we define
its “orbifold base” Df as follows: for every prime divisor E ⊂ Y , let
f∗(E) :=
∑
k∈K tk.Dk + R, where the D′ks are the pairwise distinct prime
divisors of X surjectively mapped by f onto E, while R is an effective
f -exceptional divisor of X. The multiplicity(4) mf (E) of the generic fi-
bre of f over E is then defined as: mf (E) := inf{tk, k ∈ K}, and then
Df :=
∑
E⊂Y (1 −
1
mf (E) ).E, which is an effective Q-divisor of Y (note in-
deed that this sum is finite since mf (E) = 1 for all, but finitely many of
the E′s).
Proof of Theorem 1.2.
(1). — Assume by contradiction that f(X) := Z ̸= A. After Ueno’s
theorem ([8], p. 120), there is a quotient torus q : A → B such that q(Z) ⊂
B is of dimension p > 0, and of general type. The composed map g := q◦f :
X → Z thus contradicts the specialness of X, since κ(X, g∗(KZ)) = p and
g∗(KZ) ⊂ ΩpX .
(2). — Let Z be a smooth model of the Stein factorisation Z0 of f . We
replace X by Z, which is still special since dominated by X. We may thus
assume that f is generically finite. By [7, Theorem 23], replacing Z by a
suitable finite étale cover which is still special by [3, §5.5], we may assume
that Z fibres over over a manifold of general type if Z is not birational to
A. This again contradicts the specialness of Z and thus of X.
(3). — Assume that Df ̸= 0. We shall first treat the case when Df is
ample, and reduce to this case in a second step.
Let thus Df be ample on A. By a flattening of f , followed by suitable
blow-ups of X and A, we may assume (see [3, Lemma 1.3]) that f = v ◦ f ′,
where v : A′ → A is bimeromorphic with A′ smooth, and f ′ : X → A′ is a
fibration such that its orbifold base D′ := Df ′ = D + E′, with E′ effective
and v-exceptional, D is the strict transform of Df in A′. This shows that
(4) We use here the “inf” multiplicity instead of the usual “gcd” multiplicity for reasons
explained in [3].
TOME 73 (2023), FASCICULE 5
1906 Frédéric CAMPANA
in particular f∗(KA′ + D) ⊂ ΩpX , p := dim(A). The following lemma shows
that the line bundle KA′ +D has Kodaira dimension dim(A) = p, since Df
is ample on A. This contradicts the specialness of X if Df is assumed to
be ample.
Lemma 2.1 ([3, 1.14]). — Let v : Y ′ → Y be a bimeromorphic map
between compact connected complex manifolds. Let D be an effective Q-
divisor on Y , and let D be its strict transform on Y ′ . Then:
(1) κ(Y, KY + D) ⩾ κ(Y ′, KY ′ + D).
(2) KY ′ + D = v∗(KY + ε.D) + E′ε, ∀ε > 0 small enough, E′ε effective.
(3) If κ(Y ) ⩾ 0, then κ(Y ′, KY ′ + D) = κ(Y, KY + D).
Proof.
(1). — Let U be the largest Zariski open subset of Y over which v is
isomorphic, and U ′ := v−1(U). We have an obvious equality, for each m > 0:
v∗(H0(U, m.(KY + D))) = H0(U ′, m.(KY ′ + D)). Since the complement of
U in Y has complex codimension at least 2, Hartog’s theorem shows that le
left hand side of this last equality coincides with v∗(H0(Y, m.(KY + D))).
On the other hand, the right hand side contains H0(Y ′, m.(KY ′ +D)), and
so κ(Y, KY + D) ⩾ κ(Y ′, KY ′ + D))
(2). — Let the rational numbers ai, bi be defined as follows: v∗(D) =
D +
∑
i bi.Ei, KY ′ = v∗(KY ) +
∑
i ai.Ei, where the E′is are the exceptional
divisors of v. Thus: ai > 0, bi ⩾ 0, ∀i. Then:
KY ′ + D = v∗(KY ) +
∑
ai.Ei + (1 − ε).D + v∗(ε.D) −
(
ε.
∑
bi.Ei
)
= v∗(KY + ε.D) + (1 − ε).D +
∑
(ai − ε.bi).Ei.
We thus get the claim when 0 < ε ⩽ min{1, aibi , ∀i}, with E
′
ε := (1 − ε).D +∑
(ai − ε.bi).Ei.
(3). — Let ε > 0 be sufficiently small, as above. Since KY is assumed to
be Q-effective, so is (1 − ε).v∗(KY ), and so KY ′ + D = ε.v∗(KY + D) + E′′ε ,
with: E′′ε := E′ε + (1 − ε).v∗(KY ), which is Q-effective. Thus κ(Y ′, KY ′ +
D) ⩾ κ(Y, KY + D). □
Remark 2.2. — The main property used in Lemma 2.1 is that ai > 0, ∀i,
i.e: the lift of the canonical sheaf under a modification of a smooth manifold
vanishes on the exceptional divisor. This property does not hold true for
sheaves of forms of degree less than the dimension. We thus need to use a
less direct route, which requires deeper ingredients.
We shall now reduce to the case when Df is ample.
ANNALES DE L’INSTITUT FOURIER
ALBANESE MAP OF SPECIAL MANIFOLDS: A CORRECTION 1907
Let L := OA(m.Df ) be the line bundle on A with a section vanishing on
some integral multiple m.Df of the orbifold base of f . Let T ⊂ Aut0(A)
be the connected component of the group of translations of A preserving
L. Let q : A → B := A/T be the quotient map. Then Df = 1m .q
∗(D) for
some ample effective divisor D on B (see, for example [6, Théorème 5.1]).
Since Df ̸= 0, p := dim(B) > 0.
We use Poincaré reducibility to reduce to the case that A = B × B′,
where B′ → T is a suitable finite étale cover of T . This indeed amounts to
replace X and Df by the corresponding finite étale covers, preserving all
of our hypothesis, in particular the specialness of X (by [3, §5.5], again).
We now consider the composed fibrations g := q ◦ f : X → B, and
h := q′ ◦ f : X → B′, where q′ : A → B′ is the second projection. By
[4, 2.4] the general fibre Xb′ := h−1(b′), b′ ∈ B′ of g′, is special(5) . Let
fb′ : Xb′ → Ab′ := B ×{b′} be the restriction of f to Xb′ . The orbifold base
(Ab′ , Dfb′ ) of fb′ is then nothing, but (A, Df ) ∩ Ab′ , and is thus ample on
Ab′ ∼= B. By the first part of the proof, this contradicts the specialness of
Xb′ if Dfb′ ̸= 0. Thus Df ∩ Ab′ = 0, and so Df = 0. □
BIBLIOGRAPHY
[1] C. Birkar & J. A. Chen, “Varieties fibred over abelian varieties with fibres of log
general type”, Adv. Math. 270 (2015), p. 206-222.
[2] F. Bogomolov & Y. Tschinkel, “Special elliptic fibrations”, in The Fano Confer-
ence, Univ. Torino, Turin, 2004, p. 223-234.
[3] F. Campana, “Orbifolds, special varieties and classification theory”, Ann. Inst.
Fourier 54 (2004), no. 3, p. 499-630.
[4] F. Campana & B. Claudon, “Quelques propriétés de stabilité des variétés spéciales”,
Math. Z. 283 (2016), no. 1-2, p. 581-599.
[5] J. Cao & M. Păun, “Kodaira dimension of algebraic fiber spaces over abelian vari-
eties”, Invent. Math. 207 (2017), no. 1, p. 345-387.
[6] O. Debarre, Tores et variétés abéliennes complexes, Cours Spécialisés, vol. 6, Société
Mathématique de France, Paris; EDP Sciences, Les Ulis, 1999, vi+125 pages.
[7] Y. Kawamata, “Characterization of abelian varieties”, Compositio Math. 43 (1981),
no. 2, p. 253-276.
[8] K. Ueno, Classification theory of algebraic varieties and compact complex spaces,
Lecture Notes in Mathematics, vol. 439, Springer-Verlag, Berlin-New York, 1975,
Notes written in collaboration with P. Cherenack, xix+278 pages.
(5) The statement of [4] is only given for X projective, and f the Albanese map of X,
but it is easy to check that the proof applies for X compact Kähler, and f any map with
connected fibres to an Abelian variety. Said otherwise, [4, 2.6] applies if X is compact
Kähler and if (Z, D) is the relative orbifold base of f . Indeed, since the generic orbifold
fibres (Z, D) are supposed, by contradiction, to be of general type, we deduce that Z
is Moishezon over A, hence Moishezon, since so is A. So Z becomes projective after a
suitable modification of X, and [4, 2.6] applies.
TOME 73 (2023), FASCICULE 5
1908 Frédéric CAMPANA
[9] J. Wang, “On the Iitaka Conjecture Cn,m for Kähler Fibre Spaces”, https://arxiv.
org/abs/1907.06705, 1919.
Manuscrit reçu le 15 septembre 2021,
révisé le 11 février 2022,
accepté le 12 mai 2022.
Frédéric CAMPANA
Université de Lorraine
Département de Mathématiques
F 54506, Vandœuvre-Les-Nancy (France)
frederic.campana@univ-lorraine.fr
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