ON MAXIMAL ALBANESE DIMENSIONALVARIETIESOSAMU FUJINO
Abstract. We prove that every smooth projective variety withmaximal Albanese dimension has a good minimal model. We also
treat Ueno’s problem on subvarieties of Abelian varieties.

Contents
1. Introduction
2. Preliminaries
3. Main results
3.1. Minimal model
3.2. Abundance theorem
3.3. Iitaka–Viehweg’s conjecture
3.4. Ueno’s problem
References

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1. Introduction
In this short paper, we prove that every smooth projective variety
with maximal Albanese dimension has a good minimal model. It is aneasy consequence of [BCHM]. This paper is also a supplement to [F1].
We will also treat Ueno’s problem on subvarieties of Abelian varieties.
A key point of this paper is the simple fact that there are no rational
curves on an Abelian variety. The topics treated here seem to be easy
exercises for experts.
Let us recall the following minimal model conjecture.
Conjecture 1.1 (Good minimal model conjecture (weak form)). Let
X be a smooth projective variety defined over C. Assume that KX
is pseudo-effective. Then there exists a normal projective variety X ′which satisfies the following conditions:
Date: 2009/11/12, version 1.01.
2000 Mathematics Subject Classification. 14E30.
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OSAMU FUJINO(i) X ′ is birationally equivalent to X.
(ii) X ′ has only Q-factorial terminal singularities.
(iii) KX ′ is semi-ample.
In particular, the Kodaira dimension κ(X) = κ(X, KX ) is non-negative.
We sometimes call X ′ a good minimal model of X.
Remark 1.2. By [BCHM], we can replace (ii) with the following
slightly weaker condition: (ii′ ) X ′ has at most canonical singularities.
The conjecture: Conjecture 1.1 was established in the following cases.
(A) dim X ≤ 3 (see, for example, [FA]).
(B) varieties of general type in any dimension (see, for example,[BCHM]).
(C) ∆-regular divisors on complete toric varieties in any dimension(see [I]).
(D) irregular fourfolds (see [F4]).
As stated above, in this paper, we will prove that the conjecture: Conjecture 1.1 holds for projective maximal Albanese dimensional varieties.
Notation. For a proper birational morphism f : X → Y , the exceptional locus Exc(f ) ⊂ X is the locus where f is not an isomorphism.
We will freely use the basic notation and definitions in [KM] and [U].
Acknowledgments. The author was partially supported by The Inamori Foundation and by the Grant-in-Aid for Young Scientists (A)♯20684001 from JSPS. He thanks Professor Daisuke Matsushita whose
questions made him consider this problem.
We will work over C, the complex number field, throughout thispaper.
2. Preliminaries
Let us recall the definition of maximal Albanese dimensional varieties.
Definition 2.1. Let X be a smooth projective variety. Let Alb(X)be the Albanese variety of X and α : X → Alb(X) the corresponding
Albanese map. We say that X has maximal Albanese dimension if
dim α(X) = dim X.
Remark 2.2. A smooth projective variety X has maximal Albanesedimension if and only if the cotangent bundle of X is generically generated by its global sections, that is,
H 0 (X, Ω1X ) ⊗ OX → Ω1X

ON MAXIMAL ALBANESE DIMENSIONAL VARIETIES

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is surjective at the generic point of X. It can be checked without any
difficulties.
We note that the notion of maximal Albanese dimension is birationally invariant. So, we can define the notion of maximal Albanesedimension for singular varieties as follows.
Definition 2.3. Let X be a projective variety. We say that X has
maximal Albanese dimension if there is a resolution π : X → X suchthat X has maximal Albanese dimension.
The following lemma is almost obvious by the definition of maximal
Albanese dimensional varieties and the basic properties of Albanese
mappings. We leave the proof for the reader’s exercise.
Lemma 2.4. Let X be a projective variety with maximal Albanesedimension. Let π : X → X be a resolution and α : X → Alb(X)the Albanese mapping. Let Y be a subvariety of X. Assume that Y 6⊂
π(Exc(π) ∪ Exc(β)), where β : X → V is the Stein factorization of
α : X → Alb(X). Then Y has maximal Albanese dimension.
Let us recall some basic definitions.
Definition 2.5 (Iitaka’s D-dimension and numerical D-dimension).
Let X be a normal projective variety and D a Q-Cartier Q-divisor.
Assume that mD is Cartier for a positive integer m. Let
Φ|tmD| : X 99K Pdim |tmD|
be rational mappings given by linear systems |tmD| for positive integers
t. We define Iitaka’s D-dimension(
max dim Φ|tmD| (X), if |tmD| =6 ∅ for some t,t>0κ(X, D) =−∞,otherwise.
In case D is nef, we can also define the numerical D-dimension
ν(X, D) = max{ e | D e 6≡ 0},
where ≡ denotes numerical equivalence. We note that ν(X, D) ≥κ(X, D) holds.
In this paper, we adopt the following definition of log terminal models
for klt pairs.
Definition 2.6 (Log terminal models for klt pairs). Let f : X → S bea projective morphism of normal quasi-projective varieties. Supposethat (X, B) is klt and let φ : X 99K X ′ be a birational map of normal
quasi-projective varieties over S, where X ′ is projective over S. We

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put B ′ = φ∗ B. In this case, (X ′ , B ′ ) is a log terminal model of (X, B)
over S if the following conditions hold.
(i) φ−1 contracts no divisors.
(ii) (X ′ , B ′ ) is a Q-factorial klt pair.
(iii) KX ′ + B ′ is nef over S.
(iv) a(E, X, B) < a(E, X ′ , B ′ ) for all φ-exceptional divisors E ⊂ X.
3. Main results
3.1. Minimal model. First, we recall the following elementary butvery important lemma.
Lemma 3.1 (Negative rational curves). Let X be a projective variety
and B an effective R-divisor on X such that (X, B) is log canonical.
Assume that KX + B is not nef. Then there exists a rational curve C
on X such that (KX + B) · C < 0.
Proof. It is obvious by the cone theorem for log canonical pairs. See,for example, [F3, Proposition 3.21] and [F5, Section 18]. We note that
[K4] is sufficient when (X, B) is klt.

By Lemma 3.1, we obtain the next lemma.
Lemma 3.2. Let f : X → S be a proper surjective morphism between
projective varieties. Let B be an effective R-divisor on X such that
(X, B) is log canonical. Assume that KX + B is f -nef and S contains
no rational curves. Then KX + B is nef.
Proof. If KX + B is not nef, then there exists a rational curve C on X
such that (KX + B) · C < 0 by Lemma 3.1. Since KX + B is f -nef,
f (C) is not a point. On the other hand, S contains no rational curves
by the assumption. It is a contradiction. Therefore, KX + B is nef. 
Therefore, the following lemma is obvious by Definition 2.6 andLemma 3.2.
Lemma 3.3. Let f : X → S be a proper surjective morphism between
projective varieties. Let B be an effective R-divisor on X such that
(X, B) is klt. Let (X ′ , B ′ ) be a log terminal model of (X, B) over S.
Assume that S contains no rational curves. Then (X ′ , B ′ ) is a log
terminal model of (X, B).
We give an easy consequence of the main theorem of [BCHM].
Theorem 3.4 (Existence of log terminal models). Let X be a normal
projective variety and B an effective R-divisor on X such that (X, B)
is klt. Assume that X has maximal Albanese dimension. Then (X, B)
has a log terminal model.

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Proof. Let π : X → X be a resolution and α : X → Alb(X) the
Albanese mapping of X. Since X has only rational singularities, X →S = α(X) decomposes asπf

α : X −→ X −→ S.
See, for example, [BS, Lemma 2.4.1]. Since X has maximal Albanesedimension, f : X → S is generically finite. By [BCHM, Theorem 1.2],
there exists a log terminal model f : (X ′ , B ′ ) → S of (X, B) over S.
By Lemma 3.3, (X ′ , B ′ ) is a log terminal model of (X, B).

The following corollary is obvious by Theorem 3.4.
Corollary 3.5 (Minimal models). Let X be a smooth projective variety
with maximal Albanese dimension. Then X has a minimal model.
3.2. Abundance theorem. Let us consider the abundance theorem
for maximal Albanese dimensional varieties.
Theorem 3.6 (Abundance theorem). Let X be a projective variety
with only canonical singularities. Assume that X has maximal Albanesedimension. If KX is nef, then KX is semi-ample.
Proof. By Lemma 3.7, we have κ(X, KX ) ≥ 0. Therefore, if ν(X, KX ) =0, then κ(X, KX ) = ν(X, KX ) = 0. Thus, X is an Abelian variety by
Proposition 3.8. In particular, KX is semi-ample. By Lemma 3.9, it
is sufficient to see that ν(X, KX ) > 0 implies κ(X, KX ) > 0 becauseν(X, KX ) = κ(X, KX ) means KX is semi-ample by [K2, Theorem 1.1]
and [F2, Corollary 2.5]. See also Remark 3.11 below. By Proposition
3.8, we know that ν(X, KX ) > 0 implies κ(X, KX ) > 0. We finish theproof.

Lemma 3.7. Let X be a projective variety with only canonical singularities. Assume that X has maximal Albanese dimension. Thenκ(X, KX ) ≥ 0.
Proof. Since X has only canonical singularities, we can assume that
X is smooth by replacing X with its resolution. Then this lemma
is obvious by the basic properties of the Kodaira dimension (cf. [U,Theorem 6.10]). Note that every subvariety of an Abelian variety has
non-negative Kodaira dimension (cf. [U, Lemma 10.1]).
Proposition 3.8. Let X be a projective variety with only canonicalsingularities. Assume that X has maximal Albanese dimension, KX isnef, and κ(X, KX ) = 0. Then X is an Abelian variety. In particular,
X is smooth and KX ∼ 0.

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Proof. We know that f : X → S = Alb(X) is birational by [K1,
Theorem 1], where X is a resolution of X and Alb(X) is the Albanesevariety of X. We can write
KX = f ∗ KS + E
such that E is effective and SuppE = Exc(f ). Since KX is nef andKS ∼ 0, we obtain E = 0. This means that f : X → S is an isomorphism.
Lemma 3.9 (cf. [K2, Theorem 7.3]). Let X be a projective variety with
only canonical singularities. Assume that X has maximal Albanesedimension and that KX is nef. If ν(X, KX ) > 0 implies κ(X, KX ) > 0,then ν(X, KX ) = κ(X, KX ).
Sketch of the proof. The proof of [K2, Theorem 7.3] works without any
changes. We give some comments for the reader’s convenience. We use
the same notation as in the proof of [K2, Theorem 7.3]. By Lemma 2.4,
W has maximal Albanese dimension. On the other hand, κ(W ) = 0
by the construction. By Proposition 3.8, Wmin is an Abelian variety.
In particular, KWmin ∼ 0.
Remark 3.10. In the above proof of the abundance theorem, we did
not use [K3, Theorem 8.2]. It is because we have Proposition 3.8.
Remark 3.11. On the assumption that ν(X, KX ) = κ(X, KX ), KX is
semi-ample if and only if the canonical ring of X is finitely generated.
Therefore, by [BCHM], we can check the semi-ampleness of KX without
appealing [K2, Theorem 1.1] and [F2, Corollary 2.5].
By Corollary 3.5 and Theorem 3.6, we obtain the main result of thisshort paper.
Corollary 3.12 (Good minimal models). Let X be a smooth projective
variety with maximal Albanese dimension. Then X has a good minimal model. This means that there is a normal projective variety X ′with only Q-factorial terminal singularities such that X ′ is birationallyequivalent to X and KX ′ is semi-ample.
3.3. Iitaka–Viehweg’s conjecture. By combining Corollary 3.12 withthe main theorem of [K3], we obtain the following theorem. We write
it for the reader’s convenience. For the details, see [K3].
Theorem 3.13 (cf. [K3, Theorem 1.1]). Let f : X → S be a surjective
morphism between smooth projective varieties with connected fibers andL a line bundle on S. Assume that the geometric generic fiber of f has
maximal Albanese dimension. Then the following assertions hold:ON MAXIMAL ALBANESE DIMENSIONAL VARIETIES

7(i) There exists a positive integer n such thatc ∗ (ω n ))) ≥ Var(f ).
κ(S, det(fX/S(ii) If κ(S, L) ≥ 0, thenκ(X, ωX/S ⊗ f ∗ L) ≥ κ(Xη ) + max{κ(S, L), Var(f )},where Xη is the generic fiber of f .
The next corollary is a special case of [K3, Corollary 1.2] (cf. [F1,
Theorem 0.2].
Corollary 3.14. Under the same assumptions as in Theorem 3.13,(i) κ(X, ωX/S ) ≥ κ(Xη ) + Var(f ), and(ii) if κ(S) ≥ 0, then κ(X) ≥ κ(Xη ) + max{κ(S), Var(f )}.
3.4. Ueno’s problem. The final theorem is a supplement to [U, Remark 10.13]. It is an easy consequence of Lemma 3.1.
Theorem 3.15. Let X be a projective variety and B an effective Rdivisor on X such that (X, B) is log canonical. Assume that there are
no rational curves on X. Then KX + B is nef.
Furthermore, we assume that (X, B) is klt, B is an effective Qdivisor, and KX + B is big. Then KX + B is ample.
Proof. The first half of this theorem is obvious by Lemma 3.1. If (X, B)
is klt, B is an effective Q-divisor, and KX + B is big, then KX + B issemi-ample by the base point free theorem since KX + B is nef. Then
there exists a birational morphism
f = Φ|m(KX +B)| : X → S
with f∗ OX ≃ OS for some large and divisible integer m. By theconstruction, there is an ample Cartier divisor H on S such thatm(KX + B) ∼ f ∗ H. Assume that f is not an isomorphism. Let A be
an f -ample Cartier divisor. Then there is an effective Cartier divisor
D on X such that D ∼ −A + f ∗ lH for some large integer l. Therefore,(X, B + εD) is klt and KX + B + εD is not nef for 0 < ε ≪ 1 since f is
not an isomorphism. By Lemma 3.1, there exists a rational curve C on
X such that (KX + B + εD) · C < 0. It is a contradiction because there
are no rational curves on X. Therefore, f is an isomorphism. Thus,
KX + B is ample.
Corollary 3.16 (cf. [U, Remark 10.13]). Let W be a submanifold of acomplex torus T with κ(W ) = dim W . Then KW is ample.
Proof. By [U, Lemma 10.8], there is an Abelian variety A which is acomplex subtorus of T such that W ⊂ A. Thus, W is projective. Then,by Theorem 3.15, we obtain KW is ample.
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References[BS]

M. Beltrametti, A. Sommese, The adjunction theory of complex projective
varieties, de Gruyter Expositions in Mathematics, 16. Walter de Gruyter& Co., Berlin, 1995.
[BCHM] C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimalmodels for varieties of log general type, to appear in J. Amer. Math. Soc.
[F1]
O. Fujino, Algebraic fiber spaces whose general fibers are of maximal
Albanese dimension, Nagoya Math. J. 172 (2003), 111–127.
[F2]
O. Fujino, On Kawamata’s theorem, preprint (2008).
[F3]
O. Fujino, Introduction to the log minimal model program for log canonical pairs, preprint (2008).
[F4]
O. Fujino, Finite generation of the log canonical ring in dimension four,
to appear in Kyoto Journal of Mathematics.
[F5]
O. Fujino, Fundamental theorems for the log minimal model program,
preprint (2009).
[I]
S. Ishii, The minimal model theorem for divisors of toric varieties, Tohoku
Math. J. (2) 51 (1999), no. 2, 213–226.
[K1]
Y. Kawamata, Characterization of abelian varieties, Compositio Math.
43 (1981), no. 2, 253–276.
[K2]
Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), no. 3, 567–588.
[K3]
Y. Kawamata, Minimal models and the Kodaira dimension of algebraic
fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46.
[K4]
Y. Kawamata, On the length of an extremal rational curve, Invent. Math.
105 (1991), no. 3, 609–611.
[FA]
J. Kollár et al., Flips and Abundance for algebraic threefolds, Astérisque211, (1992).
[KM]
J. Kollár, S. Mori, Birational geometry of algebraic varieties, Cambridge
Tracts in Mathematics, Vol. 134, 1998.
[U]
K. Ueno, Classification theory of algebraic varieties and compact complex
spaces, Notes written in collaboration with P. Cherenack. Lecture Notes
in Mathematics, Vol. 439. Springer-Verlag, Berlin-New York, 1975.
Department of Mathematics, Faculty of Science, Kyoto University,
Kyoto 606-8502, Japan
E-mail address: fujino@math.kyoto-u.ac.jp