Rational connectedness modulo the Non-nef locus

0.28 MB·English

by  Amaël Broustet

RATIONAL CONNECTEDNESS MODULO THE NON-NEF LOCUS
AMAËL BROUSTET AND GIANLUCA PACIENZA
9 Abstra
t. It is well known that a smooth proje
tive Fano variety is rationally
on-
0
ne
ted. Re
entlyZhang[Z2℄(andlaterHa
onandM
Kernan[HM℄asaspe
ial
aseof
0
their work on the Shokurov RC-
onje
ture) proved that the same
on
lusion holds for
2 (X,∆) −(KX +∆)
a klt pair su
h that is big and nef. We prove here a natural gen-
n
eralization of the above result by dropping the nefness assumption. Namely we show
a (X,∆) −(KX +∆)
J that a klt pair −(KXsu+
h∆t)hat is big is rationally
onne
ted modulo the
non-nef lo
us of . This result is a
onsequen
e of a more general stru
ture
8 (X,∆) −(KX +∆)
2 theorem for arbitrary pairs with pse(cid:27).

]
G
1. Introdu
tion
A
.
h An important and nowadays
lassi
al property of smooth Fano varieties is their ra-
t tional
onne
tedness, whi
h was established by Campana [Ca2℄ and by Kollár, Miyaoka
a
m and Mori [KMM℄. Singular varieties, and more generally pairs, with a weaker positivity
[ property arise naturally in the Minimal Model Program, and it was
onje
tured that a
(X,∆) −(K +∆)
1 klt pair with X big and nef is rationally
onne
ted. The
onje
ture was
v proved by Zhang [Z2℄, and was also obtained later by Ha
on and M
Kernan [HM℄ as a
4
spe
ial
ase of their work on the Shokurov RC-
onje
ture. Even in the smooth
ase, as
9
4 soon as the nefness hypothesis is dropped one
an loose the rational
onne
tedness, as
4
shown bythefollowingexample(whi
h isanaturalgeneralizationtoarbitrarydimension
.
1 of [T, Example 6.4℄).
0
PN PN+1 Z
9
0 ENx+am1pleP1N.1. LetC ⊂bePNa+h1yperplane of , Xa smooth hypersurfa
e Cof degree
Z Z
v: in and X P1 a
one over it. Let be the bloZwing up of Xat its
vertex. Noti
e that is a -bundle over the stri
t transform of . The variety has
i K = −H +aE E
X X

anoni
al divisor equal to where is the ex
eptional divisor (whi
h is
Z H X C C
r Z Z
a isomorphi
to ) and the pull-ba
k on of an hyperplane se
tion of . Sin
e
N+1 a = −1 −K
X
has multipli
ity at its vertex, an easy
omputation shows that . Thus
X Z
is big, but is not rationally
onne
ted sin
e is not.
One way to measure the la
k of nefness of a divisor is the non-emptyness of a pertur-
bation of its (stable) base lo
us. To be pre
ise, let us re
all (cid:28)rst that the stable base
SBs(D) D ∩ Bs(mD)
m≥1
lo
us ofadivisor isgivenbytheset-theoreti
interse
tion ofthe
base lo
iof allits multiples. Then one de(cid:28)nes, following[N℄ (see also [Bou℄ de(cid:28)nition3.3
D NNef(D) := ∪ SBs(mD +A)
m
and theorem A.1), the non-nef lo
us of to be the set ,
A
where is a (cid:28)xed ample divisor . One
he
ks that the de(cid:28)nition does not depend on the
Date: January 28, 2009.
Key-words : Rational
onne
tedness, non-nef lo
us, big and pse(cid:27) divisors, rational
urves.
A.M.S.
lassi(cid:28)
ation : 14J40.
1
A NNef(D) = ∅ D

hoi
e of , and that if, and only if, is nef. Noti
e that the
ountable
NNef(D) R(X,D)
union de(cid:28)ning is a (cid:28)nite one, if the ring is (cid:28)nitely generated (see Ÿ2.4
NNef(D) D
below). In any
ase is not dense, as soon as is pseudo-e(cid:27)e
tive, and we
NNef(D) ⊂ SBs(D)
always have . We prove the following.
(X,∆) −(K +∆) X
X
Theorem 1.2. Let be a pair su
h that is big. Then is rationally
NNef(−(K +∆))∪Nklt(X,∆)
X

onne
ted modulo an irredu
ible
omponent of , i.e. there
V NNef(−(K +∆))∪Nklt(X,∆)
X
exists an irredu
ible
omponent of su
h that for any
x X R x
x
general point of there exists a rational
urve passing through and interse
ting
V
.
Nklt(X,∆)
Ifthepairisklt, i.e. ifthenon-kltlo
us isempty(seeŸ2.2forthede(cid:28)nition
of the non-klt lo
us), we immediately dedu
e the following.
(X,∆) −(K +∆) X
X
Corollary 1.3. Let be a klt pair su
h that is big. Then is rationally
−(K +∆)
X

onne
ted modulo an irredu
ible
omponent of the non-nef lo
us of .
X NNef(−K ) = E
X
Noti
e that if is as in the Example 1.1 then .
Another immediate
onsequen
e of Theorem 1.2 is the following generalization of
Zhang's result [Z2, Theorem 1℄ to arbitrary pairs.
(X,∆) −(K +∆) X
X
Corollary 1.4. Let be a pair su
h that is big and nef. Then is
Nklt(X,∆)
rationally
onne
ted modulo the non-klt lo
us .
Again by Example 1.1, one
an see that the previous statement is optimal. Indeed,
X ∆ E X
take as in Example 1.1, and equal to the ex
eptional divisor . Then is not
Nklt(X,∆) = E
rationally
onne
ted, but it is rationally
onne
ted modulo .
It may be interesting to point out that, thanks to the work of Campana [Ca1℄, from
Theorem 1.2 we also dedu
e the following.
(X,∆) −(K +∆) V
X
Corollary 1.5. Let be a pair su
h that is big. Let be an irredu
ible
NNef(−(K +∆))∪Nklt(X,∆) X
X

omponent of su
h that is rationally
onne
ted modulo
V V′ V π (V′) π (X)
1 1
. Then, if is a desingularization of , the image of inside has (cid:28)nite
index.
Theorem 1.2 is a spe
ial
ase of a more general statement, whi
h may be regarded as
a generalization of the main result of Zhang in [Z1℄. Namely we have the following.
(X,∆) −(K +∆) f : X 99K Z
X
Theorem 1.6. Let be a pair su
h that is pse(cid:27). Let be
a dominant rational map with
onne
ted (cid:28)bers. Suppose that
f NNef(−(K +∆))∪Nklt(X,∆) Z.
X
(1.1) the restri
tion of to does not dominate
Z κ(Z) = 0
Then, either (i) the variety is uniruled; or (ii) . Moreover, in the
ase
−(K + ∆) Z
X
is big, under the same assumption (1.1) we ne
essarily have that is
uniruled.
X
Noti
e that the possibility (ii) in Theorem 1.6 does o

ur, as one
an see by taking
Z
to be the produ
t of a proje
tive spa
e and an abelian variety . To put Theorem 1.6
X
into perspe
tive, noti
e that in general the image of a variety with pse(cid:27) anti
anoni
al
divisor may be of general type. Indeed we have the following instru
tive example (due
to Zhang [Z2, Example, pp. 137(cid:21)138℄).
2
C g A
Example 1.7. Let be a smooth
urve of arbitrary genus . Let be an ample line
deg(A) > 2deg(K ) S := P(O (A)⊕O )
C C C
bundle with . Consider the surfa
e , together
π : S → C −K
S
with the natural proje
tion . Then : (i) is big, but not nef; (ii) for any
m > 0 | − mK |
S
integer the linear system
ontains a (cid:28)xed
omponent dominating the
C
base .
In the smooth
ase the proof of Theorem 1.6 yields a better result, whi
h is optimal
by Example 1.7.
X −K f :
X
Theorem 1.8. Let be a smooth proje
tive variety su
h that is pse(cid:27). Let
X 99K Z
be a dominant rational map with
onne
ted (cid:28)bers. Suppose that
f Cosupp(J(||−(K +∆)||)) Z.
X
(1.2) the restri
tion of to does not dominate
Z κ(Z) = 0
Then, either (i) the variety is uniruled; or (ii) . Moreover, in the
ase
−(K + ∆) Z
X
is big, under the same assumption (1.2) we ne
essarily have that is
uniruled.
As a by-produ
t of the previous result we obtain, in the smooth
ase, the following
improvement of Theorem 1.2.
X −K X
X
Theorem 1.9. Let be a smooth proje
tive variety su
h that is big. Then is
Cosupp(J(X,||−(K )||))
X
rationally
onne
ted modulo the lo
us .
X E = NNef(−K )
X
Again, noti
e that if is as in the Example 1.1 then is equal to
Cosupp(J(X,||−(K )||))
X
.
Moreover, as remarked by Zhang in [Z1℄ and [Z2℄, Theorems 1.6 and 1.8 allow to
obtain the following informations on the geometry of the Albanese map, whi
h may be
seen as generalizations of [Z1, Corollary 2℄ and [Z2, Corollary 3℄.
(X,∆) −(K +∆) Alb : X 99K
X X
Corollary 1.10. Let be a pair su
h that is pse(cid:27). Let
Alb(X) X NNef(−(K +
X
be the Albanese map (from any smooth model of ). Suppose that
∆))∪Nklt(X,∆) Alb(X)
does not dominate . Then the Albanese map is dominant with

onne
ted (cid:28)bers.
X −K
X
Corollary 1.11. Let be a smooth proje
tive variety su
h that is pse(cid:27). Let
Alb : X → Alb(X) Cosupp(J(X,|| − (K )||))
X X
be the Albanese map. Suppose that
Alb(X)
does not dominate . Then the Albanese map is surje
tive with
onne
ted (cid:28)bers.
Our approa
h to prove Theorem 1.6 is similar to [Z1℄, [Z2℄ and to [HM℄, and goes as
follows: using the hypotheses and multiplier ideal te
hniques we prove that there exists
L ∼ −(K +∆) (∆+L)
Q X |Xz
an e(cid:27)e
tive divisor su
h that the restri
tion to the general
f −(K +∆)
X
(cid:28)ber of is klt (if is not big we a
tually add a small ample to it). Then,
X Y f
supposing for simpli
ity that and are smooth, and regular, we invoke Campana's
f (K /Z + ∆ + L) ∼ −K
∗ X Q Z
positivity result [Ca3, Theorem 4.13℄ to dedu
e that is
Z
weakly positive. Then by results
ontained in [MM℄, [BDPP℄ and [N℄ the variety is
for
ed to be either uniruled or to have zero Kodaira dimension. The te
hni
al di(cid:30)
ulties

ome from the singularities of the pair, and from the indetermina
ies of the map, but
they may be over
ome by working on an appropriate resolution. In parti
ular, the
detailed and very ni
e a

ount given by Bonavero in [Bon, Ÿ7.7℄ on how to reprove [Z2℄
using the methods of [HM℄ was of great help for us.
A
knowledgements. We thank Shigeharu Takayama for bringing to our attention
the problem from whi
h this work arose.
3
2. Preliminaries
We re
all some basi
de(cid:28)nitions and results.
2.1. Notation and
onventions. We work over the (cid:28)eld of
omplex numbers. Unless
D D′ Q
otherwisespe
i(cid:28)ed, adivisorwillbeintegralandCartier. If and are -divisorsona
X D ∼ D′ D D′ Q
Q
proje
tive variety we write ,andsaythat and are -linearlyequivalent,
D −D′ D
if an integral non-zerommDultiple of is linearlyϕequiv:alXent99tKo PzeHro0.(XA,dOivi(smorD))i∗s
mD X
big if a large multiple indu
es a birational map .
D
A divisor is pseudo-e(cid:27)e
tive (or pse(cid:27)) if its
lass belongs to the
losure of the e(cid:27)e
tive
Eff(X) ⊂ N1(X)

one . For the basi
properties of big and pse(cid:27) divisors and of their
X
Y
o×nePs1se9e9K[LX1, Ÿ2.2℄.YA variety is uniruled idfimth(eYre) e=xidsitms (aXd)o−m1inant rational map
where isavarietyofdimension . Asa
ombination
of the deep results
ontainedin[MM℄ and[BDPP℄ we have the following
hara
terization
X
of uniruledness : a proper algebrai
variety is uniruled if, and only if, there exists a
X′ X KX′
smooth proje
tive variety birational to whose
anoni
al divisor is not pse(cid:27).
X
2.2. Pairs. We need to re
all some terminology about pairs. Let a normal proje
tive
∆ = a D Q K +∆ Q
P i i X
variety, an e(cid:27)e
tive Weil -divisor. If is -Cartier, we say that
(X,D) f : Y → X (X,∆)
isa pair. Consider a log-resolution of thepair , that isa proper
Y f∗∆ + E
birational morphism su
h that is smooth and is a simple normal
rossing
E f
divisor, where is the ex
eptional divisor of . There is a uniquely de(cid:28)ned ex
eptional
b E Y
divisor Pj j j on su
h that
KY = f∗(KX +∆)−(f−1)∗∆+XbjEj,
j
(f−1) ∆ ∆ Y
∗
where denotes the stri
t transform of in .
(X,∆)
We say that islog-terminalorKawamatalog-terminal(kltfor short) if forea
h
i j a < 1 b > −1
i j
and , we have and . If the previous inequalities are large, we say that
(X,∆)
the pair is log-
anoni
al. We say that the pair is klt or log-
anoni
al at a point
x D ,E
i j
if these inequalift−ie1s({axr}e)veri(cid:28)ed forNekal
th(Xo,f∆t)he divisors hxaving a non empty
interse
tion with . We
all the set of points at whi
h the pair
(X,∆)
is not klt. We refer the reader to [Kol2℄ for a detailed treatment of this subje
t.
2.3. Multiplier ideals. We willfreely use the languageof multiplieridealsas des
ribed
|D|
in [L2℄. We re
all in parti
ular that we
an asso
iate to a
omplete linear serie and
c J(c · kDk)
a rational number , an asymptoti
multiplier ideal de(cid:28)ned as the unique
maximal element for the in
lusion in c
{J( ·b(|kD|)} .
k>1
k
k > 1 b(|kD|) ⊂ J(k · kDk)
For any integer , there is an in
lusion . One important
D
property is that, when the divisor is big, the di(cid:27)eren
e between the two ideals above
k
is uniformly bounded in (see [L2, Theorem 11.2.21℄ for the pre
ise statement).
2.4. Non-nef lo
us. Thenon-neflo
us(also
alledtherestri
tedbaselo
us)ofadivisor
L L
dependonlyonthenumeri
alequivalen
e
lassof (see[Na℄). Moreover, by[ELMNP,
X
Corollary 2.10℄ if is smooth we have that
NNef(L) = ∪ Cosupp(J(X,||mL||)).
m∈N
4
L NNef(L) = ∅ L
As mentioned in the introdu
tion is nef if, and only if, and moreover is
NNef(L) 6= X
pseudo-e(cid:27)e
tive if,andonlyif, . Fortheproofsofthepropertiesmentioned
above and more details on non-nef and stable base lo
i, see [ELMNP, Ÿ1℄ and [N, III,
NNef(L)
Ÿ1℄. Note that though there is no known example, it is expe
ted that is not
R(X,D)
NZaNreisfk(iL-
)lo=sebd(|iknDg|e)neral. Nevertheless, when the alkgebra is (cid:28)nitely generated,
for a su(cid:30)
iently large integer : apply [L2, Example 11.1.3℄ and
[ELMNP, Corollary 2.10℄.
2.5. Positivity of dire
t images. Re
all the following positivity result for dire
t im-
ages, due toCampana[Ca3, Theorem 4.13℄, whi
h improves on previous results obtained
by Kawamata [Ka℄, Kollár [Kol1℄ and Viehweg [Vie1℄. We refer the reader to [Vie2, Ch.
2℄ for the de(cid:28)nition and a detailed dis
ussion of the notion of weak positivity, whi
h will
not be dire
tly used in this paper.
f : V′ → V
Theorem 2.1 (Campana). Let be a morphism with
onne
ted (cid:28)bres between
∆ Q V′
smooth proje
tive varieties. Let be an e(cid:27)e
tive -divisor on whose restri
tion to
the generi
(cid:28)bre is log-
anoni
al. Then, the sheaf
f∗OV′(m(KV′/V +∆))
m m∆
is weakly positive for all positive integer su
h that is integral.
We will rather use the following
onsequen
e of Campana's positivity result (for a
proof see e.g. [D, Se
tion Ÿ6.2.1℄).
f : V′ → V
Corollary 2.2. Let be a morphism with
onne
ted (cid:28)bres between smooth
∆ Q V′
proje
tivevarieties. Let bean e(cid:27)e
tive -divisoron whoserestri
tionto the generi
W f
(cid:28)bre is log-
anoni
al. Let be a general (cid:28)ber of and suppose that
κ(W,m(KV′/V +∆)|W) ≥ 0.
H V b > 0
Then, for every ample divisor on there exists a positive integer su
h that
h0(V′,b(KV′/V +∆+f∗H)) 6= 0.
3. Proof of the results
3.1. The proof of the main result. We will use the following standard result.
(X,∆) T f : X → T
Lemma 3.1. Let be a pair, a smooth quasi-proje
tive variety,
X f ∆
t
a surje
tive morphism, and the general (cid:28)ber of . Suppose that is big and that
(X ,∆ ) Q A B A B
t |Xt
is klt. Then there exist two -divisors and , with ample and e(cid:27)e
tive
∆ ∼ A+B (X ,(A+B) ) (X ,B )
su
h that Q , and the restri
tions t |Xt and t |Xb are klt.
Another easy fa
t whi
h we will use is the following.
a b X f : Y → X
Lemma 3.2. Let f,−1atwo idfe−a1lbsheafs of . Let E F a proper birational
Ymorphism suO
h (th−aFt ) = f−a1nad Oar(e−iEnv)e=rtifbl−e1.bLet anOd (−tEwo−e(cid:27)Fe)
t=ivfe−d1i(vais⊗orbs)on
Y Y Y
su
h that and . Then .
An important te
hni
al ingredient of the proof is provided by the following.
(X,∆) L Q X
Proposition 3.3. Let pair and a big -Cartier divisor on . Then there
Q D ∼ L Nklt(X,∆ + D) ⊂ NNef(L) ∪
Q
exists an e(cid:27)e
tive -Cartier divisor su
h that
Nklt(X,∆)
.
5
D
Proof. We will
onstru
t as the push-forward of a divisor on a well
hosen birational
(X,∆)
model of .
f : Y → X (X,∆) ∆
Y Y
Let be a log-resolution of the pair . Let be an e(cid:27)e
tive
Q Y (f ) ∆ = ∆
Y ∗ Y
-divisor on su
h that and
K +∆ = f∗(K +∆)+F
Y Y Y X Y
F ∆
Y Y
with e(cid:27)e
tive and not having
ommon
omponents with . Note that
f (Nklt(Y,∆ )) ⊂ Nklt(X,∆).
Y Y
(3.1)
E Y
By [L2, Theorem 11.2.21℄m, t>her0e is an e(cid:27)eJ
ti(vme d·kivfis∗oLrk) ⊗onO (−suE
)h⊂thba(t|mfofr∗eLa|
)h suf-
(cid:28)
iently divisible integer we have Y Y Y . Let
m > 0
be a large and su(cid:30)
iently divisible integer su
h that
1
Nklt(Y,∆ + E) = Nklt(Y,∆ ).
Y Y
(3.2) m
f : V → Y J(m·kf∗Lk)⊗O (−E) b(|mf∗L|)
Let V be a
ommon log-resolutionof Y Y and Y
∆ Q (f ) ∆ =
V V ∗ V
(∆see+[L12,EDe(cid:28)nition 9.1.12℄). Let be an e(cid:27)e
tive -divisor su
h that
Y m and
1
K +∆ = f∗(K +∆ + E)+F
V V V Y Y m V
F ∆ (V,∆ )
V V V
with e(cid:27)e
tive not having
ommon
omponents with . Again, the pair
veri(cid:28)es
1
f (Nklt(V,∆ )) ⊂ Nklt(Y,∆ + E) = Nklt(Y,∆ ).
V V Y Y
(3.3) m
G
1
Denote by the e(cid:27)e
tive divisor su
h that
O (−G −f∗E) = f−1(J(m·kf∗Lk)⊗O (−E)),
V 1 V V Y Y
G
2
and by the divisor su
h that
O (−G ) = f−1b(|mf∗L|).
V 2 V Y
J(m·kf∗Lk)⊗O (−E) ⊂ b(|mf∗L|) G +f∗E > G
Sin
e Y X Y , we have 1 V 2. Moreover, by
G
1
Lemma 3.2, the divisor veri(cid:28)es
f (G ) = Cosupp(J(m·kf∗Lk)).
(3.4) V 1 Y
|N| V
Noti
e that by
onstru
tion there exists a base-point-free linear series on su
h
f∗f∗mL ∼ N + G |N| Q N′ ∼ N
that V Y 2. Sin
e is base-point-free, there is a -divisor Q
su
h that we have
1 1
Z := Nklt(V,∆ + (G +N′)) = Nklt(V,∆ + G ) ⊂ SuppG ∪Nklt(V,∆ ).
V 1 V 1 1 V
(3.5) m m
By (3.3), (3.4) and (3.5) we get
f (Z) ⊂ Cosupp(J(m·kf∗Lk))∪Nklt(Y,∆ ) ⊂ NNef(f∗L)∪Nklt(Y,∆ ).
V Y Y Y Y
G +f∗E+N′ > G +N′ (f ) (G +f∗E+N′) > (f ) (G +N′)
Sin
e 1 V 2 , we have V ∗ 1 V V ∗ 2 , thus
1 1
Nklt(Y,∆ + ((f ) (N′ +G ))) ⊂ Nklt(Y,∆ + (E +(f ) (N′ +G ))).
Y V ∗ 2 Y V ∗ 1
m m
Sin
e we have
1 1
Nklt(Y,∆ + (E +(f ) (N′ +G ))) = f (Nklt(V,∆ + (N′ +G ))),
Y V ∗ 1 V V 1
m m
6
we dedu
e that
1
Nklt(Y,∆ + ((f ) (N′ +G ))) ⊂ NNef(f∗L)∪Nklt(Y,∆ ).
Y m V ∗ 2 Y Y
(f ) (N′+G )) ∼ mf∗L
Notethat V f∗ (NNef2(f∗LQ)) ⊂ NY N.efT(Lha)nksto(3.D1):t=o
(ofn
)lu(d1e(t(hfe)pr(oNof′i+tiGssu))(cid:30))
ient
to prove that Y Y and set Y ∗ m V ∗ 2 . Let
A Q Y W SBs(f∗L+ A)
an ample -divisor on and an irredu
ible
omponent of Y . Then
Q H X A−f∗H
there exists an ample -divisor on su
h that Y is ample. Thus
f (W) ⊂ f (SBs(f∗(L+H))) ⊂ SBs(L+H).
Y Y Y
A
This is true for every and therefore
f (NNef(f∗L)) ⊂ NNef(L).
Y Y
From the previous in
lusions, we
on
lude that
1
Nklt(X,∆+ (f ) ((f ) (N′ +G ))) ⊂ NNef(L)∪Nklt(X,∆)
Y ∗ V ∗ 2
m
D := (f ) (1((f ) (N′ +G ))) (cid:3)
and setting Y ∗ m V ∗ 2 we are done.
−(K +∆)
X
Proof of Theorem 1.6. We give the proof in the
ase is pse(cid:27) and point out
at the end how to obtain a stronger
on
lusion when it is big.
Z H X
X
We may assume that is smooth. Fix an ample divisor on , and a rational
0 < δ ≪ 1 L = −(K +∆)+
X
number . From Proposition 3.3 applied to the big divisor
δH Q D ∼ −(K + ∆)+ δH
X Q X X
we dedu
e the existen
e of an e(cid:27)e
tive -divisor su
h
Nklt(X,∆+D) Y
that does not dominatethe base . Froma result on generi
restri
tions
of multiplierideals[L2, Theorem 9.5.35℄, we have that the restri
tion to the generi
(cid:28)ber
(X ,(∆+D) )
z |Xz
is a klt pair. In
on
lusion, by the above dis
ussion and Lemma 3.1,
(X,∆)
we may repla
e our original pair by a new pair, whi
h, by abuse of notation, we
(X,∆)
will again
all , su
h that:
K +∆ ∼ δH
X Q X
(i) ;
∆ = A + B A B Q
(ii) , where and are -divisor whi
h are respe
tively ample and
A A ∼ δH
e(cid:27)e
tive, and may be taken su
h that Q 2 X;
(X ,∆ ) (X ,B )
z |Xz z |Xz
(iii) the restri
tions and to the (cid:28)ber over the general point
z ∈ Z
are klt.
µ : Y → X
Consider a proper birational morphism from a smooth proje
tive variety
Y X (X,∆)
onto providing a log-resolution of the pair and resolving the indetermina
ies
X 99K Z
of the map . In parti
ular we have the following
ommutative diagram
µ //
Y X
(3.6) A (cid:31)
A
AAA (cid:31) f
g AA (cid:15)(cid:15)(cid:31)
Z.
We write
K +Σ = µ∗(K +∆)+E,
Y X
(3.7)
Σ E E µ
where and aree(cid:27)e
tive divisorwithout
ommon
omponentsand is -ex
eptional.
E
Noti
e that from (3.7) and the fa
t that is ex
eptional we dedu
e that
κ(Y,K +Σ) = κ(X,K +∆)
Y X
(3.8)
7
whereas from (3.7) together with (iii) above we get that
(Y ,Σ ) g z ∈ Z
z |Yz
(3.9) the restri
tion to the (cid:28)ber of over the general point is klt.
More pre
isely, if we write
δ
Σ = µ∗A+B ∼ µ∗H +B
Y Q X Y
(3.10) 2
we have
(3.11)
(Y ,(B ) ) g z ∈ Z
z Y |Yz
the restri
tion to the (cid:28)ber of over the general point is klt.
We have the following result.
Q H Z
Z
Lemma 3.4. In the above setting, there exists an ample -divisor on su
h that
µ∗H ∼ 2g∗H + Γ Γ (Y ,(B + Γ) )
X Q Z z Y Yz
where is e(cid:27)e
tive and the restri
tion to the
general (cid:28)ber is klt.
F µ 0 < ε ≪ 1
Proof of Lemma 3.4. Let be the ex
eptional divisor of . For the divisor
µ∗H −εF H Z m ≫ 0
X 1
is ample. Take a very ampledivisor on , an integer large enough
m(µ∗H −εF)−g∗H
X 1
so that is base-point-free and a general member
A ∈ |m(µ∗H −εF)−g∗H |.
m X 1
µ∗H ∼ 1A + εF + 1g∗H m ≫ 0 0 < ε ≪ 1 z ∈ Z
Then X Q m m m 1. For and , and general,
the pair
1
(Y ,( A +εF +B) )
z m m |Yz
(Y ,B ) H := 1 H
is klt, sin
e z |Yz is. To
on
lude the proof it is su(cid:30)
ient to set Z 2m 1 and
Γ := 1A +εF (cid:3)
m m .
By Lemma 3.4 and (3.7) we have that
(K +B +Γ) ∼ (K +Σ) ∼ E
Y/Z Y |Yz Q Y |Yz Q |Yz
b > 0
is e(cid:27)e
tive. Therefore, by Corollary 2.2, there exists a positive integer su
h that
h0(Y,b(K +B +Γ+g∗(δH ))) 6= 0.
Y/Z Y Z
(3.12)
Consider a divisor
M ∈ |b(K +B +Γ+g∗(δH ))|.
Y/Z Y Z
(3.13)
C ⊂ Y X µ
Let be a mobile
urve
oming from , i.e. the pull-ba
k via of a general
X C ·M ≥ 0

omplete interse
tion
urve on . Then . Equivalently we have
C ·g∗K ≤ C ·g∗(δH )+b(K +B +Γ)·C.
Z Z Y Y
(3.14)
Noti
e that by (3.7), (3.10) and Lemma 3.4 we have
(K +B +Γ) ∼ µ∗(K +∆)+E +2δg∗H .
Y Y Q X Z
(3.15)
Now,
µ∗(K +∆)·C → 0 δ → 0, K +∆ ∼ δH
X X Q X
(a) as sin
e by (i) above;
E·C = 0, E µ C µ X
(b) sin
e is -ex
eptional and is the pull-ba
k via of a
urve on ;
2δ(g∗H )·C → 0 δ → 0
Z
(
) as .
8
δ → 0
Therefore letting in (3.14), from (3.15) and (a), (b) and (
) we dedu
e
C ·g∗K ≤ 0.
Z
(3.16)
C · g∗K < 0 Z
Z
If we have stri
t inequality , then is uniruled by [MM℄. Assume now
C · g∗K = 0 Z K
Z Z
that and is not uniruled. By [BDPP℄ the
anoni
al
lass is pse(cid:27)
K ≡ P + N
Z
and we
an therefore
onsider its divisorial Zariski de
omposition into a
P 1 N
positive part whi
h isnef in
odimension , and a negative one , whi
h isan e(cid:27)e
tive
R
-divisor (see [Bou℄ and [N℄). Sin
e the family of general
omplete interse
tion forms a
g(C)·K = 0
Z

onne
ting family (in the sense of [BDPP, Notation 8.1℄) and , by [BDPP,
P ≡ 0 κ (Z) = 0
σ
Theorem 9.8℄ we have that . Then the numeri
al Kodaira dimension
κ(Z) = 0
and hen
e , by [N, Chapter V, Corollary 4.9℄. This
on
ludes the proof of the
−(K +∆)
X
theorem when is only assumed to be pse(cid:27).
−(K +∆) δH
X X
If moreover is big, then we do not need to add the small ample to
it in order to apply Proposition 3.3. In this
ase we may therefore repla
e our original
(X,∆) (X,∆)
pair by a new pair, again
alled , su
h that:
K +∆ ∼ 0
X Q
(j) ;
∆ = A + B A B Q
(jj) , where and are -divisor whi
h are respe
tively ample and
e(cid:27)e
tive;
(X ,∆ ) (X ,B )
z |Xz z |Xz
(jjj) the restri
tions and to the (cid:28)ber over the general point
z ∈ Z
are klt.
Then we argue as in the pse(cid:27)
ase. Noti
e that in this
ase, from (3.8) and (j) above we
dedu
e that
κ(Y,K +Σ) = 0.
Y
(3.17)
µ∗A ∼ 2g∗A + Γ A Z Γ
Q Z Z
As in Lemma 3.4, we write , with ample on , e(cid:27)e
tive and
(Y ,(B + Γ) )
z Y Yz
the restri
tion to the general (cid:28)ber klt. As before, using Campana's
positivity result we dedu
e
h0(Y,b(K +B +Γ+g∗(A ))) 6= 0.
Y/Z Y Z
(3.18)
m > 0 H0(Z,mb(K +A ))
Z Z
Hen
e for all su(cid:30)
iently divisible, the group inje
ts into
H0(Y,mb(K +B +Γ+g∗A +f∗(K +A ))) = H0(Y,mb(K +B +µ∗A))
Y/Z Y Z Z Z Y Y
K +Σ = K +B +µ∗A
Y Y Y
(for the equality above we use again Lemma 3.4). Sin
e we
dedu
e that
0 = κ(Y,K +Σ) ≥ κ(Z,K +A ).
Y Z Z
(3.19)
Z K
Z
If the variety were not uniruled, by [BDPP℄ and [MM℄ this would imply that is
K +A
Z Z
pse(cid:27). Therefore is big, and (3.19) yields the desired
ontradi
tion. The proof
−(K +∆) (cid:3)
X
of the theorem in the
ase big is now
ompleted.
−(K + ∆)
X
Noti
e that, when is big, the proof of Theorem 1.6 shows the following
statement. whi
h will be used in the proof of Theorem 1.2.
(X,∆) κ(K +∆) = 0 ∆
X
Theorem 3.5. Let be a pair su
h that , and that is big. Let
f : X 99K Z Nklt(X,∆)
be a rational map with
onne
ted (cid:28)bers. Assume that doesn't
Z Z
dominate , Then is uniruled.
9
3.2. The proof of the
onsequen
es of the main result.
(X,∆) −(K +∆)
X
Proof of Theorem 1.2. Let be a pair with big. From Proposition 3.3
D ∼ −(K +∆)
Q X
we dedu
e that there exists an e(cid:27)e
tive divisor su
h that
Nklt(X,∆+D) ⊂ NNef(−(K +∆))∪Nklt(X,∆).
X
ν : X′ → X (X,∆+D)
Let a log-resolution of . We write
KX′ +∆′ = ν∗(KX +∆+D)+E,
(3.20)
∆′ E E ν
where and aree(cid:27)e
tivedivisorswithout
ommon
omponentsand is -ex
eptional.
E
Noti
e that from (3.20) and the fa
t that is ex
eptional we dedu
e that
κ(X′,KX′ +∆′) = κ(X,KX +∆+D) = 0
(3.21)
ν(Nklt(X′,∆′)) ⊂ NNef(−(K +∆))∪Nklt(X,∆)
X
Noti
e also that .
ρ : X′ 99K R′ := R(X′)
Consider now the rational quotient . By the main theorem
R′
of [GHS℄, the variety is not uniruled. Therefore from Theorem 3.5 we obtain that
Nklt(X′,∆′) R′
dominates .
x ∈ X y ∈ X′ ν(y) = x
Let be a general point. Let be a general point su
h that . Sin
e
f
by generi
smoothness ageneral(cid:28)berof issmooth, thus rationally
onne
ted, there isa
R y Nklt(X′,∆′)
y
single rational
urve passing through and interse
ting . Proje
ting this
X ν(Nklt(X′,∆′)) ⊂ NNef(−(K +∆))∪Nklt(X,∆)
X

urve on , and using the fa
t that ,
X R := ν(R )
x y
we have the same property for , that is : there exists a rational
urve
x NNef(−(K +∆))∪Nklt(X,∆)
X

ontaining andinterse
tinganirredu
ible
omponentof
X′ NNef(−(K +∆))∪Nklt(X,∆)
X
(it is su(cid:30)
ient to take any irredu
ible
omponent of
ν(Nklt(X′,∆′)) R′ (cid:3)

ontaining the
omponent of dominating ).
(cid:3)
Proof of Corollary 1.5. Thestatementfollowsimmediatelyfrom[Ca1, Theorem2.2℄.
Proof of Theorem 1.8. The stru
ture of the proofisidenti
altothat inthe singular
ase.
In the smooth
ase we get a better result be
ause instead of Proposition 3.3 we may
invoke the following result.
X L
Proposition 3.6. [L2, Example 9.3.57 (i)℄ Let be a smooth proje
tive variety and
Q X D ∼ L
Q
a big -Cartier divisor on . Then there exists an e(cid:27)e
tive divisor su
h that
Nklt(X,D) ⊂ J(||L||)
.
(cid:3)
The
on
lusion now follows.
f g
X → Y → Alb(X) Alb
X
Proof of Corollary 1.11. Let be the Stein fa
torization of .
Sin
e an abelian variety does not
ontain rational
urves, from the hypothesis and from
κ(Y) = 0 Alb (X)
X
Theorem 1.8 it follows that . From [U℄ it follows that is an abelian
variety. Therefore by [Ka℄ together with the universality of the Albanese map we have
Alb (cid:3)
X
that is surje
tive and with
onne
ted (cid:28)bers.
X′ → X
Proof of Corollary 1.10. It is su(cid:30)
ient to pass to a smooth model and to argue
(cid:3)
as in the proof of Corollary 1.11.
10