arXiv:1311.7396v1 [math.AG] 28 Nov 2013
general type
Caucher Birkar and Jungkai Alfred Chen
Abstract. Let (X, B) be a complex projective klt pair, and let f : X → Z be a surjective morphism onto a normal projective variety with maximal albanese dimension such that KX+ B is relatively big over Z. We show that
such pairs have good log minimal models.
1. Introduction
We work over C. The following is the main theorem of this paper.
Theorem 1.1. Suppose that (X, B) is a projective klt pair with B a Q-boundary
and that we are given a surjective morphism f : X → Z where Z is a normal projective variety with maximal albanese dimension (eg an abelian variety). If
KX+ B is big/Z, then (X, B) has a good log minimal model. Moreover, if F is
a general fibre of f , then
κ(KX + B) ≥ κ(KF + BF) + κ(Z) = dim F + κ(Z)
where KF + BF = (KX + B)|F and κ(Z) means the Kodaira dimension of a
smooth model of Z.
The proof is given in Section 5. Here we briefly outline the main steps of the proof. Since Z has maximal albanese dimension, it admits a generically finite morphism Z → A to an abelian variety. We can run an appropriate LMMP on
KX + B (and on related log divisors) over A by [4]. Since KX + B is big/Z,
such LMMP terminate. The log minimal model we obtain is actually globally a log minimal model (that is, not only over A) because A is an abelian variety (see Section 3). It requires a lot more work to show that the log minimal model is good, that is, to show that abundance holds for it. Such semi-ampleness statements are usually proved by establishing a nonvanishing result, creating log canonical centres, and lifting sections from a centre. For the nonvanishing part, we use some of the ideas of Campana-Chen-Peternell [6, Theorem 3.1] to
show that KX + B ≡ D for some Q-divisor D ≥ 0. This uses the
Fourier-Mukai transform of the derived category of A combined with the above LMMP
and vanishing theorems (see Section 4). One can replace KX + B ≡ D with
KX + B ∼Q D [7][14]. Next, we create a log canonical centre and lift sections
from it using the extension theorem of Demailly-Hacon-Pˇaun [9, Theorem 1.8].
Date: December 2, 2013. 2010 MSC: 14E30.
So, our proof is somewhat similar to the proof of the base point free theorem but the tools we use are quite different.
Theorem 1.1 is interesting for several reasons. First, it is a non-trivial special case of the minimal model and abundance conjectures. Second, its proof pro-vides an application of the extension theorem of [9] which is hard to apply in other contexts because of relatively strong restrictions on log canonical centres. Third, it is closely related to the following generalized Iitaka conjecture.
Conjecture 1.2. Let (X, B) be a projective klt pair with B a Q-boundary, and
let f : X → Z be an algebraic fibre space onto a smooth projective variety Z. Then
κ(KX + B) ≥ κ(KF + BF) + κ(Z)
where F is a general fibre of f and KF + BF := (KX + B)|F
It is plausible that our proof of Theorem 1.1 can be extended to give a proof of the conjecture when Z has maximal albanese dimension. In order to do so we would need an extension theorem stronger than [9, Theorem 1.8] (see Section 6 for more details). When X is smooth, B = 0, and Z has maximal albanese dimension, the conjecture was proved by Chen and Hacon [8] using very different methods.
Acknowledgements. The authors would like to thank Hajime Tsuji and
Mi-hai Pˇaun for many discussions on Conjecture 1.2. The first author was sup-ported by the Fondation Sciences Math´ematiques de Paris when he visited Jussieu Mathematics Institute in Paris, and by a Leverhulme grant. Part of this work was done while the first author was visiting the second author at the National Taiwan University, and he would like to thank everyone for their hospitality.
2. Preliminaries
2.1. Notation and conventions. We work over the complex numbers C and our varieties are quasi-projective unless stated otherwise. A pair (X, B) consists of a normal variety X and a Q-divisor B with coefficients in [0, 1] such that
KX + B is Q-Cartier. For definitions and basic properties of singularities of
pairs such as log canonical (lc), Kawamata log terminal (klt), dlt, plt, and the log minimal model program (LMMP) we refer to [16].
2.2. Divisors and Iitaka fibrations. Let f : X → Z be a projective
mor-phism of normal varieties, and Z′
the normalization of the image of f . By a
general fibre of f we mean a general fibre of the induced morphism X → Z′
.
We say that an R-divisor D on X is big/Z if D ∼R H + E/Z where H is an
ample/Z R-divisor and E is an effective R-divisor. Note that D is big/Z iff it
is big/Z′
.
Now assume that D is Q-Cartier. When X is projective, Iitaka [13, §10] proves the existence of the so-called Iitaka fibration of D assuming κ(D) ≥ 0.
More precisely, there is a resolution h : W → X and a contraction g : W → T
such that κ(D) = dim T and κ(h∗
D|G) = 0 for a general fibre G of g. A
similar construction exists for the map f even if X is not projective. That is, there is a resolution h : W → X and a contraction g : W → T /Z such that
κ(D|F) = dim V and κ(h∗D|G) = 0 where F is a general fibre of f , G is a general
fibre of g, and V is a general fibre of T → Z. This is called the relative Iitaka fibration and its existence follows from the absolute projective case: first take an appropriate compactification of X, Z and then consider the Iitaka fibration
of the divisor D + f∗
A where A is a sufficiently ample divisor on Z, then apply [13, §10, Exercise 10.1].
2.3. Minimal models. Let f : X → Z be a projective morphism of normal varieties and D an R-Cartier divisor on X. A normal variety Y /Z together with a birational map φ : X 99K Y /Z whose inverse does not contract any divisors is called a minimal model of D over Z if:
• DY = φ∗D is nef/Z,
• there is a common resolution g : W → X and h : W → Y such that E := g∗
D − h∗
DY is effective and Supp g∗E contains all the exceptional divisors of φ.
Moreover, we say that Y is a good minimal model if DY is semi-ample/Z.
If one can run an LMMP on D which terminates with a model Y on which
DY is nef/Z, then Y is a minimal model of D over Z. When D is a log divisor,
that is of the form KX + B for some pair (X, B), we use the term log minimal
model.
3. LMMP over abelian varieties.
In this section, we discuss running the LMMP over an abelian variety which we need in the later sections.
3.1. Relative LMMP for polarized pairs of log general type. Let f : X → Z be a morphism from a normal projective variety X. Let D be a Q-Cartier
divisor on X such that D ∼Q KX+ B + L where we assume D is big/Z, L is nef
globally, and (X, B) is klt. Sometimes (X, B + L) is referred to as a polarized
pair. Since D is big/Z, we can write D ∼Q H + E/Z where H is an ample
Q-divisor (globally) and E ≥ 0. For each sufficiently small rational number
δ > 0 there is a log divisor KX+ Bδ such that (X, Bδ) is klt and over Z we have
KX + Bδ ∼Q KX + B + δE + L + δH ∼Q (1 + δ)(KX + B + L)/Z
where we make use of the facts that L + δH is ample and (X, B + δE) is klt.
Now by [4], we can run an LMMP/Z on KX+ Bδ ending up with a log minimal
model Y /Z. The LMMP is also an LMMP on KX+B +L so Y is also a minimal
model/Z of KX+ B + L. Moreover, KY+ BY + LY is semi-ample/Z by applying
3.2. LMMP over an abelian variety. In addition to the assumptions of 3.1,
suppose that Z = A is an abelian variety. Then, KY + BY + LY is nef, not
only/A, but it is so globally. If this is not the case, then there would be an
extremal ray R on Y such that (KY +BY+LY)·R < 0. Then (KY +ΘY)·R < 0
for some Θ where
KX + Θ ∼Q KX + B + L + τ E
for some sufficiently small τ > 0 and such that KX + Θ and KY + ΘY are both
klt. Thus there is a rational curve generating R which is not contracted/A. This is not possible as A is an abelian variety.
3.3. Relative LMMP for certain dlt pairs. In addition to the assumptions of 3.1, suppose that D ≥ 0, ∆ := B + N for some Q-Cartier Q-divisor N ≥ 0 with Supp N = Supp D, and that (X, ∆) is lc. We show that we can run an
LMMP/Z on KX + ∆ + L ending up with a minimal model. By assumptions,
Supp ⌊∆⌋ ⊆ Supp N = Supp D
So, for each sufficiently small rational number ǫ > 0, we can write
KX + B ′ + L := KX + B + N − ǫD − ǫN + L ∼Q (1 − ǫ)(KX + ∆ + L) where KX+ B ′
is klt. So, by (1) we can run an LMMP/Z on KX+ B
′
+ L which
ends up with a minimal model/Z of KX+ B
′
+ L hence a minimal model/Z of
KX + ∆ + L.
If Z = A is an abelian variety, then by 3.2 the minimal model just constructed is global not just over A.
4. The nonvanishing
The main result of this section is the following nonvanishing statement.
Theorem 4.1. Let f : X → A be a morphism from a normal projective variety
X to an abelian variety A. Assume that
(1) D = KX + B + L is big/A, L is globally nef, and
(2) (X, B) is klt.
Then, there is P ∈ Pic0Q(A) such that κ(D + f
∗
P ) ≥ 0.
Proof. Step 1. By Section 3, there is a sequence X = X1 99K X2 99K · · · Xr =
X′
/A of divisorial contractions and log flips with respect to KX+ B + L so that
KX′+ BX′+ LX′ is nef. Let I be a positive integer so that IDX′ is Cartier, and
put Fs := f
′
∗OX′(sIDX′) where f
′
is the map X′
→ A. For any ample divisor
H on A and any P ∈ Pic0(A), by the Kawamata-Viehweg vanishing theorem,
Rif′
∗OX′(sIDX′ + f
′∗
H + f′∗
P ) = 0 for all i > 0 and s ≥ 2 and
Hi(X′
, sIDX′ + f
′∗
H + f′∗
hence
Hi(A, F
s⊗ OA(H + P )) = 0
for all i > 0 and s ≥ 2. In other words, the sheaf Fs⊗ OA(H) is IT0 for all
ample divisors H.
Let φ : ˜A → A be a projective ´etale map, and define ˜Xi = ˜A ×AXi. Then,
KX˜ + ˜B, the pullback of KX + B, is also klt, and ˜L, the pullback of L, is nef.
The birational map ˜X 99K ˜X′
/ ˜A is decomposed into a sequence of divisorial
contractions and log flips with respect to KX˜ + ˜B + ˜L and KX˜′ + ˜BX˜′ + ˜LX˜′
is nef. Therefore, we have vanishing properties on ˜X and ˜A similar to those
above.
From now on, essentially we only need the above vanishing properties so to
simplifiy notation we replace X by X′
, KX + B by KX′ + BX′, and D by DX′:
the new D is nef but of course we may loose the nefness of L.
Step 2. It turns out that Fs satisfies a generic vanishing theorem for s ≥ 2.
That is to say that we claim that Fs satisfies the following condition of Hacon
[12, Theorem 1.2]:
(∗) let ˆM be any sufficiently ample line bundle on the dual abelian variety ˆA,
and let M on A be the Fourier-Mukai transform of ˆM , and M∨
the dual of M;
then Hi(A, F
s⊗ M
∨
) = 0 for all i > 0.
The condition (∗) implies the existence of a chain of inclusions
V0(Fs) ⊃ V1(Fs) ⊃ · · · ⊃ Vn(Fs)
where
Vi(Fs) := {P ∈ Pic0(A) | Hi(A, Fs⊗ P ) 6= 0}
and n = dim A (see the comments following [12, Theorem 1.2]).
Step 3. Assume that (∗) holds. Since Fs is a sheaf of rank h0(F, sID|F)
where F is a general fibre of f , it is a non-zero sheaf for s ≫ 0 because D is big
over A. Therefore, V0(F
s) 6= ∅ for s ≫ 0 otherwise Vi(Fs) = ∅ for all i which
implies that the Fourier-Mukai transform of Fs is zero (by base change and
the definition of Fourier-Mukai transform). This contradicts the fact that the Fourier-Mukai transform gives an equivalence of the derived categories D(A)
and D( ˆA) as proved by Mukai [17, Theorem 2.2]. Now V0(F
s) 6= ∅ implies that
H0(X, sID + f∗
P ) = H0(A, Fs⊗ OA(P )) 6= 0
for some P ∈ Pic0(A). This implies the theorem.
Step 4. It remains to prove the claim (∗) above. Let φ : ˆA → A be the
isogeny defined by ˆM which is ´etale since we work over C. By [17, Proposition
3.11],
φ∗
(M∨
Let ˆf : ˆX := X ×A A → ˆˆ A be the map obtained by base change, and
ϕ : ˆX → X the induced map. Let Gs := ˆf∗OXˆ(ϕ
∗ sID). Now Hi(A, Fs⊗ M ∨ ) = 0 for all i > 0 if and only if Hi(X, OX(sID) ⊗ f ∗ M∨ ) = 0 for all i > 0 and this in turn holds if
Hi( ˆX, O ˆ X(ϕ ∗ sID) ⊗ ϕ∗ f∗ M∨ ) = 0 for all i > 0
because OX is a direct summand of ϕ∗OXˆ and the sheaves involved are locally
free. The latter vanishing is equivalent to
Hi( ˆA, Gs⊗ φ
∗
M∨
) = ⊕h0( ˆM)Hi( ˆA, Gs⊗ ˆM ) = 0 for all i > 0
which holds by Step 1.
Remark 4.2 We will abuse notation and use the same notation for a Cartier
divisor and its associated line bundle. Going a little bit further in the above
proof, one can have a stronger statement, that is, if V0(F
s) has positive
dimen-sion, then we can choose P so that κ(D + f∗
P ) > 0. Indeed fix an s ≥ 2 and
let Wsbe an irreducible component of V0(Fs). Clearly there is a natural
multi-plication map πs: Ws× Ws → V0(F2s) which sends (Q, Q′) to Q ⊗ Q′. Let W2s
be the irreducible component containing the image. Then, dim Ws ≤ dim W2s
since for each Q ∈ Ws, we have πs(Q × Ws) = Q ⊗ Ws ⊆ W2s. Thus, dim Wks
is independent of k for k ≫ 0. For such k we have: if Q, Q′
∈ Wks, then
Q ⊗ Wks = Q
′
⊗ Wks which implies that
G = Wks⊗ W −1 ks := {Q ⊗ Q ′−1 | Q, Q′ ∈ Wks}
is an abelian subgroup of ˆA, Wks⊗ G = Wks and that Wks = Q ⊗ G for any
Q ∈ Wks. Therefore, Wks is a translation of an abelian subvariety of positive
dimension. It follows that for certain Q ∈ Wks and Q
′
∈ G the two arrows
H0(F ks⊗ Q) ⊗ H0(Fks⊗ Q) H0(F ks⊗ Q ⊗ Q ′ ) ⊗ H0(F ks⊗ Q ⊗ Q ′−1 ) //H0(F 2ks⊗ Q2)
produce two sections of F2ks⊗ Q2 hence κ(D + f
∗
P ) > 0 for P = 1
ksIQ.
5. Proof of the main theorem
Remark 5.1 Let Z be a normal projective variety and let π : Z → A be a
generically finite morphism to an abelian variety A. Let Z → Z′
→ A be the Stein factorization of π. Then by Kawamata [15, Theorem 13] (also see Ueno
[18, Theorem 3.10]), there exist an abelian subvariety A1 ⊆ A, ´etale covers
˜ Z′
→ Z′
and ˜A1 → A1, and a normal projective variety Z2
′
• we have a finite morphism Z2 ′ → A2 := A/A1, • ˜Z′ ≃ ˜ A1× Z ′ 2, • κ(Z) = κ(Z′ ) = κ(Z′ 2) = dim Z ′ 2.
Proof. (of Theorem 1.1) Step 1. We use induction on dimension of X. By as-sumptions, there is a generically finite morphism Z → A to an abelian variety.
Since KX + B is big/Z, it is also big/A. So, by Theorem 4.1, KX + B ≡ D for
some Q-divisor D ≥ 0. By Campana-Koziarz-Pˇaun [7] or Kawamata [14], we
may assume that KX + B ∼Q D. After taking appropriate resolutions, we may
in addition assume that Z is smooth, X is smooth, and that Supp(B + D) has simple normal crossing singularities.
Step 2. Let N ≥ 0 be the Q-divisor such that Supp N = Supp D, (X, B + N) is lc, and Supp ⌊B + N ⌋ = Supp D. Let ∆ := B + N. By Section 3, we can
run an LMMP on KX + ∆ which ends up with a log minimal model (Y, ∆Y).
Moreover, KY +∆Y is semi-ample/A. Let Y → Y′/A be the contraction defined
by KY + ∆Y.
Now assume that every component of ⌊∆Y⌋ is contracted/Y
′
which is
equiva-lent to saying that every component of DY is contracted/Y
′
because Supp ⌊∆Y⌋ =
Supp DY. In this case
KY′ + ∆Y′ ∼Q DY′ + NY′ = 0
which implies that KY+∆Y ∼Q 0 because KY+∆Y is the pullback of KY′+∆Y′.
Therefore,
κ(DY + NY) = κσ(DY + NY) = 0
where κσ denotes the numerical Kodaira dimension defined by Nakayama. This
in turn implies that
κ(D + N) = κσ(D + N) = 0
which gives κ(D) = κσ(D) = 0 because Supp N = Supp D. Therefore, (X, B)
has a good log minimal model (cf. [11]). Moreover, since D is big/A and since
D is contracted/Y′
, X → Z is generically finite, that is, dim F = 0. Thus,
κ(KX + B) ≥ κ(X) ≥ κ(Z) = κ(KF + BF) + κ(Z)
So, from now on we may assume that there is a component S of ⌊∆⌋ such that
SY is not contracted/Y
′
.
Step 3. Choose a small rational number ǫ > 0 and put
KX + Γ := KX + ∆ − ǫ(⌊∆⌋ − S)
Then (X, Γ) and (Y, ΓY) are both plt with ⌊Γ⌋ = S and ⌊ΓY⌋ = SY. Let Y
′′
be a log minimal model of KY + ΓY over Y
′
. Such a model exists because
KX + Γ ∼Q M for some M ≥ 0 with S ⊂ Supp M ⊂ Supp Γ which allows us
to reduce the problem to the klt case similar to Section 3. In particular, this
also shows that KY′′ + ΓY′′ is semi-ample/Y
′
Therefore, since KY′ + ∆Y′ is ample/A, by choosing ǫ to be small enough, we
can assume that KY′′+ ΓY′′ is semi-ample/A.
Step 4. By the arguments of Section 3, KY′′ + ΓY′′ is nef globally. This
implies that Nσ(KY′′ + ΓY′′) = 0 which in particular means that SY′′ is not a
component of Nσ(KY′′ + ΓY′′). Moreover, by construction, (KY′ + ∆Y′)|S
Y ′ is
ample/A hence (KY′′+ ∆Y′′)|SY ′′ is big/A which implies that (KY′′+ ΓY′′)|SY ′′
is big/A as ǫ is sufficiently small. Also, since KY′′ + ΓY′′ is a plt log divisor,
(KY′′+ ΓY′′)|S
Y ′′ is a klt log divisor. Thus by induction κ(KY′′+ ΓY′′)|SY ′′) ≥ 0.
On the other hand, KY′′+ΓY′′ ∼Q MY′′ ≥ 0 with SY′′ ⊂ Supp MY′′ ⊂ Supp ΓY′′.
Therefore, by the extension result [9, Theorem 1.8],
H0(Y′′
, m(KY′′+ ΓY′′)) → H0(SY′′, m(KY′′+ ΓY′′)|S
Y ′′)
is surjective for any sufficiently divisible m > 0. This means that κ(KY′′ +
ΓY′′) ≥ 1 hence κ(KY′′+ ∆Y′′) ≥ 1 so κ(KY′ + ∆Y′) ≥ 1 which in turn implies
that κ(KX+∆) ≥ 1. Thus κ(KX+B) ≥ 1 since ∆ = B +N and N is supported
in D.
Step 5. Since κ(KX + B) ≥ 1, KX + B has a non-trivial Iitaka fibration
g : X 99K V which we may assume to be a morphism. Let G be a general
fibre of g. Then by the definition of Iitaka fibration, κ(KG + BG) = 0 where
KG+ BG = (KX + B)|G. Since KX + B is big/A, we can write
KX + B ∼Q H + E/A
where H is ample and E ≥ 0. Thus
KG+ BG ∼Q H|G+ E|G/A
which shows that KG + BG is also big/A. So KG + BG is big/P where P is
the normalization of the image of G in A. Now since (G, BG) is klt, KG+ BG
is big/P , and P has maximal albanese dimension, by induction, (G, BG) has a
good log minimal model. Moreover, again by induction,
0 = κ(KG+ BG) ≥ dim G − dim P + κ(P )
which implies that dim G − dim P = 0 hence G → A is generically finite. Therefore, the general fibres G of g intersect the general fibres F of f at at most finitely many points. Thus F → V is also generically finite which implies that
κ(KX + B) = dim V ≥ dim F
Step 6. Let Z′
, ˜Z′
, Z′
2, ˜A1, etc, be as in Remark 5.1 which are constructed for
the morphism Z → A. Let ˜Z = Z ×Z′Z˜′ and ˜X = X ×Z′Z˜′. Put Z
′
1 := ˜A1. The
induced morphisms ˜Z′ → Z′
, ˜Z → Z, and ˜X → X are all ´etale. By replacing
Z′
with ˜Z′
, Z with ˜Z, and X with ˜X, we can assume that Z′
= Z′
2× Z
′
1 where
Z′
1 admits an ´etale finite morphism onto an abelian subvariety A1 ⊆ A and Z2′
On the other hand, as pointed out above, (G, BG) has a good log minimal
model, say (G′
, BG′). Since Z2 admits a finite morphism into an abelian variety,
it does not contain any rational curve hence in particular the induced map
G′
99K Z2 is a morphism. Moreover, since κ(KG+ BG) = 0, KG′ + BG′ ∼Q 0.
Applying the canonical bundle formula [1], we deduce that κ(Q) = 0 where the
contraction G′
→ Q is given by the Stein factorization of G′
→ Z2. Since Z2
is of general type, Q is a point otherwise Z2 would be covered by a family of
subvarieties of Kodaira dimension zero which is not possible. Therefore, the
general fibres G of g map into the fibres of Z → Z2.
Since G → A is generically finite, G → Z is also generically finite hence
dim G is at most the dimension of a general fibre of Z → Z2. Therefore,
dim Z − κ(Z) = dim Z − dim Z2 ≥ dim G
which implies that
dim Z − dim G ≥ κ(Z) hence
κ(KX + B) = dim V = dim X − dim G
= dim Z + dim F − dim G ≥ dim F + κ(Z)
Step 7. Finally, we will construct a good log minimal model for (X, B). Run
an LMMP/V on KX+B with scaling of some ample divisor. Since κ(KG+BG) =
0 and since (G, BG) has a good log minimal model, the LMMP terminates near
G hence we arrive at a model X′
such that (KX′ + BX′)|G′ ∼Q 0 where G
′
(by abuse of notation) denotes the birational transform of G (cf. [2, Theorem 1.9]). But then continuing the LMMP we end up with a good log minimal model of (X, B) over V (cf. [2, Theorem 1.5]). Denoting the minimal model again by
X′
, the semi-ampleness of KX′ + BX′ over V gives a contraction X
′
→ T′
/V
such that T′
→ V is birational and that KX′ + BX′ ∼Q 0/T
′
. Therefore (X, B) has a good log minimal model (cf. [5, Proposition 3.3]).
6. Relation with the log Iitaka conjecture over abelian
varieties
Let (X, B) and X → Z be as in Conjecture 1.2 and assume that Z has maximal albanese dimension. So, there is a generically finite map Z → A into an abelian variety.
Let us explain how one might use the method of the proof of Theorem 1.1 to show that
κ(KX + B) ≥ κ(KF + BF) + κ(Z)
If one can show that κ(KX + B) ≥ 1, then the rest of the proof would consist
of some relatively easy inductive arguments.
Assume that κ(KX+B) ≤ 0. First, we borrow the main idea of [3] to consider
conjecture in dimension six. Next, we apply Fujino-Mori [10] to get a klt pair
(X′
, B′
), a morphism X′
→ Z, and a nef Q-divisor L′
such that KX′ + B ′ + L′ is big/Z, κ(KX + B) = κ(KX′ + B ′ + L′ )
and κ(KF + BF) is equal to the dimension of a general fibre of X
′
→ A. If
κ(KF+ BF) = 0, then X
′
→ A is generically finite and the proof in this case is
relatively easy. So, we can assume κ(KF + BF) ≥ 1.
By Theorem 4.1 and results of Campana-Koziarz-Pˇaun [7, Theorem 1] and Kawamata [14] we get a nonvanishing
KX′ + B ′ + L′ ∼Q D ′ ≥ 0
Just as in the proof of Theorem 1.1, we can construct a plt pair (X′
, Γ′ ) so that κ(KX′ + B ′ + L′ ) = κ(KX′ + Γ ′ + L′ ) S′ := ⌊Γ′ ⌋ 6= 0, S′ is irreducible, S′ ⊆ Supp D′ ⊆ Supp Γ′ , and (KX′+ Γ ′ + L′ )|S′ is big/Z.
Now, we apply the nonvanishing again to get nonzero sections in
H0(S′
, m(KX′+ Γ
′
+ L′
)|S′)
for some sufficiently divisible m > 0. If we could extend a section, then we
would be done. In general, one cannot extend sections if L′
is an arbitrary nef
divisor. But in our situation L′
is expected to have strong positivity properties such as being semi-positive in the analytic sense. If one can show that such a semi-positivity property holds, then it seems that some section can be extended
from S′
. At the moment we are not able to show that L′
is semi-positive. References
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DPMMS, Centre for Mathematical Sciences, Cambridge University,
Wilberforce Road, Cambridge, CB3 0WB, UK
email: c.birkar@dpmms.cam.ac.uk
National Center for Theoretical Sciences, Taipei Office, and Department of Mathematics,
National Taiwan University, Taipei 106, Taiwan
