Invariants of ample line bundles on projective varieties

33 Pages·2010·0.24 MB·English

Invariants of ample line bundles
on projective varieties
∗†‡
and their applications, II
YOSHIAKI FUKUMA
Abstract
Let X be a smooth complex projective variety of dimension n and let
L1,...,Ln−i be ample line bundles on X, where i is an integer with 0 ≤ i ≤
n−1. In the previous paper, we defined the i-th sectional geometric genus
gi(X,L1,...,Ln−i) of (X,L1,...,Ln−i). In this part II, we will investigate a
lowerboundforgi(X,L1,...,Ln−i). Moreoverwewillstudythefirstsectional
geometric genus of (X,L1,...,Ln−1).
Introduction.
This is the continuation of [13]. This paper (Part II) consists of section 3, 4,
5 and 6. Let X be a smooth complex projective variety of dimension n and let
L ,...,L be ample line bundles on X, where i is an integer with 0 ≤ i ≤ n−1.
1 n−i
In [13], we defined the ith sectional geometric genus g (X,L ,...,L ). This invari-
i 1 n−i
ant is thought to be a generalization of the ith sectional geometric genus g (X,L)
i
of polarized varieties (X,L). Furthermore in [13], we showed some fundamental
properties of this invariant. In this paper and [14], we will study projective varieties
more deeply by using some properties of the ith sectional geometric genus of multi-
polarized varieties which have been proved in [13]. In this paper, we will mainly
study a lower bound of g (X,L ,...,L ) and some properties of the case where
i 1 n−i
i = 1. The content of this paper is the following.
∗Key words and phrases. Polarized varieties, ample line bundles, nef and big line bundles,
sectional genus, ith sectional geometric genus, ith sectional H-arithmetic genus, ith sectional
arithmetic genus, adjoint bundles.
†2000 Mathematics Subject Classification. Primary 14C20; Secondary 14C17, 14C40, 14D06,
14E30, 14J30, 14J35, 14J40.
‡This research was partially supported by the Grant-in-Aid for Young Scientists (B)
(No.14740018), The Ministry of Education, Culture, Sports, Science and Technology, Japan, and
the Grant-in-Aid for Scientific Research (C) (No.17540033), Japan Society for the Promotion of
Science.
1
In section 3 we will give some results and definitions which will be used in this
paper.
In section 4, we will investigate a lower bound for the ith sectional geometric
genus of multi-polarized variety (X,L ,...,L ). In particular, we will study a
1 n−i
relation between g (X,L ,...,L ) and hi(O ).
i 1 n−i X
In section 5, we will study the nefness of K +L +···+L for t ≥ n−2. This
X 1 t
investigation will make us possible to study a lower bound for g (X,L ,...,L )
1 1 n−1
(see section 6) and some properties of g (X,L ,...,L ) (see [14]).
2 1 n−2
In section 6, we mainly consider the case where (X,L ,...,L ) is a multi-
1 n−1
polarized manifold of type n−1 by using results in section 5, and we will make a
study of the following:
(1) The non-negativity of g (X,L ,...,L ).
1 1 n−1
(2) A classification of (X,L ,...,L ) with g (X,L ,...,L ) ≤ 1.
1 n−1 1 1 n−1
(3) Under the assumption that |L | is base point free for any j with 1 ≤ j ≤ n−1,
j
we will prove that g (X,L ,...,L ) ≥ h1(O ). Moreover we will classify
1 1 n−1 X
(X,L ,...,L ) with g (X,L ,...,L ) = h1(O ).
1 n−1 1 1 n−1 X
(4) Assume that n = 3, h0(L ) ≥ 2 and h0(L ) ≥ 1. Then we will prove
1 2
g (X,L ,L ) ≥ h1(O ). Furthermore we will classify multi-polarized 3-folds
1 1 2 X
(X,L ,L ) with g (X,L ,L ) = h1(O ), h0(L ) ≥ 2 and h0(L ) ≥ 3.
1 2 1 1 2 X 1 2
In this paper we use the same notation as in [13].
3 Preliminaries for the second part
Notation 3.1 Let X be a projective variety of dimension n, let i be an integer with
0 ≤ i ≤ n−1, and let L ,...,L be line bundles on X. Then χ(Lt1 ⊗···⊗Ltn−i)
1 n−i 1 n−i
is a polynomial in t ,...,t of total degree at most n. So we can write χ(Lt1 ⊗
1 n−i 1
···⊗Ltn−i) uniquely as follows.
n−i
χ(Lt1 ⊗···⊗Ltn−i)
1 n−i
(cid:3) (cid:4) (cid:3) (cid:4)
(cid:2)n (cid:2) t +p −1 t +p −1
1 1 n−i n−i
= χ (L ,...,L ) ... .
p ,...,p 1 n−i
1 n−i p p
1 n−i
p=0 p1≥0,...,pn−i≥0
p +···+p =p
1 n−i
Definition 3.1 ([13, Definition 2.1]) Let X be a projective variety of dimension n,
let i be an integer with 0 ≤ i ≤ n, and let L ,...,L be line bundles on X.
1 n−i
(1) The ith sectional H-arithmetic genus χH(X,L ,...,L ) is defined by the fol-
i 1 n−i
lowing:
⎧
⎨ χ (L ,...,L ) if 0 ≤ i ≤ n−1,
1,...,1 1 n−i
(cid:8) (cid:9)(cid:10) (cid:11)
H
χ (X,L ,...,L ) =
i 1 n−i ⎩ n−i
χ(O ) if i = n.
X
2
(2) The ith sectional geometric genus g (X,L ,...,L ) is defined by the following:
i 1 n−i
g (X,L ,...,L ) = (−1)i(χH(X,L ,...,L )−χ(O ))
i 1 n−i i 1 n−i X
(cid:2)n−i
+ (−1)n−i−jhn−j(O ).
X
j=0
(3) The ith sectional arithmetic genus pi(X,L ,...,L ;F) is defined by the fol-
a 1 n−i
lowing:
pi(X,L ,...,L ;F) = (−1)i(χH(X,L ,...,L ;F)−h0(F)).
a 1 n−i i 1 n−i
Remark 3.1 Let X be a smooth projective variety of dimension n and let E be
an ample vector bundle of rank r on X with 1 ≤ r ≤ n. Then in [10, Definition
2.1], we defined the ith c -sectional geometric genus g (X,E) of (X,E) for every
r i
integer i with 0 ≤ i ≤ n − r. Let i be an integer with 0 ≤ i ≤ n − 1. Here we
note that if i = 1, then g (X,E) is the genus defined in [15, Definition 1.1], and
1
moreover if r = n−1, then g (X,E) is the curve genus g(X,E) of (X,E) which was
1
defined in [1] and has been studied by many authors (see [22], [23] and so on). Let
L ,...,L be ample line bundles on X. By setting E := L ⊕···⊕L , we see
1 n−i 1 n−i
that g (X,E) = g (X,L ,...,L ). In particular if i = 1, then g (X,L ,...,L )
i i 1 n−i 1 1 n−1
is equal to the curve genus of (X,E).
Definition 3.2 Let X andY besmooth projective varieties withdimX > dimY ≥
1. Thenamorphismf : X → Y iscalledafiber spaceiff issurjectivewithconnected
fibers. Let L be a Cartier divisor on X. Then (f,X,Y,L) is called a polarized (resp.
quasi-polarized) fiber space if f : X → Y is a fiber space and L is ample (resp. nef
and big).
Definition 3.3 Let (X,L ,···,L ) be an n-dimensional polarized manifold of type
1 k
k, where k is a positive integer. Then (X,L ,···,L ) is called a scroll (resp. quadric
1 k
fibration, Del Pezzo fibration) over a normal variety W if there exists a fiber space
f : X → W such that dimW = n − k + 1 (resp. n − k, n − k − 1) and K +
X
L +···+L = f∗(A) for an ample line bundle A on W. We say that a polarized
1 k
manifold (X,L) is a scroll (resp. quadric fibration, Del Pezzo fibration) over a
normal variety Y with dimY = m if there exists a fiber space f : X → Y such that
K +(n−m+1)L = f∗(A) (resp. K +(n−m)L = f∗(A), K +(n−m−1)L =
X X X
f∗(A)) for an ample line bundle A on Y.
Theorem 3.1 Let (X,L) be a polarized manifold with n = dimX ≥ 3. Then (X,L)
is one of the following types:
(1) (Pn,O (1)).
n
(cid:0)
3
(2) (Qn,O (1)).
n
(cid:2)
(3) A scroll over a smooth curve.
(4) K ∼ −(n−1)L, that is, (X,L) is a Del Pezzo manifold.
X
(5) A quadric fibration over a smooth curve.
(6) A scroll over a smooth surface.
(7) Let (X(cid:3),L(cid:3)) be a reduction of (X,L).
(7-1) n = 4, (X(cid:3),L(cid:3)) = (P4,O (2)).
4
(7-2) n = 3, (X(cid:3),L(cid:3)) = (Q3,O(cid:0) (2)).
3
(7-3) n = 3, (X(cid:3),L(cid:3)) = (P3,O(cid:2)(3)).
3
(7-4) n = 3, X(cid:3) is a P2-bundle(cid:0)over a smoothcurve and (F(cid:3),L(cid:3)| ) is isomorphic
F(cid:3)
to (P2,O (2)) for any fiber F(cid:3) of it.
2
(7-5) K +(n(cid:0)−2)L(cid:3) is nef.
X(cid:3)
Proof. See [2, Proposition 7.2.2, Theorem 7.2.4, Theorem 7.3.2 and Theorem 7.3.4].
(cid:2)
Notation 3.2 Let X be a projective manifold of dimension n.
• ≡ denotes the numerical equivalence.
• Z (X): the group of Weil divisors.
n−1
• N (X) := ({1-cycles}/ ≡)⊗R.
1
• NE(X): the convex cone in N (X) generated by the effective 1-cycles.
1
• NE(X): the closure of NE(X) in N (X) with respect to the Euclidean topol-
1
ogy.
• ρ(X) := dim N (X).
1
(cid:3)
• If C is a 1-dimensional cycle in X, then we denote [C] its class in N (X).
1
(cid:12)
• Let D be an effective divisor on X and D = a D its prime decomposition,
i i
i (cid:12)
where a ≥ 1 for any i. Then we write D = D .
i red i
i
• S denotes the symmetric group of order l.
l
Definition 3.4 ([27, (1.9)]) Let X be a projective manifold of dimension n and let
R be an extremal ray. Then the length l(R) is defined by the following:
l(R) = min{−K C | C is a rational curve with [C] ∈ R}.
X
4
Remark 3.2 By the cone theorem (see [24, Theorem (1.4)], [18] and [20]), l(R) ≤
n+1 holds.
Proposition 3.1 Let X be a projective manifold of dimension n.
∼
(1) If there exists an extremal ray R with l(R) = n+1, then PicX = Z and −KX
is ample.
∼
(2) If there exists an extremal ray R with l(R) = n, then PicX = Z and −KX
is ample, or ρ(X) = 2 and there exists a morphism cont : X → B onto a
R
smooth curve B whose general fiber is a smooth (n−1)-manifold that satisfies
conditions of (1).
Proof. See [27, Proposition 2.4]. (cid:2)
Lemma 3.1 Let (f,X,Y,L) be a quasi-polarized fiber space, where X is a normal
projective variety with only Q-factorial canonical singularities and Y is a smooth
projective variety with dimX = n > dimY ≥ 1. Assume that K +tL is f-nef,
X/Y
where t is positive integer. Then (K + tL)Ln−1 ≥ 0. Moreover if dimY = 1,
X/Y
then K +tL is nef.
X/Y
Proof. For any ample Cartier divisor A on X and any natural number p , K +
X/Y
tL+(1/p)Aisf-nef by assumption. Letmbea natural numbersuch thatm(K +
X/Y
tL+(1/p)A) is a Cartier divisor. Since m(K +tL+(1/p)A)−K is f-ample,
X/Y X
by the base point free theorem ([19, Theorem 3-1-1]),
(cid:3) (cid:3) (cid:4)(cid:4) (cid:3) (cid:3) (cid:4)(cid:4)
1 1
f∗f O lm K +tL+ A → O lm K +tL+ A
∗ X X/Y X X/Y
p p
is surjective for any l (cid:11) 0.
Let μ : X → X be a resolution of X. We put h = f ◦μ. Since
1
(cid:3) (cid:3) (cid:4)(cid:4)
1
μ∗f∗f O lm K +tL+ A
∗ X X/Y
p
(cid:3) (cid:3) (cid:3) (cid:4)(cid:4)(cid:4)
1
= h∗h O lm K +μ∗ tL+ A ,
∗ X X /Y
1 1 p
we have
(cid:3) (cid:3) (cid:3) (cid:4)(cid:4)(cid:4)
1
(1) h∗h O lm K +μ∗ tL+ A
∗ X1 X1/Y p
(cid:3) (cid:3) (cid:4)(cid:4)
1
→ μ∗O lm K +tL+ A
X X/Y
p
is surjective. We note that h O (lm(K +μ∗(tL+(1/p)A))) is weakly positive
∗ X X /Y
1 1
by [8, Theorem A(cid:3) in Page 358] because μ∗O (lm(tL+(1/p)A)) is semiample. (For
X
5
the definition of weak positivity, see [26].) Hence by [8, Remark 1.3.2 (1)] and (1)
above μ∗O (lm(K + tL + (1/p)A)) is pseudo-effective. Since p is any natural
X X/Y
number, we get (K +tL)Ln−1 = μ∗(K +tL)(μ∗L)n−1 ≥ 0.
X/Y X/Y
If dimY = 1, then we see that h O (lm(K + μ∗(tL + (1/p)A))) is semi-
∗ X X /Y
1 1
positive by [8, Theorem A(cid:3) in page 358] since semi-positivity and weak positivity
are equivalent for torsion free sheaves on nonsingular curves. Hence by (1) above
K +tL+(1/p)A is nef for any natural number p. Since p is any natural number,
X/Y
K +tL is nef. (cid:2)
X/Y
Lemma 3.2 Let X and Y be smooth projective varieties with dimX > dimY ≥ 1
and let f : X → Y be a surjective morphism with connected fibers. Then q(X) ≤
q(F)+q(Y), where F is a general fiber of f.
Proof. See [8, Theorem B in Appendix] or [3, Theorem 1.6]. (cid:2)
Lemma 3.3 Let X be a smooth projective variety, and let D and D be effective
1 2
divisors on X. Then h0(D +D ) ≥ h0(D )+h0(D )−1.
1 2 1 2
Proof. See [11, Lemma 1.12] or [21, 15.6.2 Lemma]. (cid:2)
Notation 3.3 Let X be a smooth projective variety of dimension n and let i be
an integer with 1 ≤ i ≤ n−1. Let L ,...,L be nef and big line bundles on X.
1 n−i
Assume that Bs|L | = ∅ for every integer j with 1 ≤ j ≤ n−i. Then by Bertini’s
j
theorem, for every integer j with 1 ≤ j ≤ n − i, there exists a general member
X ∈ |L | | such that X is a smooth projective variety of dimension n−j. (Here
j j X j
j−1
we set X := X.) Namely there exists an (n−i)-ladder X ⊃ X ⊃ ··· ⊃ X such
0 1 n−i
that a projective variety X is smooth with dimX = n−j.
j j
4 Properties of the sectional geometric genus
In this section we study the relationship between g (X,L ,...,L ) and hi(O ).
i 1 n−i X
Lemma 4.1 Let X be a projective variety of dimension n, and let s be an integer
with 0 ≤ s ≤ n−1. Let L ,...,L be Cartier divisors on X. Assume the following
1 s
conditions:
(a) There exists an irreducibleand reduced divisor X ∈ |L | | for any integer
k+1 k+1 X
k
k with 0 ≤ k ≤ s−2. (Here we put X := X.)
0
(cid:12)
(b) hj(− s t L ) = 0 for any integer j and t with 0 ≤ j ≤ n−1, t ≥ 0
m=1 m m(cid:12) m m
s
for any m, and t > 0.
m=1 m
(c) h0(L | ) > 0 and there exists a member X ∈ |L | |.
s X s s X
s−1 s−1
Then
6
(cid:12)
(1) hj(− sm=k+1umLm|Xk) = 0 for any integer(cid:12)k, j and um with 1 ≤ k ≤ s−1,
0 ≤ j ≤ n−k −1, u ≥ 0 for any m, and s u > 0.
m m=k+1 m
(2) hj(O ) = hj(O ) = ··· = hj(O ) for any integer j with 0 ≤ j ≤ n−s.
X X X
1 s−1
(3) hn−s(O ) ≤ hn−s(O ).
Xs−1 Xs
Proof. (1) First we study the case where k = 1. By the above (b) and the exact
sequence
(cid:13) (cid:14) (cid:13) (cid:14) (cid:13) (cid:14)
(cid:2)s (cid:2)s (cid:2)s
0 → O −L − u L → O − u L → O − u L | → 0,
X 1 m m X m m X m m X
1 1
m=2 m=2 m=2
(cid:12)
we have hj(− s u L | ) = 0 for any integer j and u with 0 ≤ j ≤ n − 2,
m=2 m(cid:12)m X1 m
u ≥ 0 for any m, and s u > 0.
m m=2 m
Assume that (1) is true for any integer k with k ≤ l − 1, where l is an integer
with 2 ≤ l ≤ s−1. We consider the case where k = l. By the exact sequence
(cid:13) (cid:14) (cid:13) (cid:14)
(cid:2)s (cid:2)s
0 → O −L | − u L | → O − u L |
X l X m m X X m m X
l−1 l−1 l−1 l−1 l−1
(cid:13) m=(cid:14)l+1 m=l+1
(cid:2)s
→ O − u L | → 0,
X m m X
l l
m=l+1
(cid:12)
we have hj(− s u L | ) = 0 for any integer j and u with 0 ≤ j ≤ n−l−1,
m=l+1 m(cid:12)m Xl m
u ≥ 0 for any m, and s u > 0. Hence we get the assertion.
m m=l+1 m
Next we prove (2) and (3). By (1) above, we obtain hj(−L | ) = 0 for any
k+1 X
k
integer j and k with 0 ≤ k ≤ s − 1 and 0 ≤ j ≤ n − k − 1. Hence by the exact
sequence
0 → O(−L | ) → O → O → 0,
k+1 X X X
k k k+1
we get the assertion. (cid:2)
Lemma 4.2 Let X be a projective variety of dimension n, and let L be a Cartier
divisor on X. Assume that h0(L) > 0 and hn−1(−L) = 0. Then g (X,L) =
n−1
hn−1(O ), where X ∈ |L|.
X 1
1
Proof. We consider the exact sequence
0 → −L → O → O → 0.
X X
1
Then
Hn−1(−L) → Hn−1(O ) → Hn−1(O )
X X
1
→ Hn(−L) → Hn(O ) → 0
X
7
is exact. Since hn−1(−L) = 0, we see that hn(−L) − hn(O ) + hn−1(O ) =
X X
hn−1(O ). By [11, Definition 2.1 and Theorem 2.2] or [13, Corollary 2.2], we get
X
1
g (X,L) = hn(−L)−hn(O )+hn−1(O )
n−1 X X
= hn−1(O ).
X
1
Hence we get the assertion. (cid:2)
Theorem 4.1 Let X be a projective variety of dimension n, and let i be an integer
with 0 ≤ i ≤ n−1. Let L ,...,L be Cartier divisors on X. Assume the following
1 n−i
conditions:
(a) There exists an irreducibleand reduced divisor X ∈ |L | | for any integer
k+1 k+1 X
k
k with 0 ≤ k ≤ n−i−2. (Here we put X := X.)
0
(cid:12)
(b) hj(− n−i t L ) = 0 for any integer j and t with 0 ≤ j ≤ n−1, t ≥ 0
m=1 m m(cid:12) m m
n−i
for any m, and t > 0.
m=1 m
(c) h0(L | ) > 0 and there exists a member X ∈ |L | |.
n−i X n−i n−i X
n−i−1 n−i−1
Then
g (X,L ,...,L ) ≥ hi(O ).
i 1 n−i X
Proof. By Lemma 4.1 (2), we have hj(O ) = hj(O ) for every j with 0 ≤ j ≤ i.
X X
n−i−1
Therefore
(cid:2)n−i
(−1)iχ(O )− (−1)n−i−jhn−j(O )
X X
j=0
(cid:2)1
= (−1)iχ(O )− (−1)1−jhi+1−j(O ).
X X
n−i−1 n−i−1
j=0
By [13, Lemma 2.4] we also get
χ (L ,···,L )
1,...,1 1 n−i
= χ (L | ,···,L | )
1,...,1 2 X n−i X
1 1
= ···
= χ (L | ).
1 n−i X
n−i−1
Hence by [11, Definition 2.1] and Definition 3.1 (2) we have
g (X,L ,...,L ) = g (X ,L | ).
i 1 n−i i n−i−1 n−i X
n−i−1
Here we note that by Lemma 4.1 (1) we have hj(−L | ) = 0 for any integer
n−i X
n−i−1
j with 0 ≤ j ≤ i. By Lemma 4.2 we see that g (X ,L | ) = hi(O ).
i n−i−1 n−i X X
n−i−1 n−i
8
Hence by Lemma 4.1 (2) and (3) we get
g (X,L ,...,L )
i 1 n−i
= g (X ,L | )
i n−i−1 n−i X
n−i−1
= hi(O )
X
n−i
≥ hi(O ).
X
Hence we obtain the assertion. (cid:2)
Lemma 4.3 Let X be a projective variety of dimension n, and let i be an integer
with 0 ≤ i ≤ n−1. Let L ,...,L be Cartier divisors on X. Then the following
1 (cid:12)n−i
are equivalent: (Here χi(O ) := i (−1)jhj(O ).)
X j=0 X
(a) g (X,L ,...,L ) ≥ hi(O ).
i 1 n−i X
(b) (−1)iχH(X,L ,...,L ) ≥ (−1)iχi(O ).
i 1 n−i X
(c) pi(X,L ,...,L ) ≥ (−1)i(χi(O )−1).
a 1 n−i X
Proof. By definition, we get
g (X,L ,...,L )−hi(O )
i 1 n−i X
= (−1)i(χ (L ,...,L )−χ(O ))
1,···,1 1 n−i X
n(cid:2)−i−1
+ (−1)n−i−jhn−j(O )
X
j=0
= (−1)i(χH(X,L ,...,L )−χ(O ))
i 1 n−i X
n(cid:2)−i−1
+ (−1)n−i−jhn−j(O )
X
j=0
= (−1)iχH(X,L ,...,L )−(−1)iχi(O ),
i 1 n−i X
and
pi(X,L ,...,L )−(−1)i(χi(O )−1)
a 1 n−i X
= (−1)i(χH(X,L ,...,L )−1)−(−1)i(χi(O )−1)
i 1 n−i X
= (−1)iχH(X,L ,...,L )−(−1)iχi(O ).
i 1 n−i X
Hence we get the assertion. (cid:2)
Corollary 4.1 Let X be a projective variety of dimension n, and let i be an integer
with 0 ≤ i ≤ n−1. Let L ,...,L be Cartier divisors on X. Assume the following
1 n−i
conditions:
(a) There exists an irreducibleand reduced divisor X ∈ |L | | for any integer
k+1 k+1 X
k
k with 0 ≤ k ≤ n−i−1. (Here we put X := X.)
0
9
(cid:12)
(b) hj(− n−i t L ) = 0 for any integer j and t with 0 ≤ j ≤ n−1, t ≥ 0
m=1 m m(cid:12) m m
n−i
for any m, and t > 0.
m=1 m
(c) h0(L | ) > 0 and there exists a member X ∈ |L | |.
n−i X n−i n−i X
n−i−1 n−i−1
(cid:12)
Then we get the following: (Here χi(O ) := i (−1)jhj(O ).)
X j=0 X
(1) (−1)iχH(X,L ,...,L ) ≥ (−1)iχi(O ).
i 1 n−i X
(2) pi(X,L ,...,L ) ≥ (−1)i(χi(O )−1).
a 1 n−i X
Proof. By Lemma 4.3 and Theorem 4.1, we get the assertion. (cid:2)
If X is normal, then we get the following.
Corollary 4.2 Let X be a normal projective variety of dimension n ≥ 3. Let i be an
integer with 0 ≤ i ≤ n−1. Let L ,L ,...,L be ample line bundles on X such that
1 2 n−i (cid:12)
Bs|L | = ∅ for every integer j with 1 ≤ j ≤ n−i. Assume that hj(− n−i t L ) = 0
j (cid:12)k=1 k k
for any integer j and t with 0 ≤ j ≤ n − 1, t ≥ 0 for any k, and n−i t > 0.
k k k=1 k
Then
g (X,L ,...,L ) ≥ hi(O ).
i 1 n−i X
Proof. If i = n−1, then by [12, Corollary 2.9] we get g (X,L ) ≥ hn−1(O ).
n−1 1 X
If i = 0, then g (X,L ,...,L ) = L ···L ≥ 1 = h0(O ).
0 1 n 1 n X
So we may assume that 1 ≤ i ≤ n−2. For every integer k with 1 ≤ k ≤ n−i−1,
let X ∈ |L | | be a general member. Then since Bs|L | | = ∅, we see
k k X k X
k−1 k−1
that X is a normal projective variety (for example, see [6, (0.2.9) Fact and (4.3)
k
Theorem] or [2, Theorem 1.7.1]). Since L is ample with Bs|L | = ∅, we have
n−i n−i
h0(L | ) > 0 and |L | | (cid:15)= ∅. Hence by Theorem 4.1, we get the
n−i X n−i X
n−i−1 n−i−1
assertion. (cid:2)
Here we propose the following conjecture, which is a multi-polarized version on
[11, Conjecture 4.1].
Conjecture 4.1 Let n and i be integers with n ≥ 2 and 0 ≤ i ≤ n − 1. Let
(X,L ,...,L ) be an n-dimensional multi-polarized manifold of type (n−i). Then
1 n−i
g (X,L ,...,L ) ≥ hi(O ) holds.
i 1 n−i X
Proposition 4.1 Let X be a normal projective variety of dimension n ≥ 2. Let i
be an integer with 0 ≤ i ≤ n−1. Let L ,...,L ,A,B be ample Cartier divisors
(cid:12) 1 n−i−1
on X. Assume that hj(−( n−i−1t L ) − aA − bB) = 0 for any integers j, a, b
p=1 p p (cid:12)
and t with 0 ≤ j ≤ n− 1, a ≥ 0, b ≥ 0, t ≥ 0, and a+ b + t > 0, and that
p p p
Bs|L | = ∅ for 1 ≤ j ≤ n−i−1, Bs|A| = ∅, and Bs|B| = ∅. Then
j
g (X,A+B,L ,...,L ) ≥ g (X,A,L ,...,L )+g (X,B,L ,...,L ).
i 1 n−i−1 i 1 n−i−1 i 1 n−i−1
10