数理解析研究所講究録 1345 巻 2003 年 166-181

166

A generalization of the sectional genus
and the -genus of polarized varieties$\triangle$

高知大学理学部 福間 慶明 (Yoshiaki FUKUMA)
Faculty of Science, Kochi University

1Introduction
Let $X$ be a projective variety of $\dim X=n$ over the complex number field, and let
be an ample line bundle on $X$ . Then the pair $(X, L)$ is called a polarized variety.
Moreover if $X$ is smooth, then $(X, L)$ is called a polarized manifold.
When we study polarized varieties, it is useful to use their invariants. The
following invarjants are well-known.
$L$(1) The degree$L^{n}$.

(2) The sectional geuns(3) Thegenus$\Delta$$\Delta(L)$$\mathrm{A}(\mathrm{L})$.

.

Many authors studied polarized varieties by usirig these invariants. In particular,
P. Ionescu classified polarized manifolds $(X, L)$ for the case where is very ample
andand T. Pujita classified polarized manifolds with low sectional genera
and low -generp.
In order to study polarized varieties more deeply, in [7] and [10] the author
introduced the notion of the -th sectional geometric genus $g_{i}(X, L)$ and the i-th
genus
of $(X, L)$ for every integer with $0\leq i\leq n.$ The -th sectional
geometric genus is a generalization of the degree and the sectional genus of $(X, L)$ ,and the -th genus is a generalization of the genus of $(X, L)$ . Namely $g_{0}(X, L)=$
, $g_{1}(X, L)=g(L)$ , and $\Delta_{1}(X, L)=\Delta(L)$ . (See Remark 2.1 and Remark 2.2
below.) In Section 3, we give fundamental results of these invariants. In particular,, thenif
is the geometric genus of the -dimensional manifold
which is obtained by a general $(n-i)$ members of(see Theorem 3.1). Moreover
there are some relations between (X, ) and (X, ) (see Theorem 3.2 or [10]).
So we find that the -th sectional geometric genus and the -th -genus are expectedto satisfy results which are analogous to results of -dimensional geometry”. (It has
already been known that the first sectional geometric genus and the first -genus,that is, the sectional genus and the -genus reflect some properties of geometry of
curves.)$L$$J_{\lrcorner}^{n}\leq 8,$$\Delta$$i$$\Delta$$\Delta_{i}(X, L)$

$i$

$i$

$\Delta$$i$$\Delta$$L^{n}$$\mathrm{B}\mathrm{s}|L|=\emptyset$$g_{\dot{l}}(X, L)$

$i$

$|L|$

$L$

$g_{i}$$L$$\Delta_{i}$

$i$

$i$

$\Delta$$” \mathrm{i}$$\Delta$$\Delta$

187

Since the sectional genus and the -genus have been studied deeply (see e.g.
[5] , the next step we should consider is the case where $i=2,$ artd our main goalat present is to construct the theory of the second sectional geometric genus andthe second -genus of polarized varieties. Under these consideration, in Section
4, we classify $(X, L)$ by the second sectional geometriq genus and the second
genus for the case where is spanned or very ample. By the above philosophy, thesecond sectional geometric genus and the second -genus are expected to satisfy
results which are analogous to theorems in the theory of projective surfaces. In
order to propose some problems, first we define the -th sectional $H$-arithmetic genusof $(X, L)$ . We note that when $i=2,$ this invariant corresponds to theEuler-Poincar\’e characteristic of the structure sheaf of surfaces and$1-h^{1}(\mathcal{O}_{X})+!12(X, L)$ (see Remark 5.1). Hence we
can propose some problems which
are analogous to results about the Euler-Poincare characteristic of the structure
sheaf of surfaces. So in Section 5, we will propose some conjectures about thesecond sectional -arithmetic genus and the second sectional geometric genus, andwe get partial results about these conjectures. Here we note that Conjecture 4 isanalogous to the Bogomolov-Miyaoka-Yau theorem.
On account of limited space, we cannot state all facts which are known at present.
For the reader who wants to know these topics, see [7], [8], [9], [10], and [11].
This is a survey of my talk of the symposium “Local invariants of families ofalgebraic curves” at the RIMS (Kyoto). The author would like to thank Professor
Kazuhirofor givirig the opportunity to talk about this topic.
$\Delta$$)$$\Delta$$\Delta-$$L$$\Delta$$i$$\chi_{i}^{H}(X, L)$$\mathrm{y}\mathrm{P}(X, L)=$$\mathrm{H}$$\mathrm{I}\{\mathrm{o}\mathrm{m}\mathrm{o}$2Definition of the -th sectional geometric genus and the-th -genus of polarized varieties.
$i$

$i$

$\triangle$

Notation 2.1 Let X be a projective scheme of $\dim X=$ n and let L be a line
bundle on X. Then we put$\chi(tL)-arrow\sum_{j=0}^{\mathrm{n}}\chi_{j}(X, L)\frac{t^{[j]}}{j!}$where$t^{[j]}=\{$$t(t+ 1)$ $\ldots(t+ \mathrm{y}-1)$ ,1,,

if $j>0,$
if $j=0.$

Definition 2.1 ($ee Definition 2.1 in [7].) Let $(X, L)$ be a polarized variety ofwith $0\leq i\leq n$ the -th sectional geometric genusof $(X, L)$ is defined by the following:$\dim X=n.$ Then for any integbr

$i$

$i$

$n-i$

$g_{i}(X, L)=(-1)^{i}$($\chi_{n-i}$(X, $L)-$$\chi(\mathrm{G}$

$x)$

)

$+$ $\mathrm{I}(-1)^{n-i-j}h^{n-j}(\mathcal{O}_{X})$$j=0$(Here we use Notation 2.1,.)

.

1E8

Remark 2.1 (1) If $i=0$ (resp. $i=1$ ), then $g_{i}(X, L)$ is equpl to the degree (resp.
the sectional genus) of $(X, L)$ .
(2) If $i=n,$ then $g_{n}(X, L)=h^{n}(\mathcal{O}_{X})$ and$g_{n}$(X, ) i6 independent of
$L$

$L$

.

(3) If $i=2$ and $X$ is smooth, then by the Hirzebruch-Riemann-Roch theorem, we
get that$g_{2}(X, L)$$-1+h^{1}( \mathcal{O}_{X})+\frac{1}{12}(K_{X}+(n-1)L)(K_{X}+(n -2)L)L^{n-2}$$=$$+ \frac{1}{12}\mathrm{q}(X)Ln-2+\frac{n-3}{24}(2K_{X} \{ (n-2)L)L^{n-1}$.

Definition 2.2 (See [10].) Let $(X, L)$ be a polarized variety of $\dim X=n.$ For
every integer with $0\leq i\leq n,$ the -th genzts of $(X, L)$ is defined by th$ following
formula:$i$$i$$\Delta-$

0,$\Delta_{i}$(X, ) $=\{$

if $i=0,$$L$

Remark 2.2$g_{i-1}(X,L)+(n-i+1)h^{i-1}(O_{X})-h^{i-1}(L)-\Delta_{i-1}(X, L)$(1) If i $=1,$ then[5].)(2) If $i=n,$ then $\Delta n(X; L)$$\Delta 1$, if 1$\leq n.$$\mathrm{d}$(X, L) is equal to the A-genus of (X, L). (See$=h^{n}(\mathcal{O}_{X})-$

hn(L) (see [10]).

Here we define the notion of -Iadder which is used later.
$k$

Definition 2.3 Let $(X, L)$ be a polarized variety of $\dim X=n.$ Then has a-ladder if there exists an irreducible and rpduced subvariety
ofsuch that
$L$

$k$

. for

$X_{i-1}$$X_{i}$$X_{i}\in|Li-1|$

for $1\leq i\leq k,$ where $X_{0}:=X,$ $L_{0}:=L,$ and$L_{i}:=L_{i_{\tau}1}|_{X}$$1\leq i\leq k.$

Notation 2,2 Let $(X, L)$ be a polarized variety of $\dim X=n.$ Assume thathas a -ladder. We put $X_{0}:=X$ and $L_{0}:=L.$ Let $X_{i}\in|L$,
be an irreducible
and reduced member, andfor every integer with $1\leq i\leq k.$ Let: $H^{p}(X_{q}, L_{q})arrow H^{p}(X_{q+1}, L_{q+1})$ be the natural map. If $h^{0}(L_{k})>0,$ then we
take an element $X_{k+1}\in|Lk|$ and we put $L_{k+1}=L_{k}|_{X_{k+1}}$ .
$L$

$k$

$\cdot-1|$$L_{i}:=L_{i-1}|_{x_{:}}$$i$$r_{\mathrm{p},q}$Finally we define the notion of a reduction of polarized manifolds.

Definition 2.4 (1) Let $X$ (resp. ) be an -dimensional projective manifold,
and let (resp. ) be an ample line bundle on $X$ (resp. ). Then $(X, L)$is called a simple blowing up of $(Y, A)$ if there exists a birational morphism: $Xarrow Y$ such that is a blowing up at a point of and $L=\pi^{*}(A)-E,$
where is the -exceptional effective reduced divisor.
$Y$

$L$

$\pi$

$Y$

$\pi$

$E$

$n$

$A$

$\pi$

$Y$

ies

(2) Let $X$ (resp. $Y$ ) be an -dimensional projective manifold, and let(resp.
$A)$ be an ample line bundle on $X$
resp.
Then we say thatis areduction of $(X, L)$ if there exists a birational morphism :such that
71 is a composite of simple blowing ups and $(Y, A)$ is not obtained by a simple
blowing up of any polarized manifold. The morphism is called the reduction
$n$

$L$

$($

$Y)_{\backslash }$$(\mathrm{Y}, A)$$\mu$$Xarrow \mathrm{Y}$

$\mu$

map.

Remark 2.3 Let (X, L) be a polarized manifold and let (M, A) be a reduction of(X, L). Let
X $arrow M$ be the reduction map.
$\mu:$(1) We obtain that (X, ) $=g_{i}(M, A)$ for any integer
Proposition 2.6 in [7] .
$L$$g_{v}$

with $1\leq i\leq n$ (see

$i$

$)$

(2) Assume thatare smooth.

$\mathrm{B}\mathrm{s}|L|=\emptyset$$|$. Then for a general member

of

$D$

$|L|$

,

$D$

and $\mu(D)\in$$4|$(3) $\Delta 1(X, L)\leq\Delta_{1}(M, A)$ and$2\leq i\leq n$ (see [10]).

$\Delta_{i}$(X, ) $=2(M, A)$ for every integer
$L$

with

$i$

(4) If $(X, L)$ is not obtained by a simple blowing up of another polarized manifold,then $(X, L)$ is a reduction of itself.
(5) A reduction of $(X, L)$ always exists (see Chapter

3$\mathrm{I}\mathrm{I}$, (11.11) in [5]).

Fundamental properties of $g_{i}(X,$L) andized manifolds.

L) of polar-$\Delta_{i}(X,$Theorem 3.1 Let $X$ be a projective variety of $\dim X=n\geq 2$ and let
ample Cartier divisor on X. Assume that $h^{t}(-sL)=0$ for every integers
with $0\leq t\leq n$ -1 and $1\leq s.$(A) If$|L|$

has an $(n-i)$ -ladder for an integer \’i with $1\leq i\leq n,$ then$g_{i}(X_{1}, L_{1})=$. $..=g_{i}(X_{n-i}, L_{n-i})$$=h^{i}(\mathcal{O}_{X_{n-}}.)\geq h^{:}(\mathcal{O}_{X_{n-:-1}})=\cdot$(B) Ifhas an $(n-i)$ -ladder and $h^{0}(L_{n-i})$ $>0$ for an integer
then
$|L|$

$\Delta_{i}(X, L)=\sum_{j=0}^{n-i}\dim \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(r_{i-1,j})$In particular,$\Delta_{i}(X, L)$$\geq\Delta_{i}(X, L_{1})$$\geq$

..

$i$

..

$L$

$t$

be $an$and$s$$\mathrm{g}\mathrm{i}(\mathrm{X}, L)=$$=h^{i}(\mathcal{O}_{X})$.

with $1\leq i\leq n$ ,.

$\geq\Delta_{i}(\mathrm{X},i, Ln-i)\geq 0.$(Here we use Notation 2.2. )A sketch of the $pmf$. (A) (See also [9].)We note that for every integer

$k$

with

$0\leq k\leq n-i-1$$\chi_{n-k-i}(X_{k}, L_{k})$ $=\chi_{n-k-1-}\mathrm{i}(X_{k+1}, L_{k+1})$

.

(1)

170

By the assumption we obtain that$\sum_{k=0}^{i-1}(-1)^{k}h^{k}(\mathcal{O}_{X})=\cdot\cdot=\sum_{k=0}^{i-1}(-1)^{k}h^{k}(\mathcal{O}_{X_{n-i}})$$h^{i}(\mathcal{O}_{X})=..=h^{i}(\mathcal{O}_{X_{n-i-1}})$$\leq h^{i}(\mathcal{O}_{X_{n-}}\dot{.})$

,

(2)

.

(3)

By the definition of the -th sectional geometric genus of $(X, L)$ , and by (1), (2),and (3), we obtain the assertion.
(B) (See also [10].) If $i=n,$ thenby Remark 2.2 (2).
(X, )sequence
By the exact
$i$

$L$

$\Delta_{n}$$=h^{n}(\mathcal{O}_{X})-h^{n}(L)$$H^{n-1}(L)--*H^{n-1}(L_{1})arrow H^{n}(\mathcal{O}_{X})arrow H^{n}(L)arrow 0,$

we get that.
If $1\leq i\leq n-1,$ then by [10], we obtain that$\Delta_{n}(X, L)$$\Delta$,(X, )$L$$=$$=\dim \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(r_{n-1,0})$$(-1)^{i-1} \sum_{j=0}^{i-1}\chi_{n-j}(X, L)+(n-i+1)(-1)^{i-1}(\sum_{k=0}^{i-1}(-1)^{k}h^{k}(\mathcal{O}_{X}))$

$i-1$

.

$+(-1)^{i}(1(-1)^{k}h^{k}(L))$$k=0$

Here we note that$(-1)^{i-1} \sum_{j=0}^{i-1}\chi_{n-j}(X, L)$$=$$(-1)^{\dot{\mathrm{t}}}-1 \sum_{j=0}^{i-1}\chi_{i-j}(X_{n-i}, L_{n-i})$$=$$(-1)^{i-1}(\chi(X_{n-i}, L_{n-i})-\chi(\mathcal{O}_{X_{n-:}}))$.

By (2) and the following exact sequence0$arrow$$H^{0}(\mathcal{O}_{X_{j}})arrow H^{0}(L_{j})arrow H^{0}(L_{j+1})$$arrow$$H^{1}(\mathcal{O}_{X_{j}})arrow$$arrow$$H^{i-1}(\mathcal{O}_{X_{j}})arrow H^{iarrow 1}(L_{j})arrow H^{i-1}(L_{j+1})$..
.

we get that$(n-i+1)(-1)^{i-1}( \sum_{k=0}^{i-1}(-1)^{k}h^{k}(\mathcal{O}_{X}))+$ $(-1)^{i}( \sum_{k=0}^{i-1}(-1)^{k}h^{k}(L))$$=$$(-1)^{i-1}( \sum_{k=0}^{i-1}(-1)^{k}h^{k}(\mathcal{O}_{X_{n-:}}))+(-1)|.(\sum_{k=0}^{i-1}(-1)^{k}h^{k}(L_{n-i}))$$+ \sum_{j=0}^{n-i-1}\dim \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(r_{\dot{\iota}-1,j})$

.

171

Since
$(-1)^{i-1}( \chi(X_{n-i}, L_{n-i})-\chi(\mathcal{O}_{X_{n-\dot{i}}}))+(-1)^{i-1}(\sum_{k=0}^{i-1}(arrow 1)^{k}h^{k}(\mathcal{O}_{X_{n-:}}))$$i-1$$+(-1)^{i}(1^{(-1)}khk(Ln-i))$$k=0$$=h^{i}(\mathcal{O}_{X_{n-i}})$$-h^{i}(L_{n-i})$$=\dim \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(r_{i-1_{\}}n-i})$we obtain the assertion.

,$\square$If $X$ is smooth andis ample and spanned, thenhas an $(n-1)$ -ladder,$h^{0}(Ln-1)$ $>0,$ and $h^{t}(-sL)=0$ for every integers
and with $0\leq t\leq n-1$ and$1\leq s.$ Hence by using Theorem 3.1,
we get the following.
$L$

$L$

$t$

$s$

Corollary 3.1 Let $(X, L)$ be a polarized manifold of d\’im $X=n.$ Assume that
. Then $g_{i}(X, L)$
and $\Delta_{i}(X, L)\geq 0$ for every integer with$\dim \mathrm{B}\mathrm{s}|L|=\emptyset$$\geq h^{i}(\mathcal{O}_{X})$$i$$1\leq i\leq n.$

By the above observation, we propose the following problem.

Problem 3.1 Let (X, L) be a polarized manifold of $\dim X=n.$(1) $.DoesQ$ an inequality $g_{i}(X, L)\geq h^{i}($ ’$X)$ hold for every integer$(2)$

Does an inequality$\mathrm{X}_{i}(X, L)\geq 0$$i$with $0\leq i\leq n$hold for every integer with $0\leq i\leq n2$$i$Here we note the following.
(a) If $i=0,$ then (1) is true.

(b) There exists an example of $(X, L)$ with $\Delta_{i}(X, L)<0$ in general. In detail, see[10].

Theorem 3.2 Let $(X, L)$ be a polarized manifold of $\dim X=n,$ and let
integer. Assume that
.

$i$

be an

$\mathrm{B}\mathrm{s}|L|=\emptyset$(1) If $1\leq i\leq n,$ then $\Delta_{i}(X, L)=0$ if anti only if $g_{i}(X, L)=0.$(2) If $\Delta_{i}(X, L)=0$ for an integer with $1\leq i\leq n-$ , then $\Delta_{:+1}(X, L)=0.$$i$(3) If$\Delta_{i}(X, L)$$=1$

for an integer

$i$

$1$

with $2\leq i\leq n$ , then $g_{i}(X, L)=1.$

172

A sketch of the proof. (In detail, see [10].) (1) Assume that $g_{i}(X, L)$ $=0.$ Then by
Theorem 3.1 (A) we havefor every integer with $0\leq j\leq n-i.$ Hencefor every with $0\leq j\leq n-i.$ Therefore $\Delta_{i}(X, L)=0.$$=0.$ Then$=0$ for every integef
Assume thatwith
$1\leq j\leq n-\mathrm{i}$ by Theorem 3.1 (B). In particular, $H^{i-1}(L_{n-i})$ $arrow H^{i-1}(L_{n-i+1})$is$=h^{i}(L_{n-i})$ . Then we can show thatstirjective. Hence$=0.$
Therefore $g_{i}(X, L)=0.$$=0.$ Then
(2) Assume thatby Theorem 3.1 (B). In$-h^{0}(K_{X_{n-:}}-L_{n-i})=0$ by Remark 2.2 (2) and the Serre duality.
particular) we get that $h^{0}(K_{X_{n-i}})=h^{0}(K_{X_{n-}}. -L_{n-i})$ $=0.$ Therefore
Since$h^{0}(K_{X_{n-\mathrm{i}-1}}+L_{n-i-1})=0.$
Since,1 $|=/$), we get that $h^{0}(K_{X_{n-i-1}})$ $=0.$$h^{i+1}(\mathcal{O}Xn-i-1)=0.$
Hence
and
Here we note that$j$$h^{i}(\mathcal{O}_{X_{\mathrm{j}}})=0$$j$$\dim \mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}(r_{i-1,j})=0$$\Delta_{i}(X, L)$$\Delta_{i}(\mathrm{X}, L_{j})$$j$$h^{i}(\mathcal{O}_{X_{n-i}})$$h^{i}(\mathcal{O}_{X_{n-i}})$$\Delta_{i}(X, L)$$\Delta_{i}(X_{n-i}, L_{n\neg\iota})=0$$h^{0}(K_{X_{n-:}})$$\mathrm{B}\mathrm{s}|L_{n-i}|=l$$,$$\mathrm{B}\mathrm{s}|L_{n-i}$$\Delta_{i+1}(X_{n-i-1}, L_{\mathrm{t}-i-1},)=0$$0=h^{i+1}(\mathcal{O}_{X_{n-\mathrm{i}-1}})\geq h^{i+1}(\mathcal{O}_{X_{n-:-2}})=$$..=h^{i+1}(\mathcal{O}_{X})$.

Hence by Theorem 3.1 (B)$\Delta_{i+1}(X, L)=$...

$=\Delta_{i+1}$ $(X_{n-i-1}, L_{n-i-1})=0_{\mathrm{t}}$(3) If $1=\Delta_{i}(X, L)$ $>g_{f}$ (X, ),, then $g_{i}(X, L)=0$ and by (1) we get that $\Delta_{i}(X, L)=$. But this is impossible. Therefore we find that $g_{i}(X, L)\geq\Delta_{i}(X, L)$ . If $h^{0}(K_{X_{n-i}}-$
$L$

$0$

then we can prove that
Hence $h^{0}(K_{X_{n-:}}-L_{n-i})=0.$ By Th$L_{n-i})$$\neq 0,$and this is a contradiction.
en 3.1 (B), we get that$\Delta_{i}(X, L)$$\Delta_{i}(X, L)$$\geq i\geq 2$$\geq$$\Delta_{i}(X_{n-i}, L_{n-i})$$=$$h^{i}(\mathcal{O}_{X_{n-}}.)-h^{i}(L_{n-i})$$=$$h^{i}(\mathcal{O}_{X_{n-\mathrm{i}}})$$=g_{i}(X, L)$.

Therefore $g_{i}(X, L)=$ A{(X, ) $=1.$
These complete the proof of Theorem 3.2.
Before we study behavior of the -th sectional geometric genus and the -th
genus of polarized manifolds under deformation, we define the notion of deformation
family.
$L$$\square$

$i$

$i$

$\Delta-$

Definition 3.1 If 7:
is a proper surjective smooth morphistn onto a cannected but possibly non-cotnpact manifold together with an -ample line bundle
on such that$=L,$ then we say thatandis
a deformation family of $(X, L)$ .
$\mathcal{X}arrow T$$T$$\mathcal{L}$$\mathcal{X}$$f^{-1}(\mathrm{O})=X$$\mathrm{j}|_{f^{-1}(0)}$$f$$(f : \mathcal{X}arrow T, \mathcal{L})$Proposition 3.1 Let (X, L) be a polarized manifold of $\dim X=n.$ For every
integer i with $0\leq i\leq n,$ (X, L) is a defomation invariant.
$g_{i}$

173

Proof. Let

be % indeterminate. Then $\chi(sL)$ is a deformation invariant (seeChapter III, \S 7 in [4] or Chapter III 12.9 in [12] . Hence $\chi_{n-i}(X, L)$ is a deformationis also a deformation invariant for any integer
invariant. On the other hand(see Part , 10.5 in [3]). Therefore by definition we obtain that for every integer
with $0\leq i\leq n,$ $g_{i}(X, L)$ is a deformation invariant. cl
$s$

$)$

$h^{k}(\mathcal{O}_{X})$

$k$

$i$

$\mathrm{I}$

Proposition 3.2 Let $(f : 1 arrow T, \mathcal{L})$ be a deformation family of $(X, L)$ . For everinteger with $0\leq i\leq n,$
is a lower semicontinuous function on $t\in T$
$i$

$y$

$\Delta_{i}(X_{t}, \mathcal{L}_{t})$Pmof. As in the proof of Theorem 3.1 (B), we obtain that$\Delta_{i}(X_{t}, L_{\mathrm{t}})$$=$$(-1)^{i-1} \sum_{j=0}^{i-1})$C$n-j(\mathrm{X}, L_{t})$$+(n-i+1)(-1)^{i-1}( \sum_{k=0}^{i-1}(-1)^{k}h^{k}(\mathcal{O}_{X_{t}}))$

$i-1$

’

$+(-1)^{i}\mathrm{c}1^{(-1)h}$$k(Lt))$

.

$k=0$

We note thatandare deformation invariants. On the otherhand
is an upper semi-continuous function on $t\in T$ (Fora proof, see, e.g., Part , 10.4 in [3].) Henceis a lower
$t\in
Tr$
semi-continuous function on
Therefore we get the assertion.
$\chi_{k}(X_{t}, L_{t})$$h^{k}(\mathcal{O}_{X_{t}})$$(-1)^{q} \sum_{j=0}^{q}(-1)^{j}h^{j}(L_{t})$$(-1)^{i}( \sum_{k=0}^{i-1}(-1)^{k}h^{k}(L_{t}))$$\mathrm{I}$$\square$

4

Classification of polarized manifolds by the second sectional geometric genus and the second genus$\Delta$

Theorem 4.1 Let $(X, L)$ be a polarized manifold of $\dim X=n\geq 3.$ Assume that
. Thenif and only if $(X, L)$ is one of the folloing$g_{2}(X, L)=h^{2}(\mathcal{O}_{X})$$\mathrm{B}\mathrm{s}|L\{=\emptyset$

types.

.

(1)$(\mathrm{P}^{n}, \mathcal{O}_{\mathrm{P}^{n}}(1))$(2)$(\mathbb{Q}^{n}, \mathcal{O}_{\mathbb{Q}^{n}}(1))$.

(3) A scroll over a smooth curve.
(4) $K_{X}\sim-(n-1)$ L, that is, $(X, L)$ is a $Del$ Pezzo manifold.

(5) A quadric fibmtion over a smooth curve.
(6) A scroll over a smooth surface$S$.

(7) Let $(M, A)$ be a reduction of $(X, L)$ .
(7-1) $n=4,$ (Af, ).
(7-3) $n=3,$.
(7-3) $n=3,$.
(7-4) $n=3$ , $M$ is a-bundle over a smooth curve $C$ withfor any fiber $F$ of it$A$$=(\mathrm{P}^{4}, \mathcal{O}_{\mathrm{P}^{4}}(2))$$(M, A)=(\mathbb{Q}^{3}, \mathcal{O}_{\mathbb{Q}^{3}}(2))$$(M, A)=(\mathrm{P}^{3},\mathcal{O}_{\mathrm{P}^{3}}(3))$$I^{2}$$(\mathrm{P}^{2}, \mathcal{O}_{\mathrm{P}^{2}}(2))$$(\Gamma^{\gamma}, A|_{F})=$

74

A sketch of the proof. (In detail, see Theorem 3.3, Corollary 3.4, and Remark 3.4.1
in [7].) Here we note that$g_{2}(X, L)$

So if

$=$

$g_{2}(X_{n-3}, L_{n-3})$$=$$h^{0}(\acute{K}_{X_{n-3}}+L_{n-3})$ $-h^{0}(K_{X_{n-3}})+h^{2}(\mathcal{O}_{X_{n-3}})$$=$$h^{0}(K_{X_{n-3}}+L_{n-3})$ $-h^{0}(K_{X_{n-3}})+h^{2}(\mathcal{O}_{X})$.

. Hencethen $h^{0}(K_{X_{n-3}}+L_{n-3})$ $=0$ by$h^{0}(K_{X}+(n-2)L)=0$ . Therefore by a Sommese’s result (Proposition 13.2.4 in [1])and the adjunction theory, we get the assertion.
$g_{2}(X, L)=h^{2}(\mathcal{O}_{X})_{?}$$\mathrm{B}\mathrm{s}|L_{n-3}|=\emptyset$$\square$

Theorem 4.2 Let $(X, L)$ be a polarized manifold of $\dim X=n\geq 3.$ Asserrne thatis very ample and $g_{2}(X, L)=h^{2}(\mathcal{O}_{X})+1.$ Let $(M, A)$ be a reduction of $(X, L)$ .
Then $(M, A)$ is one of the following.
$L$(1) $(M, A)$ is a Mukai manifold.
(2) $(M, A)$ is a $Del$ Pezzo fibration over a smooth curve C. Let: $Marrow C$$C$
be its morphism. Then there exists an ample line bundle $H$ onsuch that’$K_{M}+(n-2)A=f$ (H) . In this case $(g(C), \deg H)=(1,1)$ .
$f$(3) $(M, A)$ is a quadric fibration over a smooth surface S. Let 7 : $Marrow S$ be its
morphism. Then there eists an ample line bundle $H$ on such that $K_{M}+1$$(n-2)A=f^{*}(Ks+H)$ . In this case $(S, H)$ is one of the following types:
is a
(3.1)
-bundle,: $Sarrow B$ , over a smooth elliptic curve $B$ , and$H=3C_{0}-$ F, where(resp. $F$ ) denotes the minimal section of with
$C_{0}^{2}=1$ (resp. afiber of ).
(3.2)
is an abelian surface, $H^{2}=2,$ and $h^{0}(H)=1.$(3.3)
is a hyperelliptic surface, $H^{2}=2,$ and $h^{0}(H)=1.$
$S$

$S$

$\mathrm{P}^{1}$

$p$

$S$

$C\mathrm{d}$

$p$

$S$

$S$

(4) $(M, A)=(X, L)_{f}n=\dim X\geq 4_{J}$ and $(X, L)$ is a scroll over nor
protective variety of $\dim Y=3.$ If $\dim X\geq 5,$ then $Y$ is smooth and there
exists an ample vector bundle of rank $n-2$ on $Y$ such that$a$$m\iota al$$\mathrm{Y}$$X=\mathrm{P}_{Y}(\mathcal{E})$$\mathcal{E}$and$L=H(\mathcal{E})$$(\mathrm{Y}, c_{1}(\mathcal{E}))$(4.1)

is the tautological line bundle on X. In this caseis one of the following.

$(\mathrm{Y}, c_{1}(\mathcal{E}))$(4.1.1)(4.1.2)(4.1.3)(4.1.4)(4.2), where$H(\mathcal{E})$is a Mukai manifold. In this case,$(Y, \mathcal{E})\mathrm{Z}$$(\mathrm{Y}, 5)$$(\mathrm{P}^{3}, \mathcal{O}_{\mathrm{f}\mathrm{f}}(1)^{\oplus 4})$$(\mathrm{Y}, \mathcal{E})$is one of the folloing:.
.
is the tangent bundle of$\cong(\mathrm{P}^{3}, \mathcal{O}_{\mathrm{P}^{3}}(2)\oplus O_{\mathrm{p}3}(1)^{\oplus 2})$$(\mathrm{Y}, \mathcal{E})\cong(\mathrm{P}^{3},7\mathrm{p})_{\mathrm{S}}$

where
.

$T_{\mathrm{P}^{3}}$$\mathrm{P}^{3}$.

$(\mathrm{Y}, \mathcal{E})\cong(\mathbb{Q}^{3}, \mathcal{O}_{\mathbb{Q}^{3}}(1)^{\oplus 3})$is a $Del$ Pezzo fibration over a smooth curve such thatis the type (2) above. In this case $\dim X=5.$$(\mathrm{Y}, c_{1}(\mathcal{E}))$$(Y, c_{1}(\mathcal{E}))$

175

A sketch of the proof. (Ill detai, see Theorem 3.6 in [7].) By the same argument as
the proof of Theorem 4.1, we get that $h^{0}(K_{X_{n-3}})=0$ and ( $K_{X_{n-3}}+$ Ln_3) $=1.$
By using a Beltrametti-Sommese’s result (Remark 3.4 in [2]), we find that the nef
value of $(M, A)$ is greater than or equal to $n-2.$ By the adjunction theory, we canpick up possible types of $(X, L)$ . By calculating $g_{2}(X, L)$ in each case, we get theassertion.
$h^{0}$$\square$

Theorem 4.3 Let $(X, L)$ be a polarized manifold of $\dim X=n\geq 3.$ Assume that$=0$
. Then$L)=0.$
if and only if$Bs|L|=\emptyset$$\Delta_{2}(X, L)$$\mathrm{g}2\{\mathrm{X},$Proof. By Theorem 3.2 (1) we get the assertion. Cl
Theorem 4.4 Let $(X, L)$ be a polarized manifold of $\dim X=n\geq 3$ and let $(M, A)$
be a reduction of $(X, L)$ . Assume thatis very ample. If $\Delta 2(X, L)$ $=1,$ then$(X, L)$ is one of the types (1), (2), (3.1), (3.3), and (4) in Theorem 4.2. Furthermore
if $(X, L)$ is one of the types (1) $(2)$ , (3.1), (3.3), (4,1.1), (4.1.2), (4.1.3), (4.1.4), and$=1.$(4.2) in Theorem 4.2, then
$L$

$)$

$\Delta_{2}(X, L)$

A sketch of the proof.

(In detail, see [10].) By Theorem 3.2 (3), we get that
implies $g_{2}(X, L)=1.$ Hence
g2{X, )
If $g_{2}(X, L)=h^{2}(\mathcal{O}_{X})$ , thena scroll over a smooth surface with
But by calculating, we find that this case is impossible.
If $g_{2}(X, L)=h^{2}(\mathcal{O}_{X})+1,$ then we can pick up possible types of $(X, L)$ by using
Theorem 4.2. By calculating $\Delta 2(X, L)$ in each case, we get the assertion. Cl$\Delta_{2}(X, L)=1$$L$$h^{2}(\mathcal{O}_{X})\leq$$\leq h^{2}(\mathcal{O}_{X})+1.$$S$$(X, L)\dot{\mathrm{t}}^{\mathrm{S}}$$h^{2}(\mathcal{O}s)=1.$

5$\Delta_{2}(X, L)$

Problems of polarized manifqlds which are analogous totheorems of projective surfaces.

First we define the following.

Definition 5.1 (See [11].) Let $(X, L)$ be a polarized variety of $\dim X=n,$ and let
be arc integer with $0\leq i\leq n.$ Then thesectional -arithmetic genus$(X,L)$
is defined by the following.
of
$i$

$H$

$\dot{t}*th$$\chi_{\dot{\mathrm{t}}}^{H}(X, L):=\chi_{n\sim i}$$\chi_{i}^{H}(X, L)$(X, ).
$L$

Remark 5.1 (1) $)(’(X, L)=1-h^{1}(\mathcal{O}_{X})+\cdots+(-1)^{i-1}hi-1(\mathcal{O}_{X})+(-1)^{i}g_{i}(X, L)$for every integer with $1\leq i\leq n.$$i$$,(2)$

If $X$ is smooth andNotation 2.2.) Namelyof Hirzebruch ([13]).

$\mathrm{B}\mathrm{s}|L|=\emptyset$. (Here we use
in the senseis the arithmetic genus of, then$\chi_{i}^{H}(X, L)$$\chi_{i}^{H}(X, L)=\chi(\mathcal{O}_{X_{n-i}})$$X_{n-i}$be a reduction of $(X, L)$ . By Remark 2.3 (1) and Remark 5.1 (1),
we obtain that $\chi_{i}^{H}(X, L)=\chi_{i}^{H}(M, A)$ for every integer with $1\leq i\leq n.$(3) Let$(M, \mathrm{A})$

$i$

178

(4) I called this invariant the -th sectional Todd genus$i$$\mathrm{T}\mathrm{d}_{i}(X, L)$

before. But

from now on, I call this invariant like the above.

By Theorem 3.1 and Remark 5.1 (2), we can expect that the second sectional
geometric genus $g_{2}(X, L)$ and the second sectional -arithmetic genus
reflect the “2-dimensional geometric . So it is natural to consider the following.
“Can we get results which are analogous to theorems related to the geometric genus
and the Euler-Poincare’ characteristic of the structure sheaf of projective surfaces ”
In this section, we consider this.
First we consider the case where
and we use Notation 2.2.
(A) In this case by Theorem 3.1 (resp. the Lefschetz theorem, Remark 5.1 (2),and the adjunction formula) we get that $g_{2}(X, L)=h^{2}(()_{X_{n-2}})$ (resp.
,, and $(K_{X}+(n-2)L)^{2}L^{n-2}=K_{X_{n-2}}^{2})$ .
(B) Moreover if $(X, L)$ is not a scroll over a smooth surface, then there is thefollowing correspondence between
and $\kappa(K_{X}+(n-2)L)$ (see [11]).
$\chi_{2}^{H}(X, L)$$\mathrm{H}$$?$$\mathrm{B}\mathrm{s}|L|=\emptyset$$h^{1}(\mathcal{O}_{X})=$$h^{1}(\mathcal{O}_{X_{n-2}})$$\chi_{2}^{H}(X, L)=\chi(\mathcal{O}_{X_{n-2}})$$\kappa(X_{n-2})$

Value of$\kappa(X_{n-2})$$-\infty$

Value of $\kappa(K_{X}+(n-2)L)$$\Leftrightarrow$$\Leftrightarrow*$$021$$-\mathrm{o}\mathrm{o}$$\geq 012$$\Leftrightarrow^{**}\Leftrightarrow^{**}\Leftrightarrow^{**}$(We note that the directionin $(*)$ and the directionin $(**)$ need the$(X,L)$
assumption thatis not a scroll over a smooth surface.)(C) Let $(X, L)$ be a polarized manifold which is not a scroll over a smooth surface,
let $(M, A)$ be a reduction of $(X, L)$ , and we put $M_{n-2}:=\mu(X_{n-2})$ , where:$Xarrow M$ is the reduction map. Then
$K_{M_{n-2}}=(K_{M}+$(is smooth and$2)A)|M_{n-2}$ . Assume that
(We note that this condition is equivalent to$\kappa(K_{M}+(n-2)A)\geq 0.$
the condition that $\kappa(K_{X}+(n-2)L)\geq 0$ by above.)
Hence by the adjunction theory $K_{hI}+$ $(n -2)$ A is
. In particular
.
is
Hence
: $X_{n-2}arrow M_{n-2}$ is the minimalization of $X_{n-2}$ .
From (A), (B), and (C), we infer that there are the following correspondence
between invariants of smooth projective surfaces and invariants of $(X, L)$ .
$\Leftarrow$$\Rightarrow$$\mu$$M_{narrow 2}$$\kappa(X_{n-2})$$\mathrm{v}\mathrm{r}$-$\geq 0.$$\mathrm{T}\backslash \mathrm{h}\mathrm{e}\mathrm{I}\mathrm{t}$$K_{M_{n-2}}$$\mathrm{n}\mathrm{e}\mathrm{f}$$\mathrm{n}\mathrm{e}\mathrm{f}$$\mu|_{X_{n-2}}$$S$

Invariants

\dot

$0$

$\mathrm{f}$$h^{2}(\mathcal{O}_{S})$$S$$\Leftrightarrow$$\Leftrightarrow$$h^{1}(\mathcal{O}_{S})$$\Leftrightarrow$$\chi(\mathrm{C}s)$$\Leftrightarrow$$K_{S}^{2}$$\Leftrightarrow$’$K_{\tilde{S}}^{2}$$\Leftrightarrow$

Invariants of $(X, L)$$g_{2}(X, L)$$h^{1}(\mathcal{O}_{X})$$1\mathrm{i}(X, L)$$(K_{X}+(n-2)L)^{2}L^{n-2}$$(K_{M}+(n-2)A)^{2}A^{n-2}$$\kappa(K_{X}+(n-2)L)=k$$\kappa(S)=k$$\Leftrightarrow^{***}\Leftrightarrow**$$\kappa(S)=2$$\kappa(K_{X}+(n-2)L)\geq 2$(In $(*)$ , we assume that $\kappa(K_{X}+(n-2)L)\geq 0$ and let(resp. $(M,$ $A)$ ) be theminimalization of (resp. a reduction of ( $X$ , )). In $(**)k$ is an integer with$\tilde{S}$

$S$

$L$

177

, 0, or 1, and we assume that $(X, L)$ is not a scroll over a smooth surface.
In $(***)$ we assqme that $(X, L)$ is not a scroll over a smooth surface. )By considering these correspondences, we can propose some problems which areanalogous td the case of smooth projective surfaces. For example there are thefollowing five theorems of projective sprfaces.
$k=-\mathrm{o}\mathrm{o}$Theorem 1 (Castelnuovo’s theorem) Let
some that(resp.
). Then$\mathrm{k}(\mathrm{S})\geq 0$$\mathrm{k}(\mathrm{S})=2$$S$

be a smooth projective surface. As(resp.
) $>0)$ .

$\chi(\mathcal{O}_{S})\geq 0$$\chi(\mathrm{C})_{S}$Theorem 2 (Noether’s inequality) Let be a smooth projective surface of general type and let be the minimal model of S. Then.
$S$$\overline{S}$$K_{\tilde{S}}^{2}\geq 2p_{g}(\tilde{S})-$ $4$

Theprem 3 (Debarre’s inequality) Let $S$ be a smooth projective surface of gen-eral type with $q(S)>0,$ and let$\tilde{S}$be the minimal model of S. Then

$K_{\tilde{S}}^{2}\geq 2p_{g}(\tilde{S})$.

Theorem 4 (Bogomolov-Miyaoka-Yau’s inequality) Let S be a smooth pmjective surface of general type. Then.
$9\chi(\mathcal{O}_{S})\geq K_{S}^{2}$

Theorem 5 (Inequality of CastelnuovO-Beauville) Let be a smooth projective surface of general type. Then $p_{g}(S)\geq 2q(S)-4$ (that is, $\chi(\mathcal{O}_{S})\geq q(S)-3$
.
$S$

$)$

By using the above correspondences, we can give the following conjedctures. We
note that for $k=1$ , . . . ’ 5, Conjecture corresponds Theorem above.
$k$

$k$

Conjecture 1 Let (X, L) be a polaized manifold of $\dim X=n\geq 3.$ Assume that$\kappa(K_{X}+(n-2)L)\geq 0$ (resp.
). Then $\chi_{2}^{H}(X, L)\geq 0$ (resp. $>0$ ) .
$\geq 2$

Conjecture 2 Let $(X, L)$ be a polar ized manifold of $\dim X=n\geq 3.$ Assume that$\kappa(K_{X}+(n-2)L)\geq 2.$ Let (Af, ) be a reductionof $(X, L)$ . Then $(K_{M}+(n-$$A$$2)A)^{2}A^{n-2}\geq 2g_{2}(M, 4)$ $-4.$

Let $(X, L)$ be a polarized manifold of $\dim X=n\geq 3.$ Assume that$\kappa(K(+(n-2)L)\geq 2$ and $q(X)>0.$ Let
be a reduction of $(X, L)$ . Then$\backslash ^{K_{M}+(n-2)A)^{2}A^{n-2}}/\geq 2g_{2}(M, A)$ .
$\mathrm{C}\mathrm{o}\mathrm{f}^{1}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}3$$(M, \mathrm{A})$Conjecture 4 Let (X, L) be a polarized manifold of $\dim X=n\geq 3.$ Assume that
Then $9\chi_{2}^{H}(X, L)\geq(K_{X}+(n-2)L)^{2}L^{n-2}$ .

$\kappa(K_{X}+(n-2)L)\geq 2.$

Conjecture 5 Let (X, L) be a polarized manifold of $\dim X=n\geq 3.$ Assume that$\kappa(K_{X}+(n-2)L)\geq 2.$ Thm $g_{2}(X,$L) $\geq 2q(X)-4$ (that is,$L)\geq q(X)-3$ ) .
$\chi_{2}^{H}(X,$

For Conjecture 1 we get the following result.

Theorem 5.1 Let $(X, L)$ be a polarized manifold of $\dim X=n\geq 4.$ Assume that$\kappa(X)\geq 0.$

Then $\chi_{2}^{H}(X, L)>0.$

178

For the proof, see Corollary 3.5.2 in [8] .
Here we note that in my preprint [11], we study Conjecture 1 and Conjecture
4. Furthermore we propose more stronger conjecture than .Conjecture 4. See mypreprint [11] in detail.
Next we consider the following situation. Let $(X, L)$ be a polarized manifold andlet : $Xarrow C$ be a fiber space over a smooth projective curve C. Then we consider
a polarized version of the following theorems.
$\square$$f$

Theorem 6 (Beauville’s inequality) Let
be a smooth projective surface and$SarrowC$
let :be a fiber space over a smooth projective curve C. Then$(g(F) ・1)(g(C) ・1)$ , where $F$ is a general fiber of .
$S$

$f$

$\chi(\mathcal{O}_{S})\geq$$f$

Theorem 7 (Arakelov’s inequality) Let be a smooth projective surface and let: $Sarrow C$ be a fiber space over a smooth projective curve C. Let $S’$ be a relatively
minimal model of S. Assume that $g(F)\geq 2_{f}$ whereis a general fiber of . Then$\geq 8(g(F)-1)(g(C)-1)$ .
$S$

$f$

$F$

$f$

$K_{S}^{2},$First we prove a polarized version of Theorem 7.

Theorem 5.2 Let $(X, L)$ be a polarized manifold of $\dim X=n\geq 3$ and let :be a fiber space over a smooth projective curve C. Let $(M, A)$ be a reductiona fiber space : $Marrow C$ such that
, whereof $(X, L)$ . ( Then there$XarrowM$:
is the reduction map.) Assume that $g(Aph)\geq 2$ andis notscroll over a smooth curve, whereis a general fiber of . Then$f$

$X_{\neg}arrow C$$e$$\dot{m}ts$$h$$f=h\mathrm{o}\pi$$\pi$$(F_{h},A_{F_{h}})$$(K_{M}+(n-2)A\rangle^{2}A^{n-2}$

$a$

$h$

$F_{h}$$\geq 8(g(A_{F_{h}})-1)(g(C)-1)$.

Proof. First we calculate $(K_{M}+(n-2)A)^{2}A^{n-2}$ ,$(K_{M}+(n-2)A)^{2}A^{n-2}$$=$( $K_{M/C}+$ (h $-2$ ) $A$ ) $(K_{M}+(n-2)A)A^{n-2}$$+(2g(C) arrow 2)(K_{F_{h}}+(n-2)A_{F_{h}})A_{F_{h}}^{n-2}$$=(K_{M/C}+(n-2)A)^{2}A^{n-2}+2(2g(C)-2)(K_{F_{h}}+(n -2)A_{F_{h}})A_{F_{h}}^{n-2}$$=(K_{M/C}+(n-2)A)^{2}A^{n-2}+2(2g(C)-2)(2g(A_{F_{h}})-2)$ ,

(4)

where
is a general fiber of .
Since $g(A_{F_{h}})\geq 2$ andis not a scroll over a smooth curve, by Theorem
1.1.1, Theorem 1.1.2, and Theorem 1.1.3 in [6] , we get that $K_{h\mathrm{f}/C}+(n-2)A$ is.
$(K_{M/c+}(n-2)A)^{2}A^{n-2}\geq
0.$
So we get the assertion by (4).
Therefore
The following example shows that the assumption thatis not a scroll
over a smooth curve is necessary.
$h$$F_{h}$$(F_{h}, A_{F_{h}})$$\mathrm{n}\mathrm{e}\mathrm{f}$$\square$$(F_{h}, A_{F_{h}})$Example 5.1 Let $F$ andare smooth projective curves with $g(F)\geq 2$ and weput $S:=F\cross C.$ Let : $Sarrow C$ be the second projection. Let be an ample vector
$C$

$\pi$

$\mathcal{E}$179bundle on of rank $n-$ l. We put,, and $f:=tv$
whereis the tautological line bundle on $X$ and : $Xarrow S$ be the projection. Let
be a general fiber of . Then
is a scroll over , $g(L_{F_{f}})2$ , $(X, L)$ isa reduction of itself, $K_{S}^{2}=8(g(F)-1)(g(C)-1)$ $=8(g(L_{F_{f}})-1)(g(C)-1)$ , and.
$S$$L:=H(\mathcal{E})$$X:=\mathrm{P}_{S}(\mathcal{E})$$H(\mathcal{E})$$\circ p,$

$p$

$f$

$F_{f}$$F$$(F_{f}, L_{F_{f}})$$2$$(K_{X} \mathrm{I}\mathrm{F} (n-2)L)^{2}L^{n-2}=K_{S}^{2}-c_{2}(\mathcal{E})<K_{S}^{2}$

Next we give a conjecture which is a polarized version of Theorem 6.
Conjecture 6 Let $(X, L)$ be a polarized manifold of $\dim X=n\geq 3$ and let
be a fiber space over a smooth protective curve C. Then$(g(L|_{F})-1)(g(C)-1\}$ , where $F$ is a general fiber of .

$Xarrow C$$f$:$\chi_{2}^{H}(X, L)\geq$$f$

For Conjecture 6, we get the following result.
Theorem 5.3 Let $(X, L)$ be a polar ized manifold of $\dim X=n\geq 4.$ Assume that$\kappa(X)\geq 0$ and there exists
a fiber space : $Xarrow C$ over a smooth curve C. Let
be a general fiber of . Then
$f$

$F$

$f$

$\chi_{2}^{H}(X, L)\geqarrow(g(L|_{F})-1)(g(C)-1)+\frac{n^{2}-5n+5}{12}L^{n}31$.

Proof Let $(M, A)$ be a reduction of $(X, L)$ . Then there exists a fiber space :$h$such that
following.
$C$$f=h\mathrm{o}\pi$, where$\pi$$Marrow$: $Xarrow M$ is the reduction map. Here we note the

Proposition 5.1 Let $X$ be a smooth protective variety of $\dim X=n\geq 3$ such thatis not uniruled. Let be an ample line bundle on X. Then

$X$

$L$

$c_{2}(X)L^{n-2}\geq-$ $\mathrm{C})$ $L^{n}-(n-1)K_{X}L^{n-1}$.

For the proof, see Proposition 3.4 in [8].
By Remark 2.1 (3), Remark 5.1 (3), and Proposition 5.1, We get that$\square$$\chi_{2}^{H}(X, L)$

$=$

$=$

$\chi_{2}^{H}$(M, $A$ )$\frac{1}{12}(K_{M}+(n-1)A)(K_{M}+(n-2)A)A^{n-2}+\frac{1}{12}c_{2}(M)A^{n-2}$$n-3$$+(2K_{M}\overline{24}+(n-2)A)A^{n-1}$$\geq$$\frac{1}{12}K_{M}(K_{M}+(n-2)A)A^{n-2}+\frac{1}{12}(n-1)(\frac{n}{2}-2)A^{n}$$n-3$$+(2K_{M}+(n-2)A)A^{n-1}\overline{24}$$=$$\frac{1}{12}K_{M/C}(K_{M}+(n-2)A)A^{n-2}+\frac{1}{12}h^{*}(K_{C})(K_{M}+(narrow 2)A)A^{n-2}$$+ \frac{1}{12}(n-1)(\frac{n}{2}-2)$

A $n+ \frac{n-3}{24}(2K_{M}+(n-2)A)A^{n-1}$ .

180

Since $\kappa(X)\geq 0,$ we obtain that $K_{M}A^{n-1}\geq 0,$ $K_{M}+(n-2)A$ is nef by the
adjunction theory, andfor sufficiently large $m$ . Therefore
is semipositive by a Kawamata’s theorem [14]. Henceis pseud0-effective by$K_{M/C}(K_{M}+(n-2)A)A^{n-2}\geq0.$[6].
Therefore
Remark 1.3.2 in
So we get that$h_{*}(K_{M/C}^{\otimes m})\neq 0$$h_{*}(K_{M/C}^{\otimes m})$$K_{hI/C}^{\otimes m}$$\mathrm{x}\mathrm{P}(X, L)$$=\chi_{2}^{H}(M, A)$$\geq$$\frac{1}{12}K_{M/C}(K_{M}+(n-2)A)A^{n-2}+\frac{1}{12}h’(K_{C})(K_{AI} +(n-2)A)A^{n-2}$$n-3$$+ \frac{1}{12}(72-1)(\frac{n}{2}-2)A^{n}+\overline{24}(2K_{M}+(n-2)A)A^{n-1}$$\geq$$\frac{1}{12}(2g(C)-2)(K_{F_{h}}+(n-2)A|_{F_{h}})(A|_{F_{h}})^{n-2}+\frac{1}{12}(n-1)(\frac{n}{2}-2)A^{n}$$n-3$$+(2K_{M}\overline{24}+(n-2)A)A^{n-1}$$\geq$$\frac{1}{3}(g(C)-1)(g(A|_{F_{h}})-1)+\frac{n^{2}-5n+5}{12}$whereis agenerala general fiber of .
On the other hand, since

A ,
$n$

$h$

$F_{h}$$A^{n}\geq L^{n}$

and $g(L|_{F})=g(A|_{F_{h}})$ ; we get the assertion.

口

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projective varieties, de Gruyter Expositions in Math. 16, Walter de Gruyter,
Berlin, NewYork, (1995).
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system for threefolds, Ann. Scola Norm. Sup. Pisa 16 (1995), 1-24.
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$7^{\tau}ic$

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A generalization of the$\Delta- ge\gamma\iota us$$II_{f}$of quasi-polar ized varieties,

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Department of Mathematics
Faculty of Science
Kochi University
AkebonO-cho, Kochi 780-8520
Japan$\mathrm{E}$-mail: fukuma@math.kochi-u.x.jp