Sectional invariants of scroll over a smooth projective variety

Sectional Invariants of Scrolls Over a Smooth Projective Variety

YOSHIAKIFUKUMA(*)

ABSTRACT- L etXbe a smooth complex projective variety of dimensionnand letEbe an ample vector bundle of rankronX. Then we calculate theith sectional Euler numberei(PX(E);H(E)) and theith sectional Betti numberbi(PX(E);H(E)) for i2n 3 or iˆ1, and the ith sectional Hodge number of type (j;i j) h_i^{j;i j}(PX(E);H(E)) fori2n 1and0ji, wherePX(E) is the projective space bundle associated withEandH(E) is its tautological line bundle. Moreover we define a new invariantv(X;E) of (X;E) forrn 1. This invariant is thought to be a generalization of curve genus. We will investigate some properties of this invariant.

1. Introduction.

LetX be a projective variety of dimensionndefined over the field of complex numbers, and letLbe an ample (resp. a nef and big) line bundle on X. Then (X;L) is called apolarized(resp. quasi-polarized)variety. IfXis smooth, then we say that (X;L) is a polarized (resp. quasi-polarized) manifold. In order to study polarized varieties, it is important to use an invariant of (X;L). There are the following three invariants of (X;L) which are well-known.

The degreeLⁿ.

The sectional genusg(L).

TheD-genusD(L).

By using these invariants, many authors studied polarized varieties. In particular, P. Ionescu classified polarized manifolds by the degree under the assumption thatLis very ample withLⁿ8 ([13], [14], and [15]), and

(*) Indirizzo dell'A.: Department of Natural Science, Faculty of Science, Kochi University, Akebono-cho, Kochi 780-8520, Japan.

E-mail: fukuma@kochi-u.ac.jp

T. Fujita classified polarized varieties by the D-genus and the sectional genus ([3]).

In [5], in order to study polarized varieties more deeply, the author introduced the notion of theith sectional geometric genusg_i(X;L) of (X;L) for every integeriwith 0in. This is a generalization of the degree and the sectional genus of (X;L). Namely g0(X;L)ˆLⁿ and g1(X;L)ˆg(L).

Here we recall the reason why this invariant is called theith sectional geometric genus. Let (X;L) be a polarized manifold of dimension n2 with BsjLj ˆ ;, where BsjLjis the base locus of the complete linear system jLj. L etibe an integer with 1in. L etX_{n i}be the transversal inter- section of general n i members of jLj. In this case Xn i is a smooth projective variety of dimension i. Then we can prove that g_i(X;L)ˆhⁱ(O_{Xn i}), that is,g_i(X;L) is the geometric genus ofX_{n i}.

Hence we can expect that g_i(X;L) has analogous properties of the geometric genus ofi-dimensional varieties.

In [6] and [7], we introduced the notion of theith sectional H-arith- metic genusx^H_i (X;L)of(X;L). By definition we can prove that if BsjLj ˆ ;, then x^H_i (X;L)ˆx(O_{Xn i}), where X_{n i} is the transversal intersection of general n i members of jLj. Namely x^H_i (X;L) is the Euler-Poincare characteristic of the structure sheaf ofX_{n i}. (x(O_{Xn i}) is called the arith- metic genus ofXn iin the sense of Hirzebruch. (See [12, 15.5, Section 15, Chapter IV]. We also callx(OXn i) the H-arithmetic genus ofX_{n i}.)

In [8], we also introduced someith sectional invariants of (X;L), that is, the ith sectional Euler number e_i(X;L), the ith sectional Betti number b_i(X;L) and theith sectional Hodge numberh^{j;i j}_i (X;L) of type (j;i j) for every integer j with 0ji, and we investigated some properties of these. In particular we proved that polarized manifolds' version of the Hodge duality and the Hodge decomposition hold (see [8, Theorem 3.1]).

In this paper we consider theith sectional Euler number and theith sectional Betti number of scrolls over a smooth projective variety. In this paper we say that a polarized manifold (P;H) is ascroll over a smooth projective variety X if there exists an ample vector bundleEonXsuch that (P;H)(P_X(E);H(E)), whereH(E) is the tautological line bundle on P_X(E).

In section 3, we calculate e_i(P_X(E);H(E)) and b_i(P_X(E);H(E)) for i2n 3 and iˆ1 (see Theorems 3.1 and 3.2). We also calculate h^{j;i j}_i (PX(E);H(E)) for i2n 1 and 0ji(see Theorem 3.3). In par- ticular, by using these results, we can calculate these invariants of (P_X(E);H(E)) completely fornˆ1 or 2 (see Corollaries 3.1, 3.3 and 3.4).

In section 4, we will define a new invariant of generalized polarized manifolds. Here a generalized polarized manifoldmeans the pair (X;E) whereXis a smooth projective variety andEis an ample vector bundle on X. L etr:ˆrank(E). Here we state the history of invariants of (X;E). First in [2], Fujita introduced the c₁-sectional genus and the O(1)-sectional genus of (X;E). Next, in [1], for the case whererˆn 1, Ballico defined an invariant of (X;E) which is called thecurve genus cg(X;E)of(X;E) (see Definition 2.3), and several authors (in particular Lanteri, Maeda, Som- mese, and so on) studied this invariant.

As a generalization of the curve genus, for any ample vector bundleE withrn 1, Ishihara ([16, Definition 1.1]) defined an invariantg(X;E), which is called thecr-sectional genus of(X;E), and in [9] we investigated some properties aboutg(X;E). We note that ifn rˆ1, theng(X;E) is the curve genus. This invariant means the following: If a general element of H⁰(E) has a zero locusZwhich is smooth of expected dimensionn r, then g(X;E)ˆg(Z;detEj_Z), that is,g(X;E) is the sectional genus of (Z;detEj_Z).

Recently Fusi and Lanteri generalized this invariant. See [11] for detail.

In this paper, we will introduce a new invariantv(X;E) of generalized polarized manifolds (X;E) withrn 1, which is defined by using a result in section 3 (see Definition 4.1). Here we note thatv(X;E) is equal to the curve genus ifrˆn 1. We will investigatev(X;E) and give some results about this invariant (see Theorems 4.1 and 4.2, and Proposition 4.1).

The author would like to thank the referee for giving useful comments and suggestions.

Notation and Conventions.

We say thatXis avarietyifXis an integral separated scheme of finite type. In particularXis irreducible and reduced ifXis a variety. Varieties are always assumed to be defined over the field of complex numbers. In this article, we shall study mainly a smooth projective variety. The words

``line bundles'' and ``Cartier divisors'' are used interchangeably. The tensor products of line bundles are denoted additively.

O(D): invertible sheaf associated with a Cartier divisorDonX.

O_X: the structure sheaf ofX.

x(F): the Euler-Poincare characteristic of a coherent sheafF. hⁱ(F):ˆdimHⁱ(X;F) for a coherent sheafF onX.

hⁱ(D):ˆhⁱ(O(D)) for a Cartier divisorD.

q(X)(ˆh¹(O_X)): the irregularity ofX.

bi(X):ˆdimHⁱ(X;C).

KX: the canonical divisor ofX.

Pⁿ: the projective space of dimensionn.

Qⁿ: a smooth quadric hypersurface inP^n‡1. (orˆ): linear equivalence.

det (E):ˆ ^^rE, whereEis a vector bundle of rankronX.

PX(E): the projective space bundle associated with a vector bundleEonX.

H(E): the tautological line bundle onP_X(E).

E^_: the dual of a vector bundleE.

c_i(E): thei-th Chern class of a vector bundleE.

c_i(X):ˆc_i(T_X), where T_X is the tangent bundle of a smooth projective varietyX.

For a real numbermand a non-negative integern, let [m]ⁿ:ˆ m(m‡1) (m‡n 1) if n1;

1 if nˆ0:

(

[m]_n:ˆ m(m 1) (m n‡1) if n1;

1 if nˆ0:

(

Then fornfixed, [m]ⁿand [m]_nare polynomials inmwhose degree aren.

For any non-negative integern,

n! :ˆ [n]_n if n1;

1 if nˆ0:

Assume thatmandnare integers withn0. Then we put m

n :ˆ[m]_n n!

We note that m

n ˆ0 if 0m<n, and m 0 ˆ1.

2. Preliminaries.

NOTATION2.1. (1) LetYbe a smooth projective variety of dimensioni, letT_Y be the tangent bundle ofY, and letV_Y(ˆV¹_Y) be the dual bundle of TY andV_Y^j :ˆ ^^jVY. For every integerjwith 0ji, we put

h_i;j(c₁(Y); ;c_i(Y)):ˆx(V^j_Y)

ˆ Z

Y

ch(V^j_Y)Td(T_Y):

(Here ch(V^j_Y) (resp. Td(TY)) denotes the Chern character ofV^j_Y(resp. the Todd class ofTY). See [10, Example 3.2.3 and Example 3.2.4].)

(2) Let (M;L) be a polarized manifold of dimensionm. For every in- tegersiandjwith 0jim, we put

C_jⁱ(M;L):ˆX^j

lˆ0

( 1)^l m i‡l 1 l

c_{j l}(M)L^l;

w^j_i(M;L):ˆh_i;j(C₁ⁱ(M;L); ;C_iⁱ(M;L))L^{n i}:

(3) Let M be a smooth projective variety of dimensionm. For every integersiandjwith 0jim, we put

H₁(i;j):ˆ

X

i j 1 sˆ0

( 1)^sh^s(V^j_M) ifj6ˆi;

0 if jˆi;

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:

H₂(i;j):ˆ

X^{j 1}

tˆ0

( 1)^{i t}h^t(V^{i j}_M ) if j6ˆ0;

0 if jˆ0:

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DEFINITION 2.1. (See [8, Definition 3.1].) Let (M;L) be a polarized manifold of dimensionm, and letiandjbe integers with 0imand 0ji.

(1) Theith sectional Euler number ei(M;L)of(M;L) is defined by the following:

e_i(M;L):ˆXⁱ

lˆ0

( 1)^l m i‡l 1 l

c_{i l}(M)L^{m i‡l}:

(2) Theith sectional Betti number b_i(M;L)of(M;L) is defined by the following:

b_i(M;L):ˆ

e0(M;L) if iˆ0;

( 1)ⁱ e_i(M;L) X^{i 1}

jˆ0

2( 1)^jb_j(M)

!

if 1im:

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(3) The ith sectional Hodge number h_i^{j;i j}(M;L) of type (j;i j) of

(M;L) is defined by the following:

h_i^{j;i j}(M;L):ˆ( 1)^{i j}nw^j_i(M;L) H1(i;j) H2(i;j)o :

REMARK2.1. (1) Ifiˆ0, then

e₀(M;L)ˆb₀(M;L)ˆh^0;0₀ (M;L)ˆL^m: (2) Ifiˆ1, then

e1(M;L)ˆ2 2g(L);

b₁(M;L)ˆ2g(L);

h^1;0₁ (M;L)ˆh^0;1₁ (M;L)ˆg(L):

(3) Ifiˆm, then

e_m(M;L)ˆe(M);

bm(M;L)ˆbm(M);

h_m^{j;m j}(M;L)ˆh^{j;m j}(M);

h^{m j;j}_m (M;L)ˆh^{m j;j}(M):

PROPOSITION2.1. Let(M; L)be a polarized manifold of dimensionm.

(1)For every integer i with0im, the following hold:

(1.1) b_i(M;L)ˆP_i

kˆ0h^{k;i k}_i (M;L).

(1.2) h^{j;i j}_i (M;L)ˆh^{i j;}_i ^j(M;L).

(1.3) h^i;0_i (M;L)ˆh^0;i_i (M;L)ˆg_i(M;L).

(2)Assume that L is base point free. Then for every integers i and j with 1im and0ji the following hold.

(2.1) bi(M;L)bi(M).

(2.2) h^{j;i j}_i (M;L)h^{j;i j}(M).

PROOF. See [8, Theorem 3.1 (3.1.1), (3.1.3), (3.1.4) and Proposition 3.3 (2) and (3)]. p

PROPOSITION2.2. For every integersa,k,landrwith0l, X^l

jˆ0

( 1)^j r‡j a j

r k l j

ˆ( 1)^l k a‡l l

:

PROOF. See [8, Proposition 2.5]. p

NOTATION2.2. LetXbe a smooth projective variety of dimensionn1 and letE be an ample vector bundle of rankronX. We put P:ˆP_X(E), H:ˆH(E) andm:ˆdimP. Thenmˆn‡r 1. In this paper we assume thatr2.

REMARK2.2. LetX,E,P,H,m,nandrbe as in Notation 2.2.

(1) By [18, (2.1) Proposition], we have

bj(P)ˆbj(X)‡bj 2(X)‡ ‡bj 2r‡2(X):

(2) Leti be an integer withim. Thenn‡r 1iand we obtain ri n‡1.

(2.1) Ifi2n 2 andi 1j, then

j 2r‡2(i 1) 2(i n‡1)‡2

ˆ2n 1 i 1:

So by (1) above, ifi2n 2 andi 1j, then by (1) we have

b_j(P)ˆ X^l

kˆ0

bj 2k(X) if jˆ2l;

X^l

kˆ0

b_{j 2k}(X) if jˆ2l‡1:

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By the same argument as this, ifi2n 1, then we see that

b_i(P)ˆ X^l

kˆ0

b_{i 2k}(X) if iˆ2l;

X^l

kˆ0

bi 2k(X) if iˆ2l‡1:

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:

(2.2) Assume that im 1. If iˆ2n 3 and i 1j, then j 2r‡22. If this equality holds, then iˆn‡r 1ˆm. But this contradicts the assumption. Hencej 2r‡21, and by (1) above we have

b_j(P)ˆ X^l

kˆ0

b_{j 2k}(X) if jˆ2l;

X^l

kˆ0

b_{j 2k}(X) if jˆ2l‡1:

8>

>>

>>

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>>

>>

>:

By the same argument as this, if im 1, n1 (resp. n2) and iˆ2n 2 (resp.iˆ2n 3), then we see thatb_{2n 2}(P)ˆP_{n 1}

kˆ0b_{2n 2 2k}(X) (resp.b_{2n 3}(P)ˆP_{n 2}

kˆ0b_{2n 3 2k}(X)).

DEFINITION2.2. LetX be a smooth projective variety of dimensionn and letEbe a vector bundle of rankronX.

(1) TheChern polynomial c_t(E) is defined by the following:

ct(E)ˆ1‡c1(E)t‡c2(E)t²‡ :

(2) For every integerjwithj0, thejth Segre class s_j(E)ofEis defined by the following equation:c_t(E^_)s_t(E)ˆ1, wherec_t(E^_) is the Chern poly- nomial ofE^_ands_t(E)ˆP

j0s_j(E)t^j.

REMARK2.3. (1) LetXbe a smooth projective variety and letF be a vector bundle onX. L et~s_j(F) be the Segre class which is defined in [10, Chapter 3]. Thens_j(F)ˆ~s_j(F^_).

(2) For every integer i with 1i,s_i(F) can be written by using the Chern classes cj(F) with 1ji. (For example, s1(F)ˆc1(F), s2(F)ˆc1(F)² c2(F), and so on.)

DEFINITION2.3. LetX be a smooth projective variety of dimensionn and letE be an ample vector bundle of rankn 1 on X. Then thecurve genus cg(X;E)of(X;E) is defined as follows:

cg(X;E)ˆ1‡1

2(K_X‡c₁(E))c_{n 1}(E):

3. Sectional invariants of scrolls over a smooth projective manifold.

THEOREM3.1. Let X,E, P, H, m, n and r be as in Notation2.2. Then the following hold.

(3.1.1)If i2n 1, then ei(P;H)ˆ(i n‡1)cn(X).

(3.1.2)If n2, then e2n 2(P;H)ˆ(n 1)cn(X)‡cn(E).

(3.1.3)If n3, then e_{2n 3}(P;H)ˆ(n 2)(c_n(X) c_n(E)) c_{n 1}(E)(K_X‡c₁(E)).

(3.1.4)e1(P;H)ˆ (n 2)sn(E) (c1… † ‡E KX)sn 1(E).

(3.1.5)e₀(P;H)ˆs_n(E).

PROOF. By [10, Example 3.2.11] and Remark 2.3 (1), for every integerl with 0lm

c_l(P)ˆX^l

jˆ0

X^j

kˆ0

r k j k

c_k pE^_

H(E)^{j k}c_{l j}…pT_X†:

(Herep:P!Xdenotes the projection.) Hence e_i(P;H)

ˆXⁱ

jˆ0

( 1)^j m i‡j 1 j

c_{i j}(P)H^{m i‡j}

ˆXⁱ

jˆ0

( 1)^j m i‡j 1 j

X^{i j}

sˆ0

X^s

uˆ0

r u s u

cu pE^_

H(E)^{m i‡j u‡s}c_{i j s}…pTX†

( )

ˆ X

0k‡ti0k;t

X

i t k jˆ0

( 1)^j r‡j‡n i 2 j

r k i t k j

!

c_k pE^_

c_t…pT_X†H(E)^{m k t}: By Proposition 2.2 and [3, (3.7) in Chapter 0], we get

e_i(P;H) …1†

ˆ X

0k‡ti0k;t

( 1)^{i t k} k‡(n i 2)‡i t k i t k

ck pE^_

ct…pTX†H(E)^{m k t}

ˆ X

0k‡ti0k;t

( 1)^{i t k} n t 2 i t k

c_k pE^_

c_t…pT_X†H(E)^{m k t}

ˆ X

0k‡ti0k;t

( 1)^{i t k} n t 2 i t k

c_k E^_

ct…TX†s_{n k t}(E):

Here we put

E(i;k;t):ˆ( 1)^{i t k} n t 2 i t k

c_k E^_

c_t…T_X†s_{n k t}(E):

In order to calculate e_i(P;H), we have only to consider the case where E(i;k;t)6ˆ0. We note that ifk‡t>n, thenc_k E^_

c_t…T_X†s_{n k t}(E)ˆ0. So we may assume thatk‡tn.

(a) The case wherei2n 1.

First we note that if (n;i)ˆ(1;1), then

e_i(P;H)ˆe₁(P;H)ˆ2 b₁(P;H)ˆ2 2g₁(P;H)ˆ2 2q(X)ˆc₁(X):

So we assume that (n;i)6ˆ(1;1).

Here we note that (n t 2) (i t k)ˆn i 2‡k n‡k 1 1 and k‡ti. Hence n t 2

i t k

6ˆ0 if and only if one of the following holds.

(a.1)i t kˆ0.

(a.2)i t k>0 andn t 2<0.

(a.1) Ifi t kˆ0, then 2n 1iˆt‡kn. Hencenˆ1 andiˆ1.

But this contradicts the assumption here.

(a.2) If i t k>0 and n t 2<0, then tˆn 1 or n because k‡tnandk0. Hence (t;k)ˆ(n 1;0), (n 1;1) or (n;0).

If (t;k)ˆ(n 1;0), then n t 2

i t k

ˆ 1

i n‡1

ˆ( 1)^{i n‡1}: If (t;k)ˆ(n 1;1), then

n t 2 i t k

ˆ 1 i n

ˆ( 1)^{i n}: If (t;k)ˆ(n;0), then

n t 2 i t k

ˆ 2 i n

ˆ( 1)^{i n}(i n‡1):

Hence if (n;i)6ˆ(1;1), then e_i(P;H)

ˆ( 1)^{2(i n‡1)}c_{n 1}(T_X)s₁(E)‡( 1)^{2(i n)}c₁(E^_)c_{n 1}(T_X)

‡( 1)^{2(i n)}(i n‡1)c_n(T_X)

ˆ(i n‡1)c_n(X):

Therefore in any case

e_i(P;H)ˆ(i n‡1)c_n(X) ifi2n 1.

(b) The case whereiˆ2n 2. Here we note thatn2 in this case.

Then n t 2

i t k

ˆ n t 2

2n t 2 k

. Here we note that (n t 2)2n t 2 k. Hence n t 2

2n t 2 k

6ˆ0 if and only if one of the following holds.

(b.1)kˆn.

(b.2)k<nandn t 2<0.

(b.1) If kˆn, then tˆ0 because t‡kn and t0, and n t 2

2n t 2 k

ˆ1. Here we note thatn t 20 in this case because n2 andtˆ0. Hence

E(2n 2;n;0)ˆ( 1)^{2n 2 n}cn(E^_)ˆcn(E):

(b.2) Ifk<nandn t 2<0, thentˆn 1 orn. Hence (t;k)ˆ(n 1;0), (n 1;1) or (n;0). By the same argument as (a.2), we obtain

E(2n 2;0;n 1)‡E(2n 2;1;n 1)‡E(2n 2;0;n)

ˆ( 1)^{2(2n 2 n‡1)}cn 1(X)s1(E)‡( 1)^{2(2n 2 n)}c1(E^_)cn 1(X)

‡( 1)^{2(2n 2 n)}(n 1)cn(X)

ˆ(n 1)cn(X):

Hence we get

e2n 2(P;H)ˆ(n 1)cn(X)‡cn(E):

(c) Assume that iˆ2n 3. Here we note that n3 in this case. Then n t 2

2n t k 3

6ˆ0 if and only if one of the following holds.

(c.1)kˆn.

(c.2)kˆn 1.

(c.3)kn 2 andt>n 2.

First we consider the case (c.1). Thentˆ0 becausek‡tnandt0.

So we have

E(2n 3;n;0)ˆ( 1)^{2n 3 0 n} n 0 2

2n 3 0 n

cn(E^_)

ˆ n 2 n 3

c_n(E)

ˆ (n 2)cn(E):

Next we consider the case (c.2). Then tˆ0 or 1, and n t 2

2n t k 3

ˆ1. Hence we haveE(2n 3;n 1;0)ˆ c_{n 1}(E)s₁(E)ˆ

ˆ c_{n 1}(E)c₁(E) andE(2n 3;n 1;1)ˆc_{n 1}(E)c₁(X).

Finally we consider the case (c.3). Then (k;t)ˆ(0;n 1), (1;n 1) or (0;n). Hence by the same argument as above

E(2n 3;0;n 1)‡E(2n 3;1;n 1)‡E(2n 3;0;n)ˆ(n 2)cn(X):

Therefore e2n 3(P;H)

ˆ (n 2)c_n(E) c_{n 1}(E)c₁(E)‡c_{n 1}(E)c₁(X)‡(n 2)c_n(X)

ˆ(n 2)(c_n(X) c_n(E))‡c_{n 1}(E)(c₁(X) c₁(E))

ˆ(n 2)(c_n(X) c_n(E)) c_{n 1}(E)(K_X‡c₁(E)):

(d) The case whereiˆ1. Then by (1) we get e₁(P;H)

ˆ X

0k‡t10k;t

( 1)^{1 t k} n t 2

1 t k

c_k E^_

c_t…T_X†s_{n k t}(E)

ˆ (n 2)s_n(E)‡(c₁ E^_

‡c₁…T_X†)s_{n 1}(E)

ˆ (n 2)s_n(E) (c₁… † ‡E K_X)s_{n 1}(E):

(e) The case whereiˆ0. Then by (1) we gete₀(P;H)ˆs_n(E).

We get the assertion of Theorem 3.1. p

By Theorem 3.1, we get the following fornˆ1 and 2.

COROLLARY3.1. Let X,E, P, H and n be as in Notation2.2.

(3.1.1)Assume that nˆ1. Then we get the following:

ei(P;H)ˆ i(2 2g(X)) if i1;

degE if iˆ0:

(

(3.1.2)Assume that nˆ2. Then we get the following:

e_i(P;H)ˆ

(i 1)c₂(X) if i3;

c2(X)‡c2(E) if iˆ2;

(c₁… † ‡E K_X)c₁(E) if iˆ1;

s2(E) if iˆ0:

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THEOREM3.2. Let X,E, P, H, m, n and r be as in Notation2.2. Then the following hold.

(3.2.1)If mi2n 1, then b_i(P;H)ˆb_i(P).

(3.2.2)If n2and m>2n 2, then b2n 2(P;H)ˆb2n 2(P)‡cn(E) 1.

(3.2.3)If n3and m>2n 3, then

b2n 3(P;H)ˆb2n 3(P)‡(n 2)cn(E)‡cn 1(E)…KX‡c1(E)† ‡2 2q(X):

(3.2.4)b1(P;H)ˆ2‡(n 2)sn(E)‡…c1(E)‡K_X†sn 1(E).

(3.2.5)b0(P;H)ˆsn(E).

PROOF. Sinceei(P;H) has been calculated in Theorem 3.1, we need to studyX^{i 1}

kˆ0

( 1)^kb_k(P). By using Remark 2.2, we calculate this.

(a) The case where i2n 1. Since b_m(P;H)ˆb_m(P), we assume that i<m.

(a.1) Assume thati is even. Theni2n. Here we note thatbj(X)ˆ0 if j>2n. Then

X^{i 1}

kˆ0

( 1)^kb_k(P)

ˆXⁿ

kˆ0

i

2 k

b_2k(X) Xⁿ

kˆ1

i

2 k‡1

b_{2k 1}(X)

ˆXⁿ

kˆ0

n k‡1

… †b2k(X) Xⁿ

kˆ1

(n k‡2)b2k 1(X)‡i 2n 2 2 e(X):

Here we note that by Remark 2.2 (2.1), we haveb_i(P)ˆXⁿ

kˆ0

b_2k(X) because i2nin this case. Sinceiis even, we have

b_i(P;H) b_i(P)

ˆ( 1)ⁱ e_i(P;H) 2X^{i 1}

jˆ0

( 1)^jb_j(P)

! Xⁿ

kˆ0

b_2k(X)

ˆ(i n‡1)cn(X) 2Xⁿ

kˆ0

(n k‡1)b2k(X)

‡2Xⁿ

kˆ1

(n k‡2)b2k 1(X) (i 2n 2)e(X) Xⁿ

kˆ0

b2k(X)

ˆ(n‡3)e(X)‡Xⁿ

kˆ0

(2k 2n 3)b_2k(X)‡Xⁿ

kˆ1

(2n 2k‡4)b_{2k 1}(X)

ˆXⁿ

kˆ0

(2k n)b_2k(X) Xⁿ

kˆ1

(2k n 1)b_{2k 1}(X):

Here we prove the following.

CLAIM3.1. (1)Xⁿ

kˆ0

(2k n)b_2k(X)ˆ0.

(2)Xⁿ

kˆ1

(2k n 1)b_{2k 1}(X)ˆ0.

PROOF. (1) Assume thatnˆ2l. Then we note that (2k n)b2k(X)ˆ0 if kˆl. So by Poincare duality

Xⁿ

kˆ0

(2k n)b2k(X)

ˆX^{l 1}

kˆ0

(2k n)b2k(X)‡ Xⁿ

kˆl‡1

(2k n)b2k(X)

ˆX^{l 1}

kˆ0

(2k n)b_2k(X)‡X^{l 1}

kˆ0

(n 2k)b_{2n 2k}(X)

ˆX^{l 1}

kˆ0

(2k n)b_2k(X) X^{l 1}

kˆ0

(2k n)b_2k(X)

ˆ0:

Ifnˆ2l‡1, then by Poincare duality Xⁿ

kˆ0

(2k n)b_2k(X)

ˆX^l

kˆ0

(2k n)b_2k(X)‡ Xⁿ

kˆl‡1

(2k n)b_2k(X)

ˆX^l

kˆ0

(2k n)b2k(X)‡X^l

kˆ0

(n 2k)b2n 2k(X)

ˆX^l

kˆ0

(2k n)b2k(X) X^l

kˆ0

(2k n)b2k(X)

ˆ0:

Hence we obtain the assertion of (1).

(2) This can be proved by the same argument as (1). p By Claim 3.1,b_i(P;H)ˆb_i(P) ifi2n 1 andiis even.

(a.2) Assume thatiis odd. Then X^{i 1}

jˆ0

( 1)^jb_j(P)

ˆXⁿ

kˆ0

i‡1

2 k

b2k(X) Xⁿ

kˆ1

i 1

2 k‡1

b2k 1(X)

ˆXⁿ

kˆ0

(n k‡1)b_2k(X) Xⁿ

kˆ1

(n k‡1)b_{2k 1}(X)

‡ i‡1

2 n 1

e(X):

By Remark 2.2 (2.1), we haveb_i(P)ˆXⁿ

kˆ1

b_{2k 1}(X) becausei2n 1 in this case. Hence by Theorem 3.1 and Claim 3.1, we have

bi(P;H) bi(P)

ˆ (i n‡1)e(X) 2X^{i 1}

jˆ0

( 1)^jb_j(P)

! b_i(P)

ˆ (n‡2)e(X)‡Xⁿ

kˆ0

(2n 2k‡2)b_2k(X) Xⁿ

kˆ1

(2n 2k‡3)b_{2k 1}(X)

ˆXⁿ

kˆ0

(n 2k)b2k(X) Xⁿ

kˆ1

(n 2k‡1)b2k 1(X)

ˆ0:

Thereforeb_i(P;H)ˆb_i(P) ifi2n 1 andiis odd.

In any case, ifmi2n 1, thenb_i(P;H)ˆb_i(P).

(b) The case whereiˆ2n 2 and 2n 2<m. Then by Remark 2.2 (2.1) we obtain

b_j(P)ˆ X^l

kˆ0

b_{j 2k}(X) if jˆ2l, X^l

kˆ0

b_{j 2k}(X) if jˆ2l‡1 8>

>>

>>

<

>>

>>

>:

for every integerjwithj<2n 2:Hence X

2n 3 jˆ0

( 1)^jbj(P)ˆX^{n 2}

kˆ0

(n k 1)b2k(X) X^{n 1}

kˆ1

(n k)b2k 1(X):

By assumption, we haven‡r 1ˆmi‡1ˆ2n 1. So by Remark 2.2 (2.2) we obtain

b_{2n 2}(P)ˆX^{n 1}

kˆ0

b_2k(X):

Therefore

b2n 2(P;H) b2n 2(P)

ˆ e2n 2(P;H) 2²ⁿX³

jˆ0

( 1)^jbj(P)

!

b2n 2(P)

ˆ(n 1)e(X)‡c_n(E) X^{n 2}

kˆ0

(2n 2k 2)b_2k(X)

‡X^{n 1}

kˆ1

(2n 2k)b_{2k 1}(X) X^{n 1}

kˆ0

b_2k(X)

ˆ(n 1)e(X)‡c_n(E) X^{n 2}

kˆ0

(2n 2k 1)b_2k(X)

‡X^{n 1}

kˆ1

(2n 2k)b_{2k 1}(X) b_{2n 2}(X)

ˆ(n 1)e(X)‡c_n(E)‡e(X) 1 X^{n 1}

kˆ0

(2n 2k)b_2k(X)

‡X^{n 1}

kˆ1

(2n 2k‡1)b_{2k 1}(X)‡b_{2n 1}(X)

ˆne(X)‡c_n(E) 1 X^{n 1}

kˆ0

(2n 2k)b_2k(X)

‡Xⁿ

kˆ1

(2n 2k‡1)b2k 1(X):

Since

X^{n 1}

kˆ0

(2n 2k)b2k(X)ˆXⁿ

kˆ0

(2n 2k)b2k(X);

we obtain

b_{2n 2}(P;H) b_{2n 2}(P)

ˆne(X)‡c_n(E) 1 Xⁿ

kˆ0

(2n 2k)b_2k(X)

‡Xⁿ

kˆ1

(2n 2k‡1)b2k 1(X)

ˆc_n(E) 1‡Xⁿ

kˆ0

(2k n)b_2k(X) Xⁿ

kˆ1

(2k n 1)b_{2k 1}(X):

By Claim 3.1, we get

b_{2n 2}(P;H)ˆb_{2n 2}(P)‡c_n(E) 1:

(c) The case where iˆ2n 3 and i<m. Then by Remark 2.2 (2.2) we obtain

b_j(P)ˆ X^l

kˆ0

b_{j 2k}(X) if jˆ2l, X^l

kˆ0

bj 2k(X) if jˆ2l‡1 8>

>>

>>

<

>>

>>

>:

for every integerjwithj<2n 3. Hence X

2n 4 jˆ0

bj(P)ˆX^{n 2}

kˆ0

(n k 1)b2k(X) X^{n 2}

kˆ1

(n k 1)b2k 1(X):

By assumption we haven‡r 1ˆmi‡1ˆ2n 2. Hence by Remark 2.2 (2.2) we obtain

b_{2n 3}(P)ˆX^{n 1}

kˆ1

b_{2k 1}(X):

Therefore

b_{2n 3}(P;H) b_{2n 3}(P)

ˆ e_{2n 3}(P;H)‡2²ⁿX⁴

jˆ0

( 1)^jb_j(P) b_{2n 3}(P)

ˆ (n 2)…e(X) cn(E)† ‡cn 1(E)…K_X‡c1(E)†

‡X^{n 2}

kˆ0

(2n 2k 2)b_2k(X) X^{n 2}

kˆ1

(2n 2k 2)b_{2k 1}(X) X^{n 1}

kˆ1

b2k 1(X)

ˆ (n 2)…e(X) c_n(E)† ‡c_{n 1}(E)…K_X‡c₁(E)†

‡X^{n 2}

kˆ0

(2n 2k 2)b_2k(X) X^{n 2}

kˆ1

(2n 2k 1)b_{2k 1}(X) b2n 3(X)

ˆ (n 2)…e(X) cn(E)† ‡cn 1(E)…K_X‡c1(E)†

2e(X)‡2b_{2n 2}(X)‡2b_2n(X) 3b_{2n 3}(X) 2b_{2n 1}(X)

‡X^{n 2}

kˆ0

(2n 2k)b2k(X) X^{n 2}

kˆ1

(2n 2k‡1)b2k 1(X)

ˆ (n 2)…e(X) c_n(E)† ‡c_{n 1}(E)…K_X‡c₁(E)†

‡Xⁿ

kˆ0

(2n 2k)b_2k(X) Xⁿ

kˆ1

(2n 2k‡1)b_{2k 1}(X) 2e(X)‡2b_2n(X) b_{2n 1}(X)

ˆ (n 2)e(X)‡(n 2)cn(E)‡cn 1(E)…KX‡c1(E)†

‡Xⁿ

kˆ0

(n 2k)b_2k(X) Xⁿ

kˆ1

(n 2k‡1)b_{2k 1}(X)

‡(n 2)e(X)‡2b_2n(X) b_{2n 1}(X):

Sinceb2n(X)ˆ1 andb2n 1(X)ˆb1(X)ˆ2q(X), by Claim 3.1 we obtain b_{2n 3}(P;H) b_{2n 3}(P)

ˆ(n 2)c_n(E)‡c_{n 1}(E)…K_X‡c₁(E)† ‡2 2q(X):

(d) The case whereiˆ1.

Then by Theorem 3.1 (3.1.4) and the definition ofb1(P;H) b1(P;H)ˆ e1(P;H)‡2b0(P)

ˆ2‡(n 2)s_n(E)‡…c₁(E)‡K_X†s_{n 1}(E):

(e) The case whereiˆ0.

Then by Theorem 3.1 (3.1.5) and the definition ofb₀(P;H), b₀(P;H)ˆe₀(P;H)ˆs_n(E):

Therefore we get the assertion. p

COROLLARY3.2. Let X,E, P, H, m and n be as in Notation2.2. If n2 and m>i2n 2, then b_i(P;H)b_i(P).

PROOF. Ifi2n 1, then by Theorem 3.2 (3.2.1) we get the assertion.

Next we consider the case where iˆ2n 2. SinceE is ample, we have cn(E)1. Therefore by Theorem 3.2 (3.2.2), we seeb2n 2(P;H)b2n 2(P).

This completes the proof. p

COROLLARY3.3. Let X,E, P, H, m and n be as in Notation2.2.

(3.3.1)Assume that nˆ1. Then we get the following:

b_i(P;H)ˆ bi(P) if mi1;

degE if iˆ0:

(

(3.3.2)Assume that nˆ2. Then we get the following:

bi(P;H)ˆ

b_i(P) if mi3;

b2(P)‡c2(E) 1 if iˆ2;

c₁(E)(c₁… † ‡E K_X)‡2 if iˆ1;

s2(E) if iˆ0:

8>

>>

<

>>

>:

PROOF. By Theorem 3.2, we get the assertion. (Here we note that if nˆ2, then 2n 2ˆ2<n‡r 1ˆmbecause we assumer2 in this

paper.) p

REMARK 3.1. Since E is ample, we see that if nˆ1 or 2, then bi(P;H)0 for anyi.

Here we calculateh^{j;i j}_i (P;H) for the case wheremi2n 1.

THEOREM 3.3. Let X, E, P, H, m and n be as in Notation 2.2. If mi2n 1and0ji, then h^{j;i j}_i (P;H)ˆh^{j;i j}(P).

PROOF. First we note that we can take an ample line bundleAonX such thatE A^tis ample and spanned for every positive integert. Hence H(E A^t) is ample and spanned. We also note that there exists an iso- morphismf:P_X(E A^t)!PwithH(E A^t)ˆf(Hp(A^t)), where p:P!X is the projection. Therefore Hp(A^t) is also ample and spanned. By thisf, we identifyP_X(E A^t) andP. By Theorem 3.2 we have

bi(P;H(E A^t))ˆbi(P):

Hence

b_i(P;Hp(A^t))

ˆbi(P;H(E A^t))

ˆb_i(P):

On the other hand by Proposition 2.1 (1.1) and (2) we have h^{j;i j}_i (P;Hp(A^t))ˆh^{j;i j}(P)

because Hp(A^t) is ample and spanned. But since F(t):ˆ

:ˆh^{j;i j}_i (P;Hp(A^t)) h^{i;i j}(P) is a polynomial in t and F(t)ˆ0 for every positive integert, we see thatF(0)ˆ0, that is,

h^{j;i j}_i (P;H)ˆh^{i;i j}(P):

This completes the proof. p

COROLLARY3.4. Let X,E, P, H, m and n be as in Notation2.2.

(3.4.1)Assume that nˆ1. Then we get the following:

h_i^{j;i j}(P;H)ˆ h^{j;i j}(P) if mi1and0ji;

degE if iˆ0and jˆ0:

(

(3.4.2)Assume that nˆ2. Then we get the following:

h_i^{j;i j}(P;H)ˆ

h^{j;i j}(P) if mi3 and0ji;

h^0;2(P) if iˆ2 and jˆ0;

h^2;0(P) if iˆ2 and jˆ2;

h^1;1(P)‡c₂(E) 1 if iˆ2 and jˆ1;

12(c1(E)(c1… † ‡E KX))‡1 if iˆ1 and jˆ0;1;

s₂(E) if iˆ0 and jˆ0:

8>

>>

>>

>>

>>

<

>>

>>

>>

>>

>:

PROOF. If nˆ1, then by Corollary 3.3 and Theorem 3.3 we get the assertion.

Assume thatnˆ2. Then by Corollary 3.3 and Theorem 3.3 we get the assertion for the case wheremi3 and 0ji. If (i;j)ˆ(2;0) (resp.

(2;2)), then by Proposition 2.1 (1.3) and [5, Example 2.10 (8)] we have h^0;2₂ (P;H)ˆg₂(P;H)ˆh²(O_P)ˆh^0;2(P) (resp. h^2;0₂ (P;H)ˆg₂(P;H)ˆ

ˆh²(OP)ˆh^0;2(P)ˆh^2;0(P)). Moreover by Corollary 3.3 (3.3.2) and Pro- position 2.1 (1.1) we geth^1;1₂ (P;H)ˆh^1;1(P)‡c2(E) 1.

Assume that iˆ1. Then by Proposition 2.1 (1.1) we have b₁(P;H)ˆh^1;0₁ (P;H)‡h^0;1₁ (P;H). Moreover by Proposition 2.1 (1.2) we haveh^1;0₁ (P;H)ˆh^0;1₁ (P;H). Therefore by Corollary 3.3 we get the asser- tion for the caseiˆ1.

By Corollary 3.3 we get the assertion for the case whereiˆ0. p REMARK 3.2. Since E is ample, we see that if nˆ1 or 2, then h^{j;i j}_i (P;H)0 for anyiandjwith 0jim.

4. Anew invariant of generalized polarized manifolds.

Here we use Notation 2.2. Assume that iˆ2n 3, n3 andi<m.

Then by Theorem 3.2 (3.2.3) we see that (n 2)cn(E)‡cn 1(E)(KX‡c1(E)) is even because b_{2n 3}(P;H) and b_{2n 3}(P) are even (see [8, Theorem 3.1 (3.1.2)]). We put

v(X;E):ˆ1‡1

2…(n 2)cn(E)‡cn 1(E)(K_X‡c1(E))†:

Here we note thatrn 1 sincen‡r 1ˆmi‡1ˆ2n 2.

Ifrˆn 1, then

v(X;E)ˆ1‡1

2…KX‡c1(E)†cn 1(E):

Sov(X;E) is thought to be a generalization of the curve genuscg(X;E) of (X;E) (see Definition 2.3). Here we definev(X;E) again.

DEFINITION4.1. Let (X;E) be a generalized polarized manifold of di- mensionn3. Assume thatrn 1. Then the invariantv(X;E) of (X;E) is defined as follows.

v(X;E):ˆ1‡1

2…(n 2)c_n(E)‡c_{n 1}(E)(K_X‡c₁(E))†:

Sinceb2n 3(P;H) b2n 3(P)ˆ2v(X;E) 2q(X) by Theorem 3.2 (3.2.3), we can propose the following conjecture.

CONJECTURE 4.1. Let (X;E) be a generalized polarized manifold of dimension n3. Assume that rn 1. Then v(X;E)q(X).

Here we study the non-negativity ofv(X;E).

THEOREM 4.1. Let (X;E)be a generalized polarized manifold of di- mension n3. Assume that rn 1. Then v(X;E)0.

PROOF. If rˆn 1, then v(X;E) is the curve genus and then v(X;E)0 by [19, Theorem 1]. So we may assume thatrn.

IfrnandKX‡c1(E) is nef, then (KX‡c1(E))cn 1(E)0 becauseEis ample. Furthermorecn(E)1. So we obtainv(X;E)2.

IfrnandK_X‡c₁(E) is not nef, then (X;E)(Pⁿ;O_Pⁿ(1)ⁿ) by [22, Theorem 1 and Theorem 2]. In this case (K_X‡c₁(E))c_{n 1}(E)ˆ n and c_n(E)ˆ1. Hencev(X;E)ˆ0. Therefore we get the assertion. p

THEOREM 4.2. Let (X;E)be a generalized polarized manifold of di- mension n3. Assume that rn 1.

(1)If v(X;E)ˆ0, then(X;E)is one of the following.

(1.1) (Pⁿ;O_Pⁿ(1)ⁿ).

(1.2) (Pⁿ;O_Pⁿ(1)^{n 1}).

(1.3) (Pⁿ;O_Pⁿ(1)^{n 2} O_Pⁿ(2)).

(1.4) (Qⁿ;O_Qⁿ(1)^{n 1}).

(1.5) XP_P¹(F)for some vector bundleF of rank n onP¹, and E  ⁿ_jˆ1¹(H(F)‡pO_P¹(bj)), where p:X!P¹ is the bundle projection.

(2) If v(X;E)ˆ1, then(X;E)is one of the following.

(2.1) XP_C(F)for some vector bundleF of rank n on a smooth elliptic curve C, andE_F O_Pⁿ¹(1)^{n 1}for any fiber F of the bundle projection X!C.

(2.2) K_X‡det(E)ˆ O_X andrˆn 1.

PROOF. If v(X;E)1, then by the proof of Theorem 4.1, one of the following cases occurs:

(4.2.1)rˆn 1.

(4.2.2)rnandK_X‡c₁(E) is not nef.

If rˆn 1 and v(X;E)ˆ0 (resp. v(X;E)ˆ1), then 0ˆv(X;E)ˆ

ˆcg(X;E) (resp. 1ˆv(X;E)ˆcg(X;E)) and by [19, Theorem 1 and Theo- rem 2] we get the above types.

IfrnandK_X‡c1(E) is not nef, then (X;E)(Pⁿ;O_Pⁿ(1)ⁿ) by [22, Theorem 1 and Theorem 2] and then v(X;E)ˆ0. Hence we get the as- sertion. p

REMARK 4.1. (1) If (X;E) is the type (2.2) in Theorem 4.2, then a classification of (X;E) has been obtained by [20].

(2) Ifv(X;E)1, then we see thatv(X;E)q(X).

PROPOSITION 4.1. Let (X;E) be a generalized polarized manifold of dimension n3. Assume that rn 1 and E is spanned. Then

v(X;E)q(X).

PROOF. Set (P;H):ˆ(PX(E);H(E)). ThenHis ample and spanned. So by Proposition 2.1 (2.1) we obtain

b_{2n 3}(P;H)b_{2n 3}(P):

On the other hand

b_{2n 3}(P;H) b_{2n 3}(P)

ˆ2v(X;E) 2q(X):

Hence we get the assertion. p

REMARK4.2. (1) By Theorem 4.2 we can determine the type of (X;E) such thatv(X;E)ˆq(X) andq(X)1.

(2) For every integer q2, there exists an example of (X;E) with v(X;E)ˆqˆq(X) (This was given by the referee.): Let C be a smooth projective curve withg(C)ˆq2. We note that there exist vector bundlesF andGonC with rankF ˆnand rankG ˆn 1 such thatE:ˆH(F)p(G) is an ample vector bundle of rankn 1 onX, whereX:ˆPC(F),H(F) is the tautological line bundle ofF and p:X!C is the bundle projection. Then we can easily check thatv(X;E)ˆcg(X;E)ˆq2.

(3) We see that if E is ample and spanned, rankE ˆn 1 and v(X;E)ˆq(X)2, then (X;E) is isomorphic to the type in (2) above.

This has been proved by [17, Theorem].

Here we give the following conjecture which was pointed out by the referee:

CONJECTURE4.2. Let X be a smooth projective variety of dimension n3and letEbe an ample vector bundle on X withrankE ˆr. Assume that rn 1 and q(X)2. If E is spanned and v(X;E)ˆq(X), then rˆn 1.

REMARK4.3. We consider the case wherern3. (The following (a) and (b) were also pointed out by the referee.)

(a) IfK_X‡c₁(E) is not ample, then by [4, Main Theorem] (X;E) is one of the following 6 types and we can calculatev(X;E) andq(X):

(a.1) (Pⁿ;O_Pⁿ(1)^n‡1). In this case (v(X;E);q(X))ˆ(1‡(1=2) (n 2)(n‡1);0).

(a.2) (Pⁿ;O_Pⁿ(1)ⁿ). In this case (v(X;E);q(X))ˆ(0;0).

(a.3) XP_C(F) for a vector bundleF of ranknon a smooth curve C, and E H(F)p(G) for a vector bundle G on C with rankG ˆn, wherep:X!Cis the bundle projection. In this case (v(X;E);q(X))ˆ(g(C)‡(n 1)(g(C) 1‡c1(F)‡c1(G));

g(C)).

(a.4) (Pⁿ;O_Pⁿ(1)^{n 1} O_Pⁿ(2)). In this case (v(X;E);q(X))ˆ

ˆ(n 1;0).

(a.5) (Pⁿ;T_Pⁿ). In this case (v(X;E);q(X))ˆ(1‡(1=2)(n 2) (n‡1);0).

(a.6) (Qⁿ;O_Qⁿ(1)ⁿ). In this case (v(X;E);q(X))ˆ(n 1;0).

Here we consider the case where (X;E) is the type (a.3) above. Then 1cn(E)ˆc1(F)‡c1(G) becauseEis ample. Henceg(C) 1‡c1(F)‡

‡c1(G)0 and we get v(X;E)g(C)ˆq(X). Moreover if g(C)1, then we see thatv(X;E)q(X)‡(n 1)>q(X).

We also note that ifEis spanned by global sections, thencn(E)2 by [21, (3.4) Theorem] and we obtainv(X;E)q(X)‡(n 1)>q(X).

(b) Next we assume that K_X‡c₁(E) is ample. Then we can prove the following:

PROPOSITION 4.2. (1) If KX‡c1(E) is ample, then v(X;E)1‡

‡1

2(n 1).

(2)If KX‡c1(E)is ample andEis spanned, then v(X;E)n.

PROOF. (1) By assumption, we have c_n(E)1 and (KX‡c1(E))cn 1(E)1. Hence we get the assertion of (1).

(2) Since Eis spanned and KX‡c1(E) is ample, we see thatcn(E)2 by [21, (3.4) Theorem]. Hence (n 2)cn(E)‡(K_X‡c1(E))cn 1(E) 2(n 2)‡2 because the term on the left is even. Therefore we get the assertion of (2). p

The above results suggest that for the case where rn and v(X;E)6ˆq(X) there are some gaps for the value of v(X;E) q(X), de- pending onn. We will investigate this in a future paper.

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Manoscritto pervenuto in redazione il 10 luglio 2007.