YoshiakiFukuma Onthesecondsectionalgeometricgenusofquasi-polarizedmanifolds

(de Gruyter 2004

On the second sectional geometric genus of quasi-polarized manifolds

Yoshiaki Fukuma

(Communicated by A. Sommese)

Abstract.LetðX;LÞbe a quasi-polarized manifold of dimX¼n. In a previous paper we gave a new invariant (thei-th sectional geometric genus) ofðX;LÞ, which is a generalization of the degree and the sectional genus ofðX;LÞ. In this paper we study some properties of the second sectional geometric genus.

Key words.Polarized manifolds, sectional genus, sectional geometric genus, Chern class.

2000 Mathematics Subject Classification. 14C20, 14C17, 14J60

0 Introduction

LetX be a projective variety of dimX ¼nover the complex number fieldC, and let Lbe a nef and big (resp. an ample) line bundle onX. Then we call the pairðX;LÞa quasi-polarized(resp.polarized)variety, andðX;LÞis called a quasi-polarized (resp.

polarized)manifoldifXis smooth. In [6], we gave a new invariant ofðX;LÞwhich is called thei-th sectional geometric genus g_iðX;LÞofðX;LÞfor 0cicn. We note that g_iðX;LÞis a generalization of the degree Lⁿ and the sectional genusgðLÞ. (Namely g₀ðX;LÞ ¼Lⁿ andg₁ðX;LÞ ¼gðLÞ.) Here we recall the reason why we call this in- variant the sectional geometric genus. Let ðX;LÞbe a quasi-polarized manifold of dimension nd2 with BsjLj ¼q, where BsjLj is the base locus of jLj. Let i be an integer with 1cicn, and let Y be the transversal intersection of generalniele- ments ofjLj. In this case Y is a smooth projective variety of dimensioni. Then we can prove thatgiðX;LÞ ¼hⁱðOYÞ, that is,giðX;LÞis the geometric genus ofY.

In [6] we study some fundamental properties of thei-th sectional geometric genus.

We find that we can generalize some problems about the sectional genus to the case of the sectional geometric genus. For example, in [6] we proposed the following conjecture:

Conjecture 0.1.LetðX;LÞbe a quasi-polarized manifold ofdimX¼n and let i be an integer with0cicn.Then g_iðX;LÞdhⁱðOXÞ.

Here we note that if i¼0, then this is true becauseg₀ðX;LÞ ¼Lⁿd1¼h⁰ðOXÞ. If i¼1, then this is Fujita’s conjecture. (See [3, (13.7)] or [1, Question 7.2.11].) Namely we can find that an inequality gðLÞdh¹ðOXÞ is a generalization of an inequality Lⁿd1. In [6] we proved that this conjecture is true if BsjLj ¼q. Moreover we clas- sified polarized manifoldsðX;LÞwhich satisfy the following properties:

(A) dimXd3, BsjLj ¼q, andg₂ðX;LÞ ¼h²ðOXÞ,

(B) dimXd3,Lis very ample, andg2ðX;LÞ ¼h²ðOXÞ þ1.

In a future paper, we will classify polarized manifolds ðX;LÞ such that L is very ample andg2ðX;LÞ h²ðOXÞc5. In [7] we study the conjecture for the case where 0cdim BsjLjcn1.

Furthermore in [6] we proved the following which is analogous to a theorem of Sommese ([11, Theorem 4.1]):

Theorem 0.2([6, Corollary 3.5]).LetðX;LÞbe a polarized manifold ofdimX¼nd3.

Assume that L is spanned.Then the following are equivalent:

(1) g2ðX;LÞ ¼h²ðOXÞ, (2) h⁰ðKXþ ðn2ÞLÞ ¼0, (3) kðKXþ ðn2ÞLÞ ¼ y,

(4) ðX;LÞis one of the types from(1)to(7-4)in Theorem1.13below.

In this way, it is interesting and very important to study the sectional geometric genus, and we hope that by using this invariant we can study polarized manifolds more deeply.

In this paper, we mainly study the second sectional geometric genus of (quasi-) polarized manifolds. The contents of this paper are the following: In Section 1, we prepare for some results which are used later. In Section 2, we give an explicit for- mula of the second sectional geometric genus of quasi-polarized manifolds. In Sec- tion 3, we study the second sectional geometric genus of polarized manifolds and we obtain the following:

(1) We give a lower bound ofg₂ðX;LÞfor dimXd4 and kðXÞd0. (Theorem 3.5 (1).) In particular we get thatg₂ðX;LÞdh¹ðOXÞ. (Corollary 3.5.2 (1).)

(2) We give some numerical conditions ofðX;LÞwithg2ðX;LÞ ¼0 if dimXd4 and kðXÞd0. (Corollary 3.5.4.)

(3) We prove thatg2ðX;2LÞd0 for dimX ¼3. (Theorem 3.7 and Corollary 3.7.1.) (4) We give a classification of ðX;LÞ with dimX ¼3 and g₂ðX;2LÞ ¼0. (Proposi-

tion 3.10 and Proposition 3.11.)

(5) We study the case where dimXd3,KX is nef andkðXÞd0. (Theorem 3.5 (2), Corollary 3.5.2 (2), and Proposition 3.9.)

The author would like to thank the referee for giving very valuable comments on the first version of this paper. In particular, the assertion of Theorem 3.5 was im- proved by the referee’s comment.

Notation and Conventions.In this paper, we shall study mainly a smooth projective varietyX over the complex number fieldC. The words ‘‘line bundles’’ and ‘‘Cartier divisors’’ are used interchangeably.

OðDÞ: invertible sheaf associated with a Cartier divisorDonX. OX: the structure sheaf ofX.

wðFÞ: the Euler–Poincare´ characteristic of a coherent sheafF.

hⁱðFÞ ¼dimHⁱðX;FÞfor a coherent sheafFonX. hⁱðDÞ ¼hⁱðOðDÞÞfor a divisorD.

jDj: the complete linear system associated with a divisorD.

K_X: the canonical divisor ofX.

kðDÞ: Iitaka dimension of a Cartier divisorDonX. kðXÞ: Kodaira dimension ofX.

Pⁿ: projective space of dimensionn.

Qⁿ: hyperquadric surface inP^nþ1.

@(or¼): linear equivalence.

1: numerical equivalence.

1 Preliminaries

Definition 1.1. Let X be a normal projective variety of dimX¼n, and let E be a vector bundle onX. LetU¼ ðh1;. . .;hn1Þbe anðn1Þ-tuple of numerically e¤ec- tiveQ-divisors onX. ThenEis said to beU-semistableif

d_UðFÞcd_UðEÞ for every nonzero subsheafFofE, where

d_UðGÞ:¼c1ðGÞh1. . .hn1

rankG for any torsion free sheafGonX.

Theorem 1.2(Harder–Narashimhan filtration).Let X be a normal projective variety of dimX ¼n and letEbe a torsion free sheaf on X.LetU¼ ðh1;. . .;hn1Þbe anðn1Þ- tuple of numerically e¤ectiveQ-divisors on X.Then there exists a unique filtration

S_U:0¼E0YE1Y YEs¼E that has the following properties:for any integer i with1cics (1) Gr_iðSUÞ:¼Ei=Ei1is a torsion freeU-semistable sheaf, (2) d_UðGriðSUÞÞis a strictly decreasing function on i.

Proof.See [10, Theorem 2.1]. r

Remark 1.2.1.We say that the above filtrationS_U ofEis theHarder–Narashimhan filtration ofEwith respect toU.

Definition 1.3.(1) The coherent subsheafE1in Theorem 1.2 is said to be themaximal U-destabilizing subsheaf ofE.

(2) LetX be a normal projective variety of dimX ¼n and let U¼ ðh1;. . .;h_n2Þ be aðn2Þ-tuple of numerically e¤ectiveQ-divisors onX. A torsion free sheafEon X is said to begenericallyU-semipositiveif for every numerically e¤ectiveQ-divisor DonX,d_ðU;DÞððEÞ₁Þc0, whereEdenotes the dual ofE, andðEÞ₁is the maximal ðU;DÞ-destabilizing subsheaf ofE. (See [10, Section 6].)

Theorem 1.4.Let X be a normal projective variety ofdimX ¼n such that X is smooth in codimension two.Let NAðXÞHfPicðXÞ=numerical equivalencegnRbe the ample cone.LetEbe a torsion free sheaf on X,with its first Chern class being a numerically e¤ective Q-divisor.Assume that E is generically B-semipositive,where B¼ ðh1;. . .; hn2Þand hiANAðXÞ_Qfor each i.Then

c2ðEÞh1. . .hn2d0:

Proof.See [10, Theorem 6.1]. r

Theorem 1.5.Let X be a smooth projective variety ofdimX¼n.Let H₁;. . .;H_n2be ample Cartier divisors on X.ThenW_X¹ is genericallyðH1;. . .;Hn2Þ-semipositive unless X is uniruled. (For the definition that X is uniruled,see Definition1.15below.)

Proof.See [10, Corollary 6.4]. r

Theorem 1.6(Hirzebruch–Riemann–Roch). Let X be a smooth complete variety and letEbe a locally free sheaf on X.Then

wðEÞ ¼ ð

X

chðEÞtdðTXÞ;

whereTX is the tangent bundle of X, chðEÞ(resp.tdðTXÞ)is the Chern character ofE (resp. the Todd class ofTX),and Ð

X denotes the degree of the zero-dimensional com- ponent ofðchðEÞtdðTXÞÞV½X.

Proof.See [9, Chapter Four]. r Notation 1.7. Let ðX;LÞ be a quasi-polarized manifold of dimX ¼nd3 and BsjLj ¼q. (Here BsjLjdenotes the base locus ofjLj.) We putX₀:¼X andL₀:¼L.

LetX_iAjLi1jbe a smooth member ofjLi1jandL_i¼Li1j_Xi for 1cicn1.

Definition 1.8.LetðX;LÞbe a quasi-polarized variety of dimX ¼n, and letwðtLÞbe the Euler–Poincare´ characteristic oftL. Here we put

wðtLÞ ¼Xⁿ

j¼0

w_jðX;LÞt^½j j! ;

where t^½j¼tðtþ1Þ. . .ðtþ j1Þ for jd1 and t^½0¼1. Then the sectional genus gðLÞofðX;LÞis defined by the following:

gðLÞ ¼1w_n1ðX;LÞ:

Remark 1.8.1.IfX is smooth, then the sectional genus ofðX;LÞcan be expressed by the following formula:

gðLÞ ¼1þ1

2ðKXþ ðn1ÞLÞLⁿ¹; whereK_X is the canonical divisor ofX.

Definition 1.9 (See [6, Definition 2.1]). Let ðX;LÞ be a quasi-polarized variety of dimX ¼n. Then for an integer 0cicn thei-th sectional geometric genus giðX;LÞ ofðX;LÞis defined by the following formula:

giðX;LÞ ¼ ð1Þⁱðw_niðX;LÞ wðOXÞÞ þXⁿⁱ

j¼0

ð1Þ^nijh^njðOXÞ:

(Here we use notation in Definition 1.8.)

Remark 1.9.1.(1) Sincew_niðX;LÞAZ,g_iðX;LÞis an integer by definition.

(2) Ifi¼0 (resp.i¼1), theng_iðX;LÞis equal to the degree (resp. the sectional genus) ofðX;LÞ.

(3) Ifi¼n, theng_nðX;LÞ ¼hⁿðOXÞ, andg_nðX;LÞis independent ofL.

Theorem 1.10.LetðX;LÞbe a quasi-polarized manifold ofdimX ¼n.Let i be an inte- ger such that0cicn1.Then

giðX;LÞ ¼ ⁿⁱ¹X

j¼0

ð1Þ^j ni j

h⁰ðKXþ ðnijÞLÞ þXⁿⁱ

k¼0

ð1Þ^nikh^nkðOXÞ:

Proof.See [6, Theorem 2.3]. r Theorem 1.11. Let X be a variety of dimX ¼n and let L₁;L₂;A₁;. . .;An2 be nef Q-bundles on X.Then

ðL1L₂A₁. . .An2Þ²dðL1L₁A₁. . .An2ÞðL2L₂A₁. . .An2Þ:

Proof.See [3, (0.4.6)], or [1, Proposition 2.5.1]. r Definition 1.12. (1) LetX (resp.Y) be an n-dimensional projective manifold, andL (resp.A) an ample line bundle onX(resp.Y). ThenðX;LÞis called asimple blowing up ofðY;AÞif there exists a birational morphismp:X !Y such thatpis a blowing up at a point ofY andL¼pðAÞ E, whereEis thep-exceptional reduced divisor.

(2) Let X (resp.Y) be ann-dimensional projective manifold, andL (resp.A) an ample line bundle on X (resp. Y). Here we put ðX0;L₀Þ:¼ ðX;LÞ. Then we say that ðY;AÞis the first reduction ofðX;LÞif there exist polarized manifoldsðXj;L_jÞ for 1cjctþ1 and birational morphisms m_j:X_j!Xjþ1 for 0cjct such that ðXtþ1;Ltþ1Þ ¼ ðY;AÞ,ðXj;L_jÞ is a simple blowing up ofðXjþ1;Ljþ1Þ for any j with 0cjct, andðY;AÞis not obtained by a simple blowing up of any polarized mani- fold. The birational morphismm:¼m_t m₀:X !Y is called thefirst reduction map.

Remark 1.12.1. If ðX;LÞ is not obtained by a simple blowing up of any polarized manifold, thenðX;LÞis the first reduction of itself.

Theorem 1.13. Let ðX;LÞ be a polarized manifold of n¼dimXd3.ThenðX;LÞis one of the following types:

(1) ðPⁿ;Oð1ÞÞ, (2) ðQⁿ;Oð1ÞÞ,

(3) a scroll over a smooth curve,

(4) K_X@ðn1ÞL,that is,ðX;LÞis a Del Pezzo manifold, (5) a hyperquadric fibration over a smooth curve,

(6) a scroll over a smooth surface,

(7) letðX⁰;L⁰Þbe the first reduction ofðX;LÞ, (7-1) n¼4,ðX⁰;L⁰Þ ¼ ðP⁴;Oð2ÞÞ, (7-2) n¼3,ðX⁰;L⁰Þ ¼ ðQ³;Oð2ÞÞ, (7-3) n¼3,ðX⁰;L⁰Þ ¼ ðP³;Oð3ÞÞ,

(7-4) n¼3,X⁰is aP²-bundle over a smooth curve C withðF⁰;L⁰j_F0Þ ¼ ðP²;Oð2ÞÞ for any fiber F⁰of it,

(8) K_X⁰þ ðn2ÞL⁰is nef.

Proof.See [1, Proposition 7.2.2, Theorem 7.2.4, Theorem 7.3.2, and Theorem 7.3.4].

See also [4]. r

Remark 1.13.1.ðX;LÞis the type (1) (resp. the type (1), (2), or (3)) in Theorem 1.13 if and only ifK_XþnL(resp.K_Xþ ðn1ÞL) is not nef.

Proposition 1.14. Let ðX;LÞ be a polarized manifold of dimX ¼n, and let i be an integer with 1cicn.Let ðM;AÞbe the first reduction of ðX;LÞ. Then g_iðX;LÞ ¼ g_iðM;AÞ.

Proof.See [6, Proposition 2.6]. r

Definition 1.15.A varietyX of dimensionnis said to beuniruledif there exist a vari- etyY of dimensionn1 and a dominant rational mapP¹YdX. (Here we note thatPⁿis uniruled.)

Definition 1.16(See [1, (13.1)]). LetX be a normal and 1-Gorenstein projective vari- ety of dimX ¼n and letLbe a line bundle onX. For an integer jwith 0cjcn, the j-th pluridegree djðLÞof the pairðX;LÞis defined as

djðLÞ:¼ ðKXþ ðn2ÞLÞ^jL^nj;

whereKX is the canonical sheaf ofX.

Remark 1.16.1. Let ðX;LÞ be as in Definition 1.16. Then by easy calculations, we obtain the following:

(1) KXLⁿ¹ ¼d1ðLÞ ðn2Þd0ðLÞ,

(2) K_X²Lⁿ² ¼d2ðLÞ 2ðn2Þd1ðLÞ þ ðn2Þ²d0ðLÞ.

Lemma 1.17.Let X be a smooth projective variety of dimX ¼nd3 and let L be an ample line bundle on X. Assume that kðXÞd0.Let ðM;AÞbe the first reduction of ðX;LÞ.Let djðAÞbe the j-th pluridegree ofðM;AÞ.Then for j¼1;. . .;n,

djðAÞdðn2Þdj1ðAÞ:

Furthermore ifkðXÞd1,then the inequalities are strict.

Proof.See [1, Lemma 13.1.3]. r

2 An explicit formula for the second sectional geometric genus

In this section we will give an explicit formula for the second sectional geometric genus of quasi-polarized manifolds.

Proposition 2.1.LetðX;LÞbe a quasi-polarized manifold ofdimX ¼3.Then g2ðX;LÞ ¼ 1þh¹ðOXÞ þ 1

12ððKXþ2LÞðKXþLÞ þc2ÞL;

where c₂ is the second Chern class of X.

Proof.By the Hirzebruch–Riemann–Roch theorem (see Theorem 1.6), we get that wðLÞ ¼ 1

6L³þ1

4c₁L² 1

12ðc₁²þc₂ÞLþ 1 24c₁c₂;

where ci¼ciðTXÞ for the tangent bundle TX of X. By the Kawamata–Viehweg vanishing theorem and the Serre duality, we have

h⁰ðKXþLÞ ¼wðLÞ:

Hence

h⁰ðKXþLÞ ¼1 6L³1

4c1L²þ 1

12ðc₁²þc2ÞL 1 24c1c2: By the Hirzebruch–Riemann–Roch theorem, we obtain that

wðOXÞ ¼ 1 24c1c2: Therefore sincec1 ¼ KX, we get that

h⁰ðKXþLÞ ¼1 6L³þ1

4K_XL²þ 1

12ðK_X² þc₂ÞLwðOXÞ

¼1 6L³þ1

4KXL²þ 1

12ðK_X² þc2ÞL ð1h¹ðOXÞ þh²ðOXÞ h³ðOXÞÞ

¼ 1

12ð2L³þ3KXL²þK_X²LÞ þ 1

12c2L ð1h¹ðOXÞ þh²ðOXÞ h³ðOXÞÞ

¼ 1

12ððKXþ2LÞðKXþLÞ þc₂ÞL1þh¹ðOXÞ h²ðOXÞ þh³ðOXÞ:

So by Theorem 1.10 we get the assertion. r

Next we consider the case in which dimXd4.

Proposition 2.2. LetðX;LÞbe a quasi-polarized manifold of dimX¼nd4.Assume that L is spanned.Then

g₂ðX;LÞ ¼ 1þh¹ðOXÞ þ 1

12ððKXþ ðn1ÞLÞðKXþ ðn2ÞLÞ þc₂ÞLⁿ² þn3

24 ð2KXþ ðn2ÞLÞLⁿ¹:

Proof. Here we use Notation 1.7. ThenðXn3;Ln3Þ is a quasi-polarized manifold with dimXn3¼3 and BsjLn3j ¼q. Then we can prove that by the adjunction formula

ðKXþ ðn1ÞLÞðKXþ ðn2ÞLÞLⁿ² ¼ ðKXn3þ2Ln3ÞðKXn3þLn3ÞLn3: By the exact sequence

0!TXiþ1 !rðTXiÞ !OðLiÞj_Xiþ1!0;

we get that

cðrðTXiÞÞ ¼cðTXiþ1ÞcðOðLiÞj_Xiþ1Þ;

wherer:Xiþ1!Xiis the embedding,TXj is the tangent bundle ofXjfor j¼i,iþ1, andcðEÞdenotes the total Chern class of a vector bundleE. So we obtain that

c2ðXiÞj_Xiþ1¼c1ðXiþ1ÞOðLiÞj_Xiþ1þc2ðXiþ1Þ

¼ KXiþ1Liþ1þc₂ðXiþ1Þ:

Here we note that n3

2 ð2KXþ ðn2ÞLÞLⁿ¹¼ ðKXþLÞLⁿ¹þ þ ðKXþ ðn3ÞLÞLⁿ¹: Therefore

c2ðXÞLⁿ²þn3

2 ð2KXþ ðn2ÞLÞLⁿ¹

¼c2ðXÞj_X1L₁ⁿ³þKX1L₁ⁿ²þ ðKX1þL1ÞL₁ⁿ²þ þ ðKX1þ ðn4ÞL1ÞL₁ⁿ²

¼c₂ðX1ÞL₁ⁿ³þ ðKX1þL₁ÞL₁ⁿ²þ þ ðKX1þ ðn4ÞL1ÞL₁ⁿ²

¼

¼c2ðXn4ÞL²_n4þ ðKXn4þLn4ÞL³_n4

¼c2ðXn4Þj_Xn3Ln3þKXn3L²_n3

¼c2ðXn3ÞLn3:

We also note that h¹ðOXÞ ¼h¹ðOXn3Þ. By [6, Theorem 2.4] we get that g₂ðXn3; Ln3Þ ¼g₂ðX;LÞ. Therefore

1þh¹ðOXÞ þ 1

12ððKXþ ðn1ÞLÞðKXþ ðn2ÞLÞ þc2ÞLⁿ² þn3

24 ð2KXþ ðn2ÞLÞLⁿ¹

¼ 1þh¹ðOXn3Þ þ 1

12ððKXn3þ2Ln3ÞðKXn3þLn3Þ þc₂ðXn3ÞÞLn3

¼g2ðXn3;Ln3Þ

¼g2ðX;LÞ:

This completes the proof of Proposition 2.2. r

Corollary 2.3.LetðX;LÞbe a quasi-polarized manifold of n¼dimXd3.Then g₂ðX;LÞ ¼ 1þh¹ðOXÞ þ 1

12ððKXþ ðn1ÞLÞðKXþ ðn2ÞLÞ þc₂ÞLⁿ² þn3

24 ð2KXþ ðn2ÞLÞLⁿ¹: Proof.LetAbe an ample line bundle onX. We put

fðtÞ ¼ 1þh¹ðOXÞ þ 1

12ððKXþ ðn1ÞðLþtAÞÞðKXþ ðn2ÞðLþtAÞÞ þc2Þ ðLþtAÞⁿ²þn3

24 ð2KXþ ðn2ÞðLþtAÞÞðLþtAÞⁿ¹:

Here we note thatg2ðX;LþtAÞis a polynomial in one indeterminatetby Theorem 1.6 and Definition 1.9, and fðtÞ is also a polynomial in one indeterminate t. If BsjLþtAj ¼q, then g2ðX;LþtAÞ ¼ fðtÞby Proposition 2.2. But since there are infinitely many t with BsjLþtAj ¼q, we have g₂ðX;LþtAÞ ¼ fðtÞ for any t. In particularg₂ðX;LÞ ¼ fð0Þand we get the assertion. r

3 Properties of the second sectional geometric genus of polarized manifolds In this section, we assume that X is smooth and L is ample. We study the second sectional geometric genus of a polarized manifoldðX;LÞ. First we prove the follow- ing lemma.

Lemma 3.1. Let X be a smooth projective variety of dimX¼nd3, and let L,

H1;. . .;Hn2be ample Cartier divisors on X.We putU¼ ðH1;. . .;Hn2Þ.LetEbe a

vector bundle on X such that E is generically U-semipositive. Then EnL is also genericallyU-semipositive.

Proof. LetDbe a numerically e¤ectiveQ-divisor onX and let W¼ ðH1;. . .;H_n2; DÞ. Let

S_W:0¼ ðEÞ₀YðEÞ₁Y YðEÞ_s¼E

be the Harder–Narashimhan filtration of E with respect to W. Then by Theorem 1.2 for any integeriwith 1cicsthe following are satisfied:

ð€Þ Gr_iðSWÞ:¼ ðEÞ_i=ðEÞ_i1is a torsion freeW-semistable sheaf,

ð€€Þ d_WðGriðSWÞÞ:¼d_WððEÞ_i=ðEÞ_i1Þis a strictly decreasing function oni.

SinceEis genericallyU-semipositive, we get that c1ððEÞ₁ÞH1. . .Hn2D

rankðEÞ₁ c0:

Claim 3.1.1.The Harder–Narashimhan filtration ofðEnLÞwith respect toWis the following:

S_WnL:0¼ ðEÞ₀nLYðEÞ₁nLY YðEÞ_snL

¼EnL¼ ðEnLÞ:

Proof. (A) First we prove that Gr_iðSWnLÞ is a torsion free W-semistable sheaf.

We find that Gr_iðSWnLÞ ¼ ððEÞ_inLÞ=ððEÞ_i1nLÞ ¼ ððEÞ_i=ðEÞ_i1ÞnL is torsion free. For any subsheafF ofððEÞ_inLÞ=ððEÞ_i1nLÞ we obtain that FnLis a subsheaf ofðEÞ_i=ðEÞ_i1and by usingð€Þwe get that

c1ðFnLÞH1. . .Hn2D

rankðFnLÞ cc1ððEÞ_i=ðEÞ_i1ÞH1. . .Hn2D rankððEÞ_i=ðEÞ_i1Þ : Since

c1ðFnLÞH1. . .Hn2D

rankðFnLÞ ¼c1ðFÞH1. . .Hn2D

rankF þLH1. . .Hn2D;

we get that

c₁ðFÞH1. . .H_n2D

rankF ¼c₁ðFnLÞH1. . .H_n2D

rankðFnLÞ LH1. . .Hn2D cc₁ððEÞ_i=ðEÞ_i1ÞH1. . .Hn2D

rankððEÞ_i=ðEÞ_i1Þ LH₁. . .H_n2D

¼c1ðððEÞ_i=ðEÞ_i1ÞnLÞH1. . .Hn2D rankððEÞ_i=ðEÞ_i1Þ

¼c₁ðððEÞ_inLÞ=ððEÞ_i1nLÞÞH1. . .Hn2D rankðððEÞ_inLÞ=ððEÞ_i1nLÞÞ : ThereforeððEÞ_inLÞ=ððEÞ_i1nLÞisW-semistable.

(B) Next we prove thatd_WðGriðSWnLÞÞis a strictly decreasing function oni. By usingð€€Þ, we get that

dWðGriðSWnLÞÞ dWðGriþ1ðSWnLÞÞ

¼c₁ðððEÞ_i=ðEÞ_i1ÞnLÞH1. . .Hn2D rankðððEÞ_i=ðEÞ_i1ÞnLÞ c1ðððEÞ_iþ1=ðEÞ_iÞnLÞH1. . .Hn2D

rankðððEÞ_iþ1=ðEÞ_iÞnLÞ

¼c₁ððEÞ_i=ðEÞ_i1ÞH1. . .Hn2D

rankðððEÞ_i=ðEÞ_i1ÞnLÞ c1ððEÞ_iþ1=ðEÞ_iÞH1. . .Hn2D rankðððEÞ_iþ1=ðEÞ_iÞnLÞ

¼c₁ððEÞ_i=ðEÞ_i1ÞH1. . .Hn2D

rankððEÞ_i=ðEÞ_i1Þ c₁ððEÞ_iþ1=ðEÞ_iÞH1. . .H_n2D rankððEÞ_iþ1=ðEÞ_iÞ >0:

ThereforeS_WnLis the Harder–Narashimhan filtration ofEnLwith respect toW. This completes the proof of Claim 3.1.1. r By Claim 3.1.1, the maximal W-destabilizing subsheaf of EnL is ðEÞ₁nL. SinceEis genericallyU-semipositive andLis ample, we have

d_ðU;DÞððEnLÞ₁Þ ¼d_{ðH1;...;Hn2;DÞ}ððEÞ₁nLÞ

¼c₁ððEÞ₁nLÞH1. . .Hn2D rankððEÞ₁nLÞ

¼c1ððEÞ₁ÞH1. . .Hn2D

rankððEÞ₁nLÞ LH1. . .Hn2D

¼c1ððEÞ₁ÞH1. . .Hn2D

rankðEÞ₁ LH₁. . .Hn2Dc0:

HenceEnLis genericallyU-semipositive. r

By Theorem 1.5 and Lemma 3.1 we get the following.

Corollary 3.2. Let X be a smooth projective variety of dimX¼nd3, and let L;

H1;. . .;Hn2 be ample Cartier divisors on X. Then W¹_XnL is generically ðH1;. . .;

Hn2Þ-semipositive unless X is uniruled.

Corollary 3.3.Let X be a smooth projective variety ofdimX¼nd3,and let L be an ample divisor on X.If X is not uniruled,then c₂ðW_X¹ nLÞLⁿ²d0.

Proof. By Corollary 3.2, we get that W¹_XnL is generically ðL;. . .;LÞ-semipositive.

On the other handc1ðW_X¹ nLÞ ¼KXþnL is nef unlessX is uniruled. (See Remark 1.13.1.) Hence by Theorem 1.4, we havec2ðW_X¹ nLÞLⁿ²d0. r

Proposition 3.4.Let X be a smooth projective variety ofdimX ¼nd3.Let L be an ample Cartier divisor on X.If X is not uniruled,then

c₂ðXÞLⁿ²d n

2 Lⁿ ðn1ÞKXLⁿ¹: Proof.By [8, Example 3.2.2], we get that

c₂ðW_X¹ nLÞ ¼X²

i¼0

ni 2i

c_iðW_X¹ÞL²ⁱ

¼ n

2 L²þ n1 1

c₁ðW_X¹ÞLþc₂ðW_X¹Þ

¼ n

2 L²þ n1 1

K_XLþc₂ðW_X¹Þ:

Therefore

c₂ðW_X¹ nLÞLⁿ²¼ n

2 Lⁿþ n1 1

K_XLⁿ¹þc₂ðW_X¹ÞLⁿ²: Becausec₂ðW_X¹ÞLⁿ²¼c₂ðXÞLⁿ², by Corollary 3.3 we have

0cc2ðW_X¹ nLÞLⁿ²¼ n

2 Lⁿþ n1 1

KXLⁿ¹þc2ðXÞLⁿ²: Namely

c₂ðXÞLⁿ²d n

2 Lⁿ ðn1ÞKXLⁿ¹:

This completes the proof of Proposition 3.4. r

Theorem 3.5.LetðX;LÞbe a polarized manifold ofdimX ¼n.Assume thatkðXÞd0.

LetðM;AÞbe the first reduction ofðX;LÞ,and letgbe the number of points blown up under the first reduction map.

(1) If nd4,then

g2ðX;LÞd1þh¹ðOXÞ þLⁿþg

12 ðn²5nþ5Þ:

(2) If nd3and K_X is nef,then

g2ðX;LÞd1þh¹ðOXÞ þ 1

24ð3n²11nþ10ÞLⁿ:

Proof.(1) First we note that by Proposition 1.14,g₂ðX;LÞ ¼g₂ðM;AÞ. So we calcu- lateg₂ðM;AÞ. Here we note thatM is not uniruled becausekðMÞd0. So by Prop- osition 3.4 we have

c2ðMÞAⁿ²d n

2 Aⁿ ðn1ÞKMAⁿ¹: On the other hand,

ðKMþ ðn1ÞAÞðKM þ ðn2ÞAÞAⁿ²þc₂ðMÞAⁿ² dðKMþ ðn1ÞAÞðKMþ ðn2ÞAÞAⁿ² n

2 Aⁿ ðn1ÞKMAⁿ¹

¼ ðKM þ ðn1ÞAÞðKMþ ðn2ÞAÞAⁿ²

ðn1ÞðKMþ ðn2ÞAÞAⁿ¹þ ðn1Þðn2Þ ðn1Þn 2

Aⁿ

¼KMðKMþ ðn2ÞAÞAⁿ²þ ðn1Þ n 22

Aⁿ: ð3:5:aÞ

Letd_jðAÞbe the j-th pluridegree ofðM;AÞ, that is,

djðAÞ:¼ ðKMþ ðn2ÞAÞ^jA^nj:

By (1) and (2) in Remark 1.16.1, we get that

K_MðKMþ ðn2ÞAÞAⁿ²þ ðn1Þ n 22

Aⁿ

¼d2ðAÞ ðn2Þd1ðAÞ þðn1Þðn4Þ

2 d0ðAÞ; ð3:5:bÞ ð2KMþ ðn2ÞAÞAⁿ¹¼2d₁ðAÞ ðn2Þd0ðAÞ: ð3:5:cÞ Therefore by Corollary 2.3 and by (3.5.a), (3.5.b), and (3.5.c), we obtain that

g2ðM;AÞ ¼ 1þh¹ðOMÞ þ 1

12ðKMþ ðn1ÞAÞðKMþ ðn2ÞAÞAⁿ² þ 1

12c2ðMÞAⁿ²þn3

24 ð2KMþ ðn2ÞAÞAⁿ¹ d1þh¹ðOMÞ þ 1

12K_MðKMþ ðn2ÞAÞAⁿ² þ 1

12ðn1Þ n 22

Aⁿþn3

24 ð2KMþ ðn2ÞAÞAⁿ¹

¼ 1þh¹ðOMÞ þ 1

12ðd2ðAÞ ðn2Þd1ðAÞÞ þðn1Þðn4Þ

24 d0ðAÞ þn3

24 ð2d1ðAÞ ðn2Þd0ðAÞÞ

¼ 1þh¹ðOMÞ þ 1

12ðd2ðAÞ d1ðAÞ d0ðAÞÞ:

SincekðXÞd0, by Lemma 1.17 we get that for j¼1;. . .;n d_jðAÞdðn2Þdj1ðAÞ:

Therefore

g2ðM;AÞd1þh¹ðOMÞ þ 1

12ðd2ðAÞ d1ðAÞ d0ðAÞÞ d1þh¹ðOMÞ þ 1

12ððn2Þd1ðAÞ d1ðAÞ d0ðAÞÞ d1þh¹ðOMÞ þ 1

12ððn3Þðn2Þ 1Þd0ðAÞ

¼ 1þh¹ðOMÞ þ 1

12ðn²5nþ5Þd0ðAÞ:

Sinced₀ðAÞ ¼Lⁿþgandh¹ðOMÞ ¼h¹ðOXÞ, we get the assertion (1).

(2) Assume that nd3 and K_X is nef. In this case ðX;LÞGðM;AÞbecauseK_X is nef. We also note thatc2ðXÞLⁿ²d0 by Miyaoka’s theorem ([10, Theorem 6.6]).

Hence by Corollary 2.3

g2ðX;LÞd1þh¹ðOXÞ þ 1

12ðKXþ ðn1ÞLÞðKXþ ðn2ÞLÞLⁿ² þn3

24 ð2KXþ ðn2ÞLÞLⁿ¹: By using (1) and (2) in Remark 1.16.1 we obtain that

ðKXþ ðn1ÞLÞðKXþ ðn2ÞLÞLⁿ²¼d₂ðLÞ þd₁ðLÞ;

and

ð2KXþ ðn2ÞLÞLⁿ¹¼2d₁ðLÞ ðn2ÞLⁿ: Hence

g₂ðX;LÞd1þh¹ðOXÞ þd₂ðLÞ þd₁ðLÞ

12 þn3

24 ð2d1ðLÞ ðn2ÞLⁿÞ

¼ 1þh¹ðOXÞ þd2ðLÞ

12 þn2

12 d1ðLÞ n²5nþ6 24 Lⁿ:

By using Lemma 1.17, we obtain that

g2ðX;LÞd1þh¹ðOXÞ þ ðn2Þ²

6 n²5nþ6 24

! Lⁿ

¼ 1þh¹ðOXÞ þ3n²11nþ10 24 Lⁿ:

We get the assertion (2). r

Remark 3.5.1. In both cases of Theorem 3.5, if kðXÞd1, then the inequalities are strict by Lemma 1.17.

Corollary 3.5.2.LetðX;LÞbe a polarized manifold of dimX¼n.

(1) Ifnd4 andkðXÞd0, theng2ðX;LÞdh¹ðOXÞ.

(2) Ifn¼3,kðXÞd0, andK_X is nef, theng₂ðX;LÞdh¹ðOXÞ.

Proof.(1) By Theorem 3.5 (1), we obtain that g2ðX;LÞd1þh¹ðOXÞ þLⁿþg

12 ðn²5nþ5Þ:

Sincend4,gd0, andLⁿd1, we get that Lⁿþg

12 ðn²5nþ5Þ>0:

Henceg2ðX;LÞ>h¹ðOXÞ 1. Becauseg2ðX;LÞis an integer, we obtain the assertion (1).

(2) Assume that n¼3, kðXÞd0, and K_X is nef. Then by Theorem 3.5 (2), we obtain that

g2ðX;LÞd1þh¹ðOXÞ þ1 6L³:

SinceL³d1, we get thatg₂ðX;LÞ>h¹ðOXÞ 1. Becauseg₂ðX;LÞis an integer, we

obtain the assertion (2). r

Remark 3.5.3. (1) Let ðX;LÞ be a polarized manifold of dimX¼n such that kðXÞd0.

(1.1) Ifnd7, then by Theorem 3.5 (1) we get thatg2ðX;LÞdh¹ðOXÞ þ1.

(1.2) If KX is nef and nd5, then by Theorem 3.5 (2) we get that g2ðX;LÞd h¹ðOXÞ þ1.

(2) The inequality in Corollary 3.5.2 (2) is best possible. Namely, there exists an example ofðX;LÞsuch that dimX ¼3,kðXÞd0,K_X is nef, andg₂ðX;LÞ ¼h¹ðOXÞ.

LetX ¼C^ð3Þbe a symmetric product of a smooth projective curveCof genus three.

Letp:CCC!C^ð3Þbe the natural map and letp:CCC!Cbe the first projection. We putL¼pðpðxÞÞforxAC. ThenkðXÞd0 andK_X is nef. Further- moreg₂ðX;LÞ ¼3¼h¹ðOXÞ.

By Theorem 3.5 (1), we can give numerical conditions for polarized manifolds ðX;LÞwithg2ðX;LÞ ¼0,kðXÞd0, and dimXd4.

Corollary 3.5.4. Let ðX;LÞ be a polarized manifold of dimX¼nd4. Assume that kðXÞd0. Let g be the number of points blown up under the first reduction map. If g2ðX;LÞ ¼0,then h¹ðOXÞ ¼0and

(1) Lⁿþgc12for n¼4;

(2) Lⁿþgc2for nd5;

(2.a) If nd5and Lⁿþg¼2,then n¼5;

(2.b) If nd5and Lⁿþg¼1,then n¼5;6.

Proof.Assume thatg2ðX;LÞ ¼0. By Corollary 3.5.2 (1), we get thath¹ðOXÞ ¼0.

(1) Ifn¼4, then

Lⁿþg

12 ðn²5nþ5Þ ¼L⁴þg 12 : Because by Theorem 3.5 (1)

0¼g₂ðX;LÞd1þL⁴þg 12 ; we obtain thatL⁴þgc12 and we get the assertion (1).

(2) Ifnd5, then

Lⁿþg

12 ðn²5nþ5Þd 5

12ðLⁿþgÞ:

Hence by Theorem 3.5 (1)

0¼g2ðX;LÞd1þ 5

12ðLⁿþgÞ;

and we obtain thatLⁿþgc2 becauseLⁿþgis an integer.

(2.a) IfLⁿþg¼2, then

0¼g2ðX;LÞd1þ1

6ðn²5nþ5Þ: Namelyn²5n1c0. Sincend5, we get thatn¼5.

(2.b) IfLⁿþg¼1, then

0¼g2ðX;LÞd1þ 1

12ðn²5nþ5Þ:

Namelyn²5n7c0. Sincend5, we get that 5cnc6.

This completes the proof of Corollary 3.5.4. r

Here we consider the case wherekðXÞ ¼ y.

Proposition 3.6. Let ðX;LÞ be a polarized manifold of dimX ¼n such that kðXÞ ¼ yand X is not uniruled.Then

(1) If KXþ ððn2Þ=2ÞL is nef and nd6,then g2ðX;LÞdh¹ðOXÞ.

(2) If K_XþL is nef and n¼5,then g₂ðX;LÞdh¹ðOXÞ.

Proof.(1) By the same argument as in the proof of Theorem 3.5 (1) (see (3.5.a)), we get that

ðKXþ ðn1ÞLÞðKXþ ðn2ÞLÞLⁿ²þc₂ðXÞLⁿ² dKXðKXþ ðn2ÞLÞLⁿ²þ ðn1Þ n

22

Lⁿ becauseX is not uniruled. Furthermore

K_XðKXþ ðn2ÞLÞLⁿ²þ ðn1Þ n 22

Lⁿ

¼ K_Xþn2

2 L

2

Lⁿ²ðn2Þ²

4 Lⁿþ ðn1Þ n 22

Lⁿ

¼ KXþn2

2 L

2

Lⁿ²þn²6nþ4 4 Lⁿ: SinceKXþ ððn2Þ=2ÞLis nef andnd6, we get that

K_Xþn2

2 L

2

Lⁿ²þn²6nþ4 4 Lⁿd1 and

ð2KXþ ðn2ÞLÞLⁿ¹d0:

Hence by Corollary 2.3g₂ðX;LÞdh¹ðOXÞbecauseg₂ðX;LÞAZ.

(2) Assume that n¼5 andK_XþLis nef. Then by the same argument as in the proof of Theorem 3.5 (1) (see (3.5.a)), we get that

ðKXþ4LÞðKXþ3LÞL³þc2ðXÞL³dKXðKXþ3LÞL³þ2L⁵: Hence

g₂ðX;LÞ þ1h¹ðOXÞd 1

12K_XðKXþ3LÞL³þ1 6L⁵þ 1

12ð2KXþ3LÞL⁴

¼ 1

12KXðKXþ3LÞL³þ1 6L⁵þ 1

12KXL⁴þ 1

12ðKXþ3LÞL⁴

¼ 1

12ðKXþLÞðKXþ3LÞL³þ 1

12ðKXþ2LÞL⁴: SinceK_XþLis nef, we obtain that

ðKXþLÞðKXþ3LÞL³d0 and ðKXþ2LÞL⁴>0:

Thereforeg2ðX;LÞ>h¹ðOXÞ 1. Becauseg2ðX;LÞis an integer, we get thatg2ðX;LÞ dh¹ðOXÞ. This completes the proof of Proposition 3.6. r

Next we consider a lower bound ofg₂ðX;2LÞfor the case where dimX ¼3.

Theorem 3.7.LetðX;LÞbe a polarized manifold ofdimX¼3.

(1) Assume thatkðXÞd0.LetðM;AÞbe the first reduction ofðX;LÞ,and letgbe the number of points blown up under the first reduction map.Then

g2ðX;2LÞd1þh¹ðOXÞ þ5

6ðL³þgÞ:

(2) Assume thatkðXÞ ¼ y.Then

g2ðX;2LÞdh²ðOXÞd0:

Proof.(I) The case wherekðXÞd0.

LetðM;AÞbe the first reduction ofðX;LÞ. By Theorem 1.10, we get that g2ðX;2LÞ ¼h⁰ðKXþ2LÞ h³ðOXÞ þh²ðOXÞ:

On the other hand, since h⁰ðKXþ2LÞ ¼h⁰ðKMþ2AÞ, h³ðOXÞ ¼h³ðOMÞ, and h²ðOXÞ ¼h²ðOMÞ, we obtain thatg2ðX;2LÞ ¼g2ðM;2AÞ.

Next we calculateg₂ðM;2AÞ. SinceMis not uniruled, we havec₂ðMÞAd3A³ 2K_MA²by Proposition 3.4. Hence

ððKMþ4AÞðKMþ2AÞ þc2ðMÞÞA

¼ ðKMþ4AÞðKMþ2AÞAþc2ðMÞA

dðKMþ4AÞðKMþ2AÞA3A³2KMA²: ð3:7:aÞ

SinceK_MþAis nef andkðMÞd0, we get that

ðKMþ4AÞðKMþ2AÞA3A³2KMA² d4ðKM þ2AÞA²2ðKMþAÞA²A³

¼4ðKM þAÞA²þ4A³2ðKMþAÞA²A³

¼2ðKM þAÞA²þ3A³: ð3:7:bÞ By Lemma 1.17 we get thatðKMþAÞA²dA³. Hence

ðKMþ4AÞðKMþ2AÞA3A³2KMA²d5A³: Therefore by Proposition 2.1

g₂ðX;2LÞ ¼g₂ðM;2AÞ

¼ 1þh¹ðOMÞ þ 1

12ððKMþ4AÞðKMþ2AÞ þc₂ðMÞÞð2AÞ d1þh¹ðOMÞ þ5

6A³ ¼ 1þh¹ðOXÞ þ5

6ðL³þgÞ:

Hence we get the assertion (1).

(II) The case wherekðXÞ ¼ y.

By Theorem 1.10 and the Serre duality, we have

g₂ðX;2LÞ ¼h⁰ðKXþ2LÞ k⁰ðKXÞ þh²ðOXÞ

¼h⁰ðKXþ2LÞ þh²ðOXÞdh²ðOXÞd0:

This completes the proof of Theorem 3.7. r

Corollary 3.7.1. Let ðX;LÞ be a polarized manifold of dimX¼3. Assume that kðXÞd0. Theng₂ðX;2LÞdh¹ðOXÞd0.

Proof.By Theorem 3.7 (1), we get that

g₂ðX;2LÞd1þh¹ðOXÞ þ5

6ðL³þgÞ;

wheregis the number of points blown up under the first reduction map.

SinceL³þg>0, we get thatg₂ðX;2LÞ>h¹ðOXÞ 1. Hence we get the assertion

becauseg₂ðX;2LÞis an integer. r

Here we note that if Lis nef and big, dimX ¼3 and h⁰ðLÞd1, then we get the following:

Proposition 3.8. Let ðX;LÞ be a quasi-polarized manifold. If dimX ¼3 and h⁰ðLÞd1,then g₂ðX;LÞdh²ðOXÞd0.

Proof.By Theorem 1.10 and the Serre duality we get that g₂ðX;LÞ ¼h⁰ðKXþLÞ h⁰ðKXÞ þh²ðOXÞ:

Ifh⁰ðKXÞ ¼0, then

g2ðX;LÞ ¼h⁰ðKXþLÞ þh²ðOXÞdh²ðOXÞ:

Ifh⁰ðKXÞd1, thenh⁰ðKXþLÞ h⁰ðKXÞdh⁰ðLÞ 1d0 and so we get that g₂ðX;LÞ ¼h⁰ðKXþLÞ h⁰ðKXÞ þh²ðOXÞdh²ðOXÞ:

This completes the proof of Proposition 3.8. r

Remark 3.8.1.By the same method as in the proof of Proposition 3.8, we can prove thatgn1ðX;LÞdhⁿ¹ðOXÞifXis a smooth projective variety of dimX¼n, andLis a nef and big line bundle onX withh⁰ðLÞd1.

Here we assume that dimX¼nd3, KX is nef, and kðXÞd0. In this case, by Theorem 3.5 (2), we get that

g₂ðX;LÞd1þh¹ðOXÞ þ 1

24ð3n²11nþ10ÞLⁿ: By using this inequality, we studyðX;LÞwithg₂ðX;LÞ ¼0.

Proposition 3.9.LetðX;LÞbe a polarized manifold ofdimX ¼nd3.Assume that K_X is nef andkðXÞd0.If g₂ðX;LÞ ¼0,then n¼3,h¹ðOXÞ ¼0,and we obtain the fol- lowing:

L³ K_XL² K_X²L c₂ðXÞL gðLÞ

3 2 0 0 5

2 2 0 2 4

2 2 1 1 4

2 2 2 0 4

1 2 0 4 3

1 2 1 3 3

1 2 2 2 3

1 2 3 1 3

1 2 4 0 3

6 0 0 0 7

5 0 0 2 6

4 0 0 4 5

3 0 0 6 4

2 0 0 8 3

1 0 0 10 2

Proof. Here we note thatðX;LÞis the first reduction of itself. We also note that by Miyaoka’s theorem ([10, Theorem 6.6])

c₂ðXÞLⁿ²d0: ð3:9:1Þ Furthermore sinceK_X is nef we get that

K_X²Lⁿ²d0 and KXLⁿ¹d0: ð3:9:2Þ Assume that g2ðX;LÞ ¼0. By Corollary 3.5.2, we get that h¹ðOXÞ ¼0. Hence by Theorem 3.5 (2)

g₂ðX;LÞd1þ 1

24ð3n²11nþ10ÞLⁿ: Claim.n¼3.

Proof.Ifnd5, then

1

24ð3n²11nþ10ÞLⁿd5 4Lⁿ andg₂ðX;LÞd1. Therefore this is impossible.

Assume thatn¼4. Then 1

24ð3n²11nþ10ÞLⁿ¼ 7 12L⁴: Since

0¼g2ðX;LÞd1þ 7 12L⁴;

we obtain thatL⁴¼1. In this case by Corollary 2.3 the second sectional geometric genus ofðX;LÞis the following:

g2ðX;LÞ ¼ 1þ 1

12K_X²L²þ1

2KXL³þ 7 12þ 1

12c2ðXÞL²:

By (3.9.1) and (3.9.2), we obtain that KXL³¼0 because g2ðX;LÞ ¼0. But then ðKXþ3LÞL³¼3 and this is impossible becauseðKXþ3LÞL³is even. This completes

the proof of this claim. r

Sincen¼3 andh¹ðOXÞ ¼0, by Corollary 2.3 we get that

12¼ ððKXþ2LÞðKXþLÞ þc2ðXÞÞL¼K_X²Lþ3KXL²þ2L³þc2ðXÞL: ð3:9:3Þ By (3.9.1) and (3.9.2) we haveL³c6. Here we note thatK_XL²þ2L³is even. Hence

K_XL² is even and we obtain thatK_XL²¼0 or 2 by (3.9.3), and by (3.9.1), (3.9.2), and (3.9.3) we get the list in Proposition 3.9. (Here we note that by Theorem 1.11

K_X²L¼0 ifK_XL² ¼0 becauseK_X is nef.) r

Problem 3.9.1.Find an example ofðX;LÞsuch that dimX¼3,K_Xis nef,kðXÞd0, andg₂ðX;LÞ ¼0.

Remark 3.9.2.There exists a Calabi–Yau 3-foldXsuch that there is an ample divisor LonX withg2ðX;LÞ ¼0 andL³¼1 or 2. (See [6, Example 4.3.3].)

In Theorem 3.7 and Corollary 3.7.1, we proved that g2ðX;2LÞd0 ifðX;LÞis a polarized 3-fold. Here we study a polarized 3-foldðX;LÞwithg2ðX;2LÞ ¼0.

Proposition 3.10.Let ðX;LÞbe a polarized manifold ofdimX ¼3.If g2ðX;2LÞ ¼0, thenkðXÞ ¼ y.

Proof. Assume that kðXÞd0. Let ðM;AÞ be the first reduction ofðX;LÞ, and let g be the number of points blown up under the first reduction map. Assume that g₂ðX;2LÞ ¼0. By Theorem 3.7 (1), we obtain that L³¼1,g¼0, andh¹ðOXÞ ¼0.

HenceðX;LÞGðM;AÞ.

By (3.7.a) and (3.7.b) in the proof of Theorem 3.7, we get that g₂ðX;2LÞ ¼ 1þ1

6ðKXþ4LÞðKXþ2LÞLþ1 6c₂ðXÞL d1þ1

3ðKXþLÞL²þ1 2L³

¼ 1þ1

3K_XL²þ5 6L³: HenceKXL²¼0 becauseg2ðX;2LÞ ¼0.

In this case

gðLÞ ¼1þ1

2ðKXþ2LÞðLÞ²¼2:

By [2, Theorem (1.10) and Remark (2.2)],OðKXÞ ¼OX andDðLÞc3, whereDðLÞ ¼ 3þL³h⁰ðLÞ. So we obtain that h⁰ðLÞd1. On the other hand by Theorem 1.10 and the Serre duality,

0¼g2ðX;2LÞ

¼h⁰ðKXþ2LÞ h⁰ðKXÞ þh²ðOXÞ dh⁰ðKXþ2LÞ h⁰ðKXÞ

¼h⁰ð2LÞ 1

since OðKXÞ ¼OX. Hence h⁰ð2LÞ ¼1 since h⁰ð2LÞ>0. On the other hand by the Riemann–Roch theorem and the Kodaira vanishing theorem, we get that

h⁰ð2LÞ ¼h⁰ðKXþ2LÞ ¼L³þ2h⁰ðKXþLÞ ¼L³þ2h⁰ðLÞd3:

So this is impossible. Therefore we get the assertion. r Proposition 3.11. Let ðX;LÞ be a polarized manifold of dimX ¼3. Assume that g2ðX;2LÞ ¼0and h⁰ð2LÞd2.ThenðX;LÞis one of the following type:

(1) ðX;LÞ ¼ ðP³;O_P³ð1ÞÞ, (2) ðX;LÞ ¼ ðQ³;O_Q³ð1ÞÞ,

(3) ðX;LÞis a scroll over a smooth curve.

Proof.First we prove the following claim:

Claim 3.11.1.h⁰ðKXþ2LÞ ¼0.

Proof.By Theorem 1.10 and the Serre duality, we obtain that g₂ðX;2LÞ ¼h⁰ðKXþ2LÞ h⁰ðKXÞ þh²ðOXÞ

dh⁰ðKXþ2LÞ h⁰ðKXÞ:

If h⁰ðKXÞ>0, then 0¼g₂ðX;2LÞdh⁰ðKXþ2LÞ h⁰ðKXÞdh⁰ð2LÞ 1d1 and this is a contradiction. Hence h⁰ðKXÞ ¼0 and 0¼g₂ðX;2LÞdh⁰ðKXþ2LÞd0.

Thereforeh⁰ðKXþ2LÞ ¼0. This completes the proof of Claim 3.11.1. r Hence we obtain thatK_Xþ2Lis not nef by [5, Corollary 2.7]. SoðX;LÞis one of the

above types by Theorem 1.13 and Remark 1.13.1. r

References

[1] M. C. Beltrametti, A. J. Sommese,The adjunction theory of complex projective varieties.

de Gruyter 1995. MR 96f:14004 Zbl 0845.14003

[2] T. Fujita, Classification of polarized manifolds of sectional genus two. In:Algebraic geom- etry and commutative algebra, Vol. I, 73–98, Kinokuniya, Tokyo 1988. MR 90c:14025 Zbl 0695.14019

[3] T. Fujita,Classification theories of polarized varieties. Cambridge Univ. Press 1990.

MR 93e:14009 Zbl 0743.14004

[4] T. Fujita, On Kodaira energy and adjoint reduction of polarized manifolds.Manuscripta Math.76(1992), 59–84. MR 93i:14032 Zbl 0766.14027

[5] Y. Fukuma, On the nonemptiness of the adjoint linear system of polarized manifolds.

Canad. Math. Bull.41(1998), 267–278. MR 99j:14006 Zbl 0955.14040

[6] Y. Fukuma, On the sectional geometric genus of quasi-polarized varieties, I. To appear in Comm. Algebra.

[7] Y. Fukuma, On the sectional geometric genus of quasi-polarized varieties, II. To appear inManuscripta Math.

[8] W. Fulton,Intersection theory. Springer 1998. MR 99d:14003 Zbl 0885.14002

[9] F. Hirzebruch,Topological methods in algebraic geometry. Springer 1966. MR 34 #2573 Zbl 0138.42001

[10] Y. Miyaoka, The Chern classes and Kodaira dimension of a minimal variety. In:Alge- braic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., 449–476, North- Holland 1987. MR 89k:14022 Zbl 0648.14006

[11] A. J. Sommese, On the adjunction theoretic structure of projective varieties. In:Complex analysis and algebraic geometry(Go¨ttingen, 1985), volume 1194 ofLecture Notes in Math., 175–213, Springer 1986. MR 87m:14049 Zbl 0601.14029

Received 21 June, 2002; revised 5 September, 2002

Y. Fukuma, Department of Mathematics, Faculty of Science, Kochi University, Akebono- cho, Kochi 780-8520, Japan

Email: fukuma@math.kochi-u.ac.jp