A NOTE ON NON-NEGATIVITY OF THE ith -GENUS OF QUASI-POLARIZED VARIETIES

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1 Kyushu J. Math , doi: /kyushujm A NOTE ON NON-NEGATIVITY OF THE ith -GENUS OF QUASI-POLARIZED VARIETIES Yoshiaki FUKUMA Received 14 January 2009 Abstract. Let X, L be a quasi-polarized variety of dimension n. In this paper, we will study non-negativity of the ith -genus i X, L. We will prove the following. Assume that X is a normal Gorenstein variety such that the irrational locus of X consists of at most finite points and the dimension of the singular locus of X is less than or equal to the dimension of the base locus of L. Ifi is greater than the dimension of the base locus of L, then i X, L is non-negative. We will also give a lower bound for i X, L when X, L is a polarized abelian variety. 1. Introduction Let X be a projective variety of dimension n defined over the complex number field and let L be a line bundle on X. IfL is nef and big respectively ample, then X, L is called a quasi-polarized respectively polarized variety. Furthermore,ifX is smooth and L is nef and big respectively ample, we say that X, L is a quasi-polarized respectively polarized manifold. For this X, L, there are some invariants, for example, the sectional genus gl and the -genus L see [7]. Fujita studied polarized varieties by using these invariants, and he gave many interesting results see [7] in detail. On the other hand, in order to study polarized varieties more deeply, the author extended these invariants. In [8], we defined the ith sectional geometric genus g i X, L of X, L for every integer i with 0 i n, which is a generalization of the degree L n and the sectional genus gl of X, L. We note that g 0 X, L L n, g 1 X, L gl and g n X, L h n O X. Some properties of the ith sectional geometric genus which have been obtained in [8] alsoshow that the ith sectional geometric genus is a natural generalization of the sectional genus. On the other hand, in [12], we defined the ith -genus i X, L for every integer i with 0 i n. This gives a generalization of the -genus. Namely, if i 1, then 1 X, L is the -genus L of X, L. Furthermore,in[12] we studied some properties of i X, L. For example, if X is smooth and Bs L, then some properties of i X, L are similar to those of the -genus L of X, L see [12, Section 3] or Section 3 in this paper. So i X, L has some good properties under the assumption that X is smooth and Bs L. As the next step, we want to know whether or not the ith -genus for general quasipolarized varieties has good properties. For example, does the inequality i X, L 0 hold 2000 Mathematics Subject Classification: Primary 14C20; Secondary 14J30, 14J35, 14J40, 14K99. Keywords: Nef and big line bundle, ample line bundle, quasi-polarized variety, ith -genus, ith sectional geometric genus, Abelian variety. c 2010 Faculty of Mathematics, Kyushu University

2 18 Y. Fukuma for 2 i n? Unfortunately, the answer is negative. There exists an example of X, L with i X, L < 0see[12, Section 4]. Hence it is interesting and important to consider when the ith -genus is non-negative. For example, in [13], if X, L is a polarized toric variety, then we can show that i X, L 0see[13, Theorem 3.3]. So, in this paper, we study non-negativity of the ith -genus for quasi-polarized varieties. First, if X is a normal Gorenstein variety, L is nef and big, dim IrrX 0, dim SingX dim Bs L n 1andi dim Bs L +1, then we prove that i X, L 0 see Theorem 4.1, where SingX respectively IrrX denotes the singular locus of X respectively the irrational locus of X see Definition 2.2. Furthermore, we consider the case where K X 0 see Proposition 4.1. In particular, we investigate the case where X, L is a polarized Abelian variety, and we get a lower bound for i X, L see Theorem 4.2. In particular, we see that i X, L > 0 in this case. We will review in Section 3 the ith sectional geometric genus and the ith -genus for the reader s convenience. 2. Preliminaries Definition 2.1. Let X, L be a quasi-polarized variety of dimension n.thenl has a k-ladder if there exists a sequence of irreducible and reduced subvarieties X X 1 X k such that X i L i 1 for 1 i k, wherex 0 : X, L 0 : L and L i : L Xi.Herewenotethat dim X j n j.thenletr p,q : H p X q,l q H p X q+1,l q+1 be the natural map. Remark 2.1. Let X be a projective variety of dimension n 3andletL be a nef and big line bundle on X. Assume that X is normal and Cohen Macaulay, dim SingX m and dim Bs L m,wherem is an integer with m n 3. Then there exists a member X 1 L such that X 1 satisfies the following properties: X 1 is irreducible by Bertini s theorem see [16, Theorem 17.16]; X 1 is Cohen Macaulay by [7, Section 2 in Chapter 0]. In particular, X 1 is reduced; dim SingX 1 m and dim Bs L 1 m by Bertini s theorem see [16, Theorem 17.16]. Here L 1 : L X1. Hence we see that X 1 is normal by [3, Section 5.8 in Chapter IV] or [7, 2.8 Fact in Chapter 0]. Therefore, by carrying out this process, we see that L has an n m 2-ladder X X 1 X n m 2 such that X j is normal and Cohen Macaulay for every integer j with 1 j n m 2. Moreover, if X is Gorenstein, then so is each X j. Definition 2.2. Let X be a normal projective variety of dimension n, andletπ : X X be a resolution of singularities of X. Thenweset IrrX : i>0 SuppR i π O X and we call this set the irrational locus of X. Here we note that IrrX does not depend on the desingularization of X. THEOREM 2.1. Let X be a normal projective variety of dimension n and let L be a nef and big line bundle on X. Thenh i ω X L 0 for i>max{0, dim IrrX}, whereω X is the dualizing sheaf of X. Proof. See [22, Theorem].

3 A note on non-negativity of the ith -genus Review on the ith sectional geometric genus and the ith -genus of quasi-polarized varieties Here we are going to review the ith sectional geometric genus and the ith -genus of quasi-polarized varieties X, L for every integer i with 0 i dim X. Up until now, many investigations of X, L via the sectional genus and the -genus have been given. In order to analyze X, L more deeply, we have extended these notions. First, in [8, Definition 2.1] we gave an invariant called the ith sectional geometric genus which is thought to be a generalization of the sectional genus. Here we recall the definition of this invariant. Notation 3.1. Let X, L be a quasi-polarized variety of dimension n, andletχtl be the Euler Poincaré characteristic of tl.thenχtl is a polynomial in t of degree n, andweset n t + j 1 χtl χ j X, L. j Definition 3.1. [8, Definition 2.1] Let X, L be a quasi-polarized variety of dimension n. Then, for any integer i with 0 i n, theith sectional geometric genus g i X, L of X, L is defined by the following: g i X, L 1 i n i χ n i X, L χo X + 1 n i j h n j O X. Remark If i 0 respectively i 1, then g i X, L is equal to the degree respectively the sectional genus of X, L. 2 If i n, theng n X, L h n O X and g n X, L is independent of L. Then the ith sectional geometric genus satisfies the following properties. THEOREM 3.1. [9, Propositions 2.1 and 2.3, and Theorem 2.4] Let X be a projective variety of dimension n 2 and let L be a nef and big line bundle on X. Assume that h t sl 0 for every integers t and s with 0 t n 1 and 1 s, and L has an n i-ladder for an integer i with 1 i n. Then the ith sectional geometric genus has the following properties: 1 g i X j,l j g i X j+1,l j+1 for every integer j with 0 j n i 1 hereweuse the notation in Definition 2.1; 2 g i X, L h i O X. In particular, from Theorem 3.11 and Remark 3.12 we see that if X, L satisfies the assumption in Theorem 3.1, then the ith sectional geometric genus is the geometric genus of the i-dimensional projective variety X n i. This is the reason why we call this invariant the ith sectional geometric genus. From Theorem 3.1 we see that the ith sectional geometric genus is expected to have properties similar to those of the geometric genus of i-dimensional projective varieties. For other results concerning the ith sectional geometric genus see, for example, [8 11]. The following result will be used later. THEOREM 3.2. Let X be a projective variety with dim X n and let L be a nef and big line bundle on X.

4 20 Y. Fukuma 1 For any integer i with 0 i n 1, we have n i 1 n i g i X, L 1 n j χ n i jl+ j n i 1 n i k h n k O X. 2 Assume that X is smooth. Then for any integer i with 0 i n 1, we have n i 1 n i g i X, L 1 j h 0 n i K X + n i jl+ 1 n i k h n k O X. j Proof. 1 By the same argument as in the proof of [8, Theorem 2.2], we obtain n i n i χ n i X, L 1 n i j χ n i jl j n i 1 1 n i j n i j χ n i jl+ χo X. Hence by Definition 3.1, we get the assertion. 2 By using the Serre duality and the Kawamata Viehweg vanishing theorem, we get the assertion from 1. As the next step, we want to generalize the notion of the -genus. Several generalizations can be considered from various points of view. Here we will give a generalization of the genus from the following point of view. For the case of X, L, the following result has been obtained. THEOREM 3.3. See e.g. [7, 3 in Chapter I] Let X be a projective variety of dimension n 2 and let L be a nef and big line bundle on X. We use the notation in Definition 2.1. If L has an n 1-ladder and h 0 L n 1 >0,then n 1 X, L dim Cokerr 0,j. In particular, we have X, L X 1,L 1 X n 1,L n 1 0. Here we want to give the definition of the ith -genus which satisfies a generalization of Theorem 3.3. Now we are going to give the definition of the ith -genus. Definition 3.2. [12, Definition 2.1] Let X, L be a quasi-polarized variety of dimension n. For every integer i with 0 i n, the ith -genus i X, L of X, L is defined by the following formula: 0 ifi 0, i X, L g i 1 X, L i 1 X, L +n i + 1h i 1 O X h i 1 L if 1 i n. Remark If i 1, then 1 X, L is equal to the -genus of X, L.

5 A note on non-negativity of the ith -genus 21 2 If i n, then n X, L h n O X h n L see [12, Proposition 2.4]. 3 For every integer i with 1 i n, by the definition of the ith -genus, we have the following equality which will be used later: i 1 X, L g i 1 X, L i X, L + n i + 1h i 1 O X h i 1 L. Then, for the case of the ith -genus, we can prove the following. THEOREM 3.4. See [12, Theorem 2.8 and Corollary 2.9] and [9, Proposition 2.1] Let X be a projective variety of dimension n 2 and let L be a nef and big line bundle on X. We use the notation in Definition 2.1. Assume that h t sl 0 for every integers t and s with 0 t n 1 and 1 s. If L has an n i-ladder and h 0 L n i >0 for an integer i with 1 i n, then n i i X, L dim Cokerr i 1,j. In particular, we have i X, L i X 1,L 1 i X n i,l n i 0. The definition of the ith -genus is so complicated that many things about the ith genus are unknown. Therefore, it is important to investigate the following problems in order to understand the meaning and properties of the ith -genus. Problem 3.1. i Does the ith -genus have a property similar to that of the -genus? Concretely: i-1 Does i X, L 0 hold? i-2 Can we get the ith -genus version of the Fujita theory on the -genus? ii Are there any relationships between g i X, L and i X, L? iii Are there any relationships between i X, L and i+1 X, L? iv We must classify X, L by the value of the ith -genus. v What is the geometric meaning of the ith -genus? Remark 3.3. If X is smooth and L is nef and big, then the following facts concerning Problem 3.1 are known. 1 First we consider Problem 3.1i-1. If i 1, then 1 X, L 0see[7, 4.2 Theorem] and [6, 1.1 Theorem]. Moreover, if L is base point free, then we have i X, L 0 for every integer i with 0 i n. 2 Next we consider Problem 3.1ii. If i 1andL is merely ample, then it is known that g 1 X, L 0 if and only if 1 X, L 0see[7, 12.1 Theorem]. Therefore, we consider the case where i 2. Then under the assumption that Bs L we see that g i X, L 0 if and only if i X, L 0see[12, Theorem 3.13]. 3 Next we consider Problem 3.1iii. Then, for example, we get the following. Assume that L is base point free. If i X, L i 1, then i+1 X, L 0 see [12, Proposition 3.9]. In particular, if i X, L 0, then i+1 X, L 0. Maybe there will be several relationships between i X, L and i+1 X, L other than this. 4 For Problem 3.1iv, we get a classification of X, L by the value of 2 X, L as follows: 4.1 a classification of polarized manifolds X, L such that Bs L and 2 X, L 0see[12, Theorem 3.13 and Remark ];

6 22 Y. Fukuma 4.2 a classification of polarized manifolds X, L such that L is very ample and 2 X, L 1see[12, Theorem 3.17] and [14, Remark 2]. 5 Problem 3.1v seems to be the most difficult problem among the above problems even in the case where L is base point free or very ample. We also note that we are studying Problem 3.1i-2 now and we will explain this in a future paper. As we said in Remark 3.3, under the assumption that L is base point free, we have some answers for Problem 3.1 above. We believe that the ith -geuns has good properties similar to those of the -genus and is useful if L satisfies some special conditions. Therefore, the main purpose of our investigation for the time being is to consider the following two things. Problem 3.2. I Investigate Problem 3.1 for the case where L has base points. II If L is very ample or base point free, then find an application using results of the ith -genus such as those mentioned in Remark 3.3. For example, investigate an unsolved problem about X, L with very ample L by using results of the ith -genus. In this paper, we consider Problem 3.1i-1 under the assumption that L has base points. As we said above, if i 1, then 1 X, L, which equals the -genus, is non-negative for every nef and big line bundle L on X. This was proved by Fujita. Moreover, Fujita also gave a classification of X, L with small X, L see [7]. These results are very useful and are used in various problems. So, in order to make the ith -genus useful, it is important to study the non-negativity of the ith -genus for 2 i n. As we said in Remark 3.3, if Bs L, then i X, L 0 for every integer i with 1 i n. However, in general for each integer i with 2 i n there exists an example of X, L with i X, L < 0see[12, Section 4]. So it is interesting and important to know when i X, L is non-negative for i 2. This is the theme of this paper. 4. Non-negativity of the ith -genus In this section, we consider non-negativity of i X, L. First, we consider the case where X is normal and Gorenstein, dim IrrX 0 and dim SingX dim Bs L dim X 1. THEOREM 4.1. Let X, L be a quasi-polarized variety of dimension n. Assume that X is normal and Gorenstein, dim IrrX 0 and dim SingX dim Bs L n 1. Then, for every integer i with i dim Bs L +1, i X, L 0 holds. Proof. Set m : dim Bs L. Ifm n 1, then we have only to prove that n X, L 0. By Remark 3.22 and the Serre duality, we have n X, L h n O X h n L h 0 ω X h 0 ω X L 1. If h 0 ω X L 1 0, then we get the assertion. So we may assume that h 0 ω X L 1 0. However, since h 0 L > 0, we have h 0 ω X h 0 ω X L 1 h 0 L 1 0by[8, Lemma 1.12] or [19, Lemma]. Therefore, we get n X, L 0 in this case. Next we assume that m n 2. By Remark 2.1, L has an n m 2-ladder X X 1 X n m 2 such that X j is normal and Gorenstein for every integer j with 0 j n m 2, where X 0 : X. WesetL j : L Xj for 1 j n m 2andL 0 : L.

7 By the exact sequence A note on non-negativity of the ith -genus 23 0 H 0 O Xj H 0 L j H 0 L j+1, we see that h 0 L j+1 h 0 L j h 0 O Xj h 0 L j 1 for every integer j with 0 j n m 3. By assumption, h 0 L 0 n m holds e.g. see [5, 1.7 Lemma]. Hence h 0 L n m 2 h 0 L n m 3 1 h 0 L n m 2 2. Hence, if i m + 2, then Conditions A 1 i and A 2 i in [12, 2.7.2] are satisfied. By [17, Chapter III, Corollary 7.7] and Theorem 2.1, we have h t sl 0 for integers t and s with 0 t n 1and1 s. Hence by [9, Proposition 2.1a], Condition Bi, i in [12, 2.7.2] is also satisfied for m + 2 i n 1. Therefore, we get i X, L 0forevery integer i with i m + 2by[12, Corollary 2.92]. Here we note that if i n, thenwedo not need Conditions A 1 i and Bi, i in [12, 2.7.2] see the proof of [12, Theorem 2.82]. Assume that i m + 1. As we said above, Condition A 1 m + 2 in [12, 2.7.2] holds. By [9, Proposition 2.1a] Condition Bm + 2,m+ 1 in [12, 2.7.2] also holds. Hence by [12, Corollary 2.91], we see that m+1 X, L m+1 X n m 2,L n m 2. Next we calculate m+1 X n m 2,L n m 2. Here we note that by Theorem 3.21 we have g m+1 X n m 2,L n m 2 1 m+2 χ L n m 2 h m+2 O Xn m 2 + h m+1 O Xn m 2. By [9, Claim 2.1.1] we have 1 m+2 χ L n m 2 h m+2 L n m 2. Hence g m+1 X n m 2,L n m 2 h m+2 L n m 2 h m+2 O Xn m 2 + h m+1 O Xn m 2. Since dim Bs L n m 2 dim Bs L m dimx n m 2 2, by Bertini s theorem see [9, Theorem 1.83] we can take a general member X n m 1 of L n m 2 such that X n m 1 is generically reduced. On the other hand, by [7, 2.4, 2, Chapter 0], X n m 1 is Cohen Macaulay. Hence X n m 1 is reduced. Here we note that X n m 1 is not always irreducible. Then by using the exact sequence 0 L n m 2 O Xn m 2 O Xn m 1 0, we see that h m+2 L n m 2 h m+2 O Xn m 2 + h m+1 O Xn m 2 h m+1 O Xn m 1 0 because h m+1 L n m 2 0by[9, Claim 2.1.1]. Therefore, g m+1 X n m 2,L n m 2 h m+2 L n m 2 h m+2 O Xn m 2 + h m+1 O Xn m 2 By Remark 3.22, Since, by Remark 3.23, h m+1 O Xn m 1. m+2 X n m 2,L n m 2 h m+2 O Xn m 2 h m+2 L n m 2. m+1 X n m 2,L n m 2 g m+1 X n m 2,L n m 2 m+2 X n m 2,L n m 2 + h m+1 O Xn m 2 h m+1 L n m 2,

8 24 Y. Fukuma we get m+1 X n m 2,L n m 2 h m+1 O Xn m 1 h m+2 O Xn m 2 h m+2 L n m 2 + h m+1 O Xn m 2 h m+1 L n m 2. Here we set L n m 1 : L Xn m 1. Then by using the exact sequence we have Therefore, 0 O Xn m 2 L n m 2 L n m 1 0, h m+1 O Xn m 2 h m+1 L n m 2 + h m+1 L n m 1 h m+2 O Xn m 2 + h m+2 L n m 2 0. m+1 X n m 2,L n m 2 h m+1 O Xn m 1 h m+1 L n m 1. Since X n m 1 is Cohen Macaulay and equidimensional, by [17, Chapter III, Corollary 7.7], the Serre duality holds, that is, and h m+1 O Xn m 1 h 0 ω Xn m 1 h m+1 L n m 1 h 0 ω Xn m 1 L n m 1 1. Here we note that ω Xn m 1 is a Cartier divisor by [20, Proposition 5.73]. Let X n m 1 j Z j,wherez j is an irreducible component of X n m 1 for each j. Since dim Bs L n m 2 dim Bs L m dimx n m 2 2, there exists an element s H 0 L n m 2 such that every Z j is not an irreducible component of the divisor divs of zeros of the global section s. Letδ : H 0 L n m 2 H 0 L n m 1.Thenδs H 0 L n m 1 defines a non-zero homomorphism CLAIM 4.1. Kerρ 0. ρ : O Xn m 1 L n m 1. Proof. Since δs does not vanish identically on any Z j, we see that dim SuppKerρ < dim X n m 1. However, since X n m 1 is Cohen Macaulay, X n m 1 is locally 1-Macaulay. For the definition of locally 1-Macaulay, see [7, 2.4 Definition in Chapter 0]. Therefore, by using [4, 1.12 Corollary] we have Kerρ 0. Hence ρ is injective, and ω Xn m 1 L n m 1 1 ω Xn m 1 ω Xn m 1 and L n m 1 1 are invertible. So we have h 0 ω Xn m 1 h 0 ω Xn m 1 L n m is injective because Hence m+1 X n m 2,L n m 2 h 0 ω Xn m 1 h 0 ω Xn m 1 L n m Therefore, m+1 X, L m+1 X n m 2,L n m 2 0. This completes the proof. By Theorem 4.1 we can get the following corollary.

9 A note on non-negativity of the ith -genus 25 COROLLARY 4.1. Let X, L be a quasi-polarized manifold of dimension n. Assume that n 2 and dim Bs L n 1.Then i X, L 0 for every integer i with i dim Bs L + 1. Moreover, we obtain the following corollaries. COROLLARY 4.2. Let X, L be a polarized variety of dimension n. Assume that X is normal and Gorenstein, dim IrrX 0 and dim SingX dim Bs L n 1.Ifi X, L,then i X, L 0. Proof. Since i X, L, we have i dim Bs L +1 by the -genus inequality of Fujita [7, 4.2 Theorem]. By Theorem 4.1, we get the assertion. COROLLARY 4.3. Let X, L be a polarized manifold of dimension n 2. Assume that X, L 2.Then i X, L 0 for every integer i with 2 i n. Proof. If h 0 L 0, then X, L n + L n n But then this contradicts the assumption. Hence h 0 L 1, that is, dim Bs L n 1. By assumption, we have i 2 X, L. Hence by Corollary 4.2 we get i X, L 0foreveryi with i 2. Therefore, we get the assertion. The following examples show that Corollary 4.1 is the best possible. Example Let Y be a smooth projective variety of dimension m 2suchthatK Y is ample with h 0 K Y 0. There exists such a Y for every m 2. Let A 1,...,A n m be ample divisors on Y such that Bs A i and h m A i 0foranyi,wheren>m.WesetE : K Y A 1 A n m and X : P Y E. Letπ : P Y E Y be its projection. For 1 i n m, let E i : K Y A 1 A i 1 A i+1 A n m and D i : P Y E i. Then we see that D i HE π A i and D 1 D n m PK Y.WesetL : HE. ThenL is ample and D i + π A i L for each i. SinceD 1 D n m PK Y and A i is spanned by its global sections, we have dim Bs L m. On the other hand, by [8, Example 2.108] and [12, Lemma ] we have g m X, L h m O X and m+1 X, L 0. Hence by Remark 3.23 we see that m X, L g m X, L m+1 X, L + n mh m O X h m L n m + 1h m O X h m L. Here we note that h m O X h m O Y h 0 K Y 0and h m L h m π L h m E h m K Y A 1 A n m h m K Y 1. Hence m X, L 1 < 0 and by Theorem 4.1 we see that dim Bs L m. Therefore, dim Bs L m.thisx, L is an example with dim Bs L m and m X, L < 0. 2 By [1, Theorem 1.1 and Section 2], there exists a Calabi-Yau 3-fold X and an ample line bundle L on X such that h 0 L 1. Then we note that OK X O X, h 1 O X

10 26 Y. Fukuma h 2 O X 0, h 3 O X 1andh j L 0foreveryj 1. Hence we have 3 X, L h 3 O X h 3 L 1. On the other hand, since h 0 K X + L h 0 L 1, we see that g 2 X, L 0 by Theorem Therefore, by Remark X, L g 2 X, L 3 X, L + h 2 O X h 2 L Since dim Bs L 2, we infer that this X, L is an example with i dim Bs L and i X, L < 0. Next we consider the case where X, L is a quasi-polarized manifold with dim X 3 and K X 0. PROPOSITION 4.1. Let X, L be a quasi-polarized manifold of dimension three. Assume that K X 0. Then the following hold: 1 3 X, L 0; 2 i if OK X O X,then 2 X, L 1; ii if OK X O X,then 2 X, L 1. Proof. Here we note that L K X is nef and big by assumption. Therefore, by the Kawamata Viehweg vanishing theorem, we have h j L 0foreveryj 1. 1 By Remark 3.22, we have 3 X, L h 3 O X h 3 L h 3 O X 0. 2 By Theorem 3.22 and Remark 3.22 and 3, we get 2 X, L g 2 X, L 3 X, L + h 2 O X h 2 L g 2 X, L h 3 O X + h 2 O X h 0 K X + L 2h 3 O X + 2h 2 O X. Since K X is nef, we see that c 2 XL 0byatheoremofMiyaoka s[21, Theorem 6.6]. Hence, by the Kawamata Viehweg vanishing theorem, the Serre duality and the Hirzebruch Riemann Roch theorem see [18]or[15, Example ], we have h 0 K X + L χ L 1 6 L K XL K2 X + c 2XL c 2XK X 1 6 L3 > 0. i If OK X O X,thenh 3 O X 0. Hence 2 X, L h 0 K X + L 2h 3 O X + 2h 2 O X 1. ii If OK X O X,thenh 3 O X 1. Hence 2 X, L h 0 K X + L 2h 3 O X + 2h 2 O X 1. This completes the proof. Remark 4.1. The inequality of Proposition 4.12ii is the best possible. See Example Next, as a special case of K X 0, we consider the case where X, L is a polarized Abelian variety. Here we note that if X is an Abelian variety, then a line bundle L on X is ample if and only if L is nef and big see, e.g., [2, Proposition 4.5.2].

11 A note on non-negativity of the ith -genus 27 Let X be an Abelian variety of dimension n and let L be an ample line bundle on X.Here we study a lower bound for the ith -genus of X, L. Ifi n, then by Remark 3.22 and the Kodaira vanishing theorem we have n X, L h n O X 1. So we consider the case where 1 i n 1. THEOREM 4.2. Let X, L be a polarized Abelian variety of dimension n 2, and let i be an integer with 1 i n 1.Then L n n 2 n 2 n! + n + if i 2, i 1 i 2 i X, L L n + n if i 1. n! Proof. In this case h i L 0 for every integer i with i>0, and h j O X n j for every integer j with j 0. By the definition of the ith -genus, we see that, for every integer i with i 2, i X, L g i 1 X, L i 1 X, L + n i + 1h i 1 O X h i 1 L n g i 1 X, L i 1 X, L + n i + 1. i 1 Hence we obtain i X, L n g i 1 X, L i 1 X, L + n i + 1 i 1 g i 1 X, L g i 2 X, L + i 2 X, L n n i + 2h i 2 O X + n i + 1 i k g i 1 k X, L + i 2 X, L + 1 j n n i j i 1 j i 2 1 k g i 1 k X, L + 1 i 1 1 X, L i j n n i j i 1 j i 1 1 k g i 1 k X, L + 1 i h 0 i 1 L + 1 j n n i j i 1 j i 1 1 g k n i 1 k X, L i h 0 L.

12 28 Y. Fukuma By [9, Claim 3.A.4.1], we have i 1 k 1 1 i 1 k 1 j h j O X i 1 k 1 n 1 i 1 k 1 Therefore, by [9, Remark 3.A.3.1 1], we have n 1 i 1 k 1 j j g i 1 k X, L!Sn, Ln n! + n 1. n 1. Here Sn, denotes the Stirling number of the second kind with the type n, see [9, Definition 3.A.1]. Hence i X, L i 1 1 g k n i 1 k X, L i h 0 L i 1 1 k!sn, Ln i 1 n! + 1 k n 1 i k n + 1 i h 0 L. First, we calculate i 1 1k n n i+1+k.then i 1 1 k n i 1 n 1 k i k i k i n 1 i t t t i n 1 1 i t n t 1 1 i n i n 1 1 t t 1 1 i i 1 n 1 n 1 p+1 p p0 i 1 n 1 n 1 i p 1 p p0 n 2 n. i 1

13 A note on non-negativity of the ith -genus 29 In the last step, we have used [9, Claim 3.A.4.1]. Next we calculate i 1 1k n 1 n i+1+k.ifi 2, then i 1 1 k n 1 i 2 1 k n 1 i 2 n 1 1 k i 2 k i 2 n 1 1 i 2 t t t0 n 2. i 2 In the last step, we have used [9, Claim 3.A.4.1]. If i 1, then i 1 1 k n 1 0. Finally, we calculate i 1 1k!Sn, L n /n! + 1 i h 0 L. We note that i 1 1 k!sn, i 1 1 i 1 t n t!sn, n t t0 1 By [23, 24d on p. 34], we have where So we have n+i 1 i 1 1 n t n t!sn, n t t0 1 n+i 1 n x n kn i+1 1 k k!sn, k. n Sn, k[x] k, 1 { xx 1 x k + 1 if k 1, [x] k : 1 if k 0. 1 n n 1 k k!sn, k k1 n i 1 k k!sn, k + k1 n kn i+1 1 k k!sn, k.

14 30 Y. Fukuma Because h 0 L L n /n!, weget { n } L 1 n+i 1 1 k n k!sn, k n! + 1i h 0 L kn i+1 i 1 Ln 1 n! n i 1n+i 1 1 k k!sn, k Ln n! + 1i 1 h 0 L 1 Therefore, we get n i n i k1 1 k k!sn, k Ln n!. k1 i X, L { n i } L 1 n i 1 k n n 2 n 2 k!sn, k n! + n + if i 2, i 1 i 2 k1 { n 1 } L 1 n 1 1 k n k!sn, k + n if i 1. n! k1 Here we prove the following. LEMMA 4.1. For every integer p with p n 1, the following holds: 1 n p n p n p n p k k!sn, k 1 n p t t n. t + 1 k1 Proof. By the Stirling formula [23, 24a on p. 34], we get Hence n p 1 k k!sn, k k1 Here we prove the following. k!sn, k n p k1 t2 k k 1 k j j n. 2 j k k 1 j j n j 1 2 n p n n p n n p n p n p n n p n p+1 n p k 1 t t 1 1 n. 3 t 1 kt 1

15 A note on non-negativity of the ith -genus 31 CLAIM 4.2. Let s and t be integers with 0 s t. Then t k t + 1. s s + 1 ks Proof. If t s 0, then this is true. Assume that this equality holds for 0 t s p. We consider the case where t s p + 1. Then t ks k s s+p+1 ks k s s+p k s + p s s ks s + p + 1 s + p s + 1 s s + p + 2. s + 1 This completes the proof of Claim 4.2. By Claim 4.2 and 3, we get n p 1 k k!sn, k k1 n p+1 t2 n p Therefore, we get the assertion of Lemma t 1 n p + 1 t 1 t n p + 1 t + 1 Here, for any integer r with r 1, we set r r + 1 a r 1 r t t n. t + 1 t n. t 1 n Then, by Lemma 4.1, L n n 2 n 2 a n i n! + n + if i 2, i 1 i 2 i X, L L n a n 1 + n if i 1. n! Here we note that tl is ample and spanned by its global sections for any large t, and i X, tl 0by[12, Corollary 3.3]. Since L n > 0, we obtain LEMMA 4.2. a n i 1. Proof. First we prove the following. a n i 0. 4

16 32 Y. Fukuma CLAIM 4.3. a p + a p+1 p + 1!Sn, p + 1. Proof. By using 2 we have p+1 p + 2 a p+1 1 p+1 t t + 1 p+1 p+1 1 p+1 t p + 1 t p+1 t p + 1 t + 1 t n p t t n t n + p + 1!Sn, p + 1 p p p+1 t t n + p + 1!Sn, p + 1 t + 1 p p p t t n + p + 1!Sn, p + 1 t + 1 a p + p + 1!Sn, p + 1. This completes the proof of Claim 4.3. CLAIM 4.4. a n k 1 k is divisible by n k + 1! for every integer k with 1 k n. Proof. a By Claim 4.3, we see that a n + a n 1 n! holds. On the other hand, by Lemma 4.1 and 1 we have n n + 1 a n 1 n t t n t n n 1 t t!sn, t 1 n 1 n 1. Hence we get a n n! andtheassertionistruefork 1. b Assume that a n t 1 t is divisible by n t + 1!.Weset a n t 1 t n t + 1!b n t. Then, by Claim 4.3, we have a n t + a n t 1 n t!sn, n t. Hence a n t 1 n t!sn, n t a n t n t!sn, n t n t + 1!b n t 1 t n t!sn, n t n t + 1b n t + 1 t+1. Hence a n t 1 1 t+1 is divisible by n t! and we get the assertion of Claim 4.4. Here we go back to the proof of Lemma 4.2. We use the notation in the proof of Claim 4.4. Namely, we set b n i : a n i 1 i n i + 1!.

17 A note on non-negativity of the ith -genus 33 Then b n i is an integer by Claim 4.4. Assume that 1 i n 1. If b n i 1, then a n i < 0 since 1 i n 1andn 2. So this is impossible by 4. Hence b n i 0. i Assume that i is even. Then a n i 1 because b n i 0and 1 i 1. ii Assume that i is odd. Then a n i n i + 1!b n i 1. If b n i 0, then a n i < 0 and this is impossible by 4. Hence b n i > 0anda n i n i + 1!b n i because i n 1. Therefore, we get the assertion of Lemma 4.2. By Lemma 4.2, we get L n n 2 n 2 a n i n! + n + if i 2, i 1 i 2 i X, L L n a n 1 + n if i 1, n! L n n 2 n 2 n! + n + if i 2, i 1 i 2 L n + n if i 1. n! Hence we get the assertion of Theorem 4.2. Remark Let X, L be a polarized Abelian variety of dimension n 2. Assume that i n 1. Then, by Theorem 3.22 and Remark 3.22 and 3, n 1 X, L g n 1 X, L n X, L + h n 1 O X h n 1 L h 0 K X + L h n O X + h n 1 O X h n O X + h n 1 O X h 0 L + 2h n 1 O X 2h n O X h 0 L + 2n 2 Ln n! + 2n 2 L n n 2 n 2 n! + n + if n 3, n 1 1 n 1 2 L n + n if n 2. n! Therefore, the inequality in Theorem 4.2 is the best possible. 2 If i is odd with 1 i n 1and L n n 2 n 2 n! + n + if i 3, i 1 i 2 i X, L L n + n if i 1, n! then by the proof of Theorem 4.2 we can prove that i n 1 see ii in the proof of Theorem 4.2.

18 34 Y. Fukuma Acknowledgement. This research was partially supported by a Grant-in-Aid for Scientific Research C No , from the Japan Society for the Promotion of Science. REFERENCES [1] A. Beauville. A Calabi Yau threefold with Non-Abelian Fundamental Group London Mathematical Society Lecture Notes Series, 264. Cambridge University Press, 1999, pp [2] Ch. Birkenhake and H. Lange. Complex Abelian Varieties, 2nd edn Grundlehren der Mathematischen Wissenschaften, 302. Springer, Berlin, [3] J. Dieudonné and A. Grothendieck. Éléments de Géométrie Algébrique. Publ. Math. Inst. Hautes Études Sci. 4, 8, 11, 17, 20, 24, 28, 32. [4] T. Fujita. On L-dimension of coherent sheaves. J. Fac. Sci. Univ. Tokyo Sect. IA Math , ; , [5] T. Fujita. Theorems of Bertini type for certain types of polarized manifolds. J. Math. Soc. Japan , [6] T. Fujita. Remarks on quasi-polarized varieties. Nagoya Math. J , [7] T. Fujita. Classification Theories of Polarized Varieties London Mathematical Society Lecture Note Series, 155. Cambridge University Press, [8] Y. Fukuma. On the sectional geometric genus of quasi-polarized varieties, I. Comm. Algebra , [9] Y. Fukuma. On the sectional geometric genus of quasi-polarized varieties, II. Manuscripta Math , [10] Y. Fukuma. A lower bound for the second sectional geometric genus of polarized manifolds. Adv. Geom , [11] Y. Fukuma. On the second sectional H-arithmetic genus of polarized manifolds. Math. Z , [12] Y. Fukuma. A generalization of the -genus of quasi-polarized varieties. J. Math. Soc. Japan , [13] Y. Fukuma. On invariants of polynomial functions. Japan. J. Math , [14] Y. Fukuma. Addendum: On the sectional geometric genus of quasi-polarized varieties, I. Comm. Algebra , [15] W. Fulton. Intersection Theory, 2nd edn Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge 2. Springer, Berlin, [16] J. Harris. Algebraic Geometry Graduate Texts in Mathematics, 133. Springer, Berlin, [17] R. Hartshorne. Algebraic Geometry Graduate Texts in Mathematics, 52. Springer, New York, [18] F. Hirzebruch. Topological methods in algebraic geometry Grundlehren der mathematischen Wissenschaften, 131. Springer, Berlin, [19] J. Kollár. Shafarevich Maps and Automorphic Forms M. B. Porter Lectures. Princeton University Press, Princeton, NJ, [20] J. Kollár and S. Mori. Birational Geometry of Algebraic Varieties Cambridge Tracts in Mathematics, 134. Cambridge University Press, [21] Y. Miyaoka. The Chern classes and Kodaira dimension of a minimal variety. Adv. Stud. Pure Math , [22] A. J. Sommese. On the adjunction theoretic structure of projective varieties. Proc. Complex Analysis and Algebraic Geometry Conf Lecture Notes in Mathematics, Springer, 1986, pp [23] R. P. Stanley. Enumerative Combinatorics, Vol. I Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Yoshiaki Fukuma Department of Natural Science Faculty of Science Kochi University Akebono-cho, Kochi Japan fukuma@kochi-u.ac.jp