ON PROJECTIVE MORPHISMS OF VARIETIES WITH NEF ANTICANONICAL DIVISOR

Transcription

1 ON PROJECTIVE MORPHISMS OF VARIETIES WITH NEF ANTICANONICAL DIVISOR A Dissertation presented to the Faculty of the Graduate School at the University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by LUNHAO AO Dr. Qi Zhang, Dissertation Supervisor JULY 2012

2 c Copyright by Lunhao Ao 2012 All Rights Reserved

3 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled: ON PROJECTIVE MORPHISMS OF VARIETIES WITH NEF ANTICANONICAL DIVISOR presented by Lunhao Ao, a candidate for the degree of Doctor of Philosophy and hereby certify that, in their opinion, it is worthy of acceptance. Dr. Qi Zhang Dr. S. Dale Cutkosky Dr. Zhenbo Qin Dr. Jianguo Sun

4 ACKNOWLEDGEMENTS Looking back, I am very grateful for what I have received from the mathematics department of University of Missouri-Columbia throughout the past four years. All these years of studies are full of excitement and challenge. Firstly I wish to express my sincere, heartily, and deepest gratitude to my adviser Professor Qi Zhang for his constant help, guidance, support and encouragement. Secondly, I wish to express my sincere and heartily gratitude to Professor S. Dale Cutkosky and Professor Zhenbo Qin for their teaching, guidance and support. I also would like to thank my doctoral committee member Professor Jianguo Sun for his interest and help. I am grateful to Professor Dan Edidin, Professor Charles Li, Professor Hema Srinivasan, Professor Jan Segert, Professor Shuguang Wang for their help and support. Finally, I am thankful for the financial support from the Department of mathematics, University of Missouri-Columbia. ii

5 TABLE OF CONTENTS ACKNOWLEDGEMENTS ii ABSTRACT v CHAPTER Introduction Basic properties of ample and nef divisors and some statements Ample Divisors and Nef Divisors Ample and Nef Vector bundles Some results about mod p reduction Some early statement for mod p reduction The bend-and-break lemma Some results about projective varieties with numerically effective tangent bundles and nef anticanonical bundles Some background Some results about numerically effective vector bundles Some general results Projective surfaces with nef tangent bundles Projective 3-folds with nef tangent bundles Structure of the Albanese map Projective surfaces with nef anticanonical bundles Weak positivity Introduction and the roots out of section The main lemma of weak positivity iii

6 5.3 The modification of weak positivity A new method to prove image theorem of varieties with nef anticanonical divisors Background and introduction The main part of this new proof BIBLIOGRAPHY VITA iv

7 ABSTRACT We shall study and discuss some important properties of the projective varieties with nef anticanonical bundles and nef tangent bundles. And we shall review some background and history about the subject. Then we shall use weak positivity theorem to give a new proof of a theorem of Olivier Debarre without using mod p reduction, which gives an affirmative answer to a question raised by Fujino and Gongyo. v

8 Chapter 1 Introduction In the research of algebraic geometry, specially in the classification theory of higher dimensional varieties, it is inevitable that we have to deal with some special projective varieties such as varieties with nef anticanonical and nef tangent bundles. For many years, mathematician have searched for the geometrical structure of these varieties. By the intensive works of many mathematicians, today we have plenty of knowledge of those fascinating varieties. A compact Riemann surface always has a hermitian metric with constant curvature. The negative sign corresponds to curves of general type (genus 2), and the case of zero curvature corresponds to elliptic curves (genus 1), and the positive curvature corresponds to rational curves (genus 0). In higher dimensional cases, this situation becomes much more complicated. For example, Frankel s famous conjecture on the characterization of P n (C) on the compact Kahler manifold of dimension n with positive sectional curvature. Later, Hartshorne gave a stronger conjecture by strengthening Frankel s conjecture and claiming that P n (k) is the only nonsingular projective algebraic variety with ample tangent vector bundle defined over an algebraically closed field k. In the coming years, Mori solved Harthsorne s conjecture by using his amazing characteristic p methods. 1

9 As a natural extension, by combining algebraic and analytic tools, several authors have investigated the geometrical and topological properties of varieties with nef anticanonical and nef tangent bundles. It was well-known that the classification of those varieties in low dimensions. For example, varieties with ample anticanonical bundle are Fano varieties. In dimension 2, these are the famous surfaces: Del-Pezzo surface. We shall discuss more about these varieties in dimension two in later sections. We shall introduce the weak positivity theorem, and use it to give a new proof of the image theorem of varieties with nef anticanonical divisor of Olivier Debarre, which answers a question in Fujino and Gongyo s paper. 2

10 Chapter 2 Basic properties of ample and nef divisors and some statements 2.1 Ample Divisors and Nef Divisors Definition 2.1. (Cartier divisors) A Cartier divisor on X is a global section of the quotient sheaf M X /O X. We denote by Div(X) the group of all such, so that Div(X) = Γ(X, M X/O X) More precisely, then, a divisor D Div(X) is represented by data {(U i, f i )} consisting of an open covering {U i } of X together with elements f i Γ(U i, M X ), having the property that on U ij = U i U j one can write f i = g ij f j for some g ij Γ(U ij, O X)). (2.1) the function f i is called a local equation for D at any point x U i. Two such collections 3

11 determine the same Cartier divisor if there is a common refinement {V k } of the open coverings on which they are defined so that they are given by data {(V k, f k )} and {(V k, f k )} with f k = h k f k on V k for some h k Γ(V k, O X). The group operation on Div(X) is always written additively: if D, D Div(X) are expressed respectively by data {(U i, f i )} and {(U i, f i)}, then D + D is given by {(U i, f i f i)}. the support of a divisor D is the set of points x X at which a local equation of D at x is not a unit in O x,x. We can easily know that there is a unique invertible sheave corresponding to a Cartier divisor, people could refer to the [Ha1]. Also we call a invertible sheaf a line bundle. In this article, we use the words Cartier divisors, invertible sheaves, and line bundles interchangeably. They represent the same object. And locally free sheaves are the same as vector bundles. Definition 2.2. (Ample and very ample line bundles and divisors on a complete scheme). Let X be a complete scheme, and L is a line bundle on X. (i). L is very ample if there exists a closed embedding X P of X into some projective space P = P N such that L = O X (1) = def O P N (1) X. (ii). L is ample if L N m is very ample for some m > 0. A Cartier divisor D on X is ample or very ample if the corresponding line bundle O X (D) satifies the condition. 4

12 Definition 2.3. (Nef divisors and Line bundles). Let X be a complete scheme. A Cartier divisor D on X (with Z, Q, or R coefficients) is numerically effective, if (D C) 0 for all irreducible curves C X. Here we call numerically effective nef. A line bundle L on X is nef, if and only if the correspondent Cartier divisor D is nef. Here, L = O X (D). Definition 2.4. (Numerical equivalence). Two Cartier divisors D 1, D 2 Div(X) are numerically equivalent, they are written D 1 D 2, if (D 1 C) = (D 2 C) for every irreducible curve C X, or equivalently if (D 1 γ) = (D 2 γ) for all one-cycles γ on X. Numerical equivalence of line bundles is defined in the similar way. A Cartier divisor or Line bundle is numerically trivial if it is numerically equivalent to zero. And we define that N um(x) Div(X) is the subgroup consisting of all numerically trivial divisors. And the Néron Severi group of X is the group N 1 (X) = Div(X)/Num(X) of numerical equivalence classes of Cartier divisors on X. We have a useful criterion for the ample line bundles by using the fancy technique of cohomology. 5

13 Theorem 2.5. Let L be a line bundle on a complete scheme X. The following are equivalent: (i). L is ample. (ii). Given any coherent sheaf F on X, there exists a positive integer m 1 = m 1 (F) having the property that H i (X, F L N m ) = 0 for all i > 0, m m 1 (F). (iii). Given any coherent sheaf F on X, there exists a positive integer m 2 = m 2 (F) such that F L N m is generated by its global sections for all m m 2 (F). (iv). There is a positive integer m 3 > 0 such that L N m is very ample for every m m 3. Another fundamental criterion is that ampleness is characterized numerically. Theorem 2.6. (NaKai-Moishezon criterion) A Cartier divisor D on a projective scheme X is ample if and only if, for every irreducible subvariety Y X, we have D dim(y ) Y > 0 (including the irreducible components of X). At here, D dim(y ) Y > 0 means the intersection number between these subvarieties. The same result holds when D is a Q-Cartier divisor. Corollary 2.7. (Numerical nature of ampleness). If D 1, D 2 Div(X) are numerically equivalent Cartier divisors on a projective variety or scheme X, then D 1 is ample if and only if D 2 is ample. S. Kleiman also gave a useful criterion for checking if a divisor D on X is ample, i.e., D is ample if and only if there exists a positive number ε such that (D C) ε 6

14 C for every effective curve C in X. Please refer to [Kl1]. C is the norm of C in the R-linear space N 1 (X) Z R. 2.2 Ample and Nef Vector bundles Definition 2.8. (Construction). Let X be an algebraic variety or scheme, and let E be a vector bundle of rank e on X. We denote by π : P (E) X the projective bundle of one-dimensional quotients of X. P (E) is realized as the scheme P (E) = P roj OX (Sym(E)), where Sym(E) = S m E denotes the symmetric algebra of E. (Serre line bundle). The line bundle O P (E) (1) on P (E) is called Serre line bundle, also as a tautological quotient of π E: π E O P (E) (1) 0. For m 0 we have π O P (E) (m) = S m E. There is a natural surjective morphism π E O P (E) (1). Definition 2.9. (Ample and nef vector bundles). A vector bundle E on X is ample if the Serre line bundle O P (E) (1) is an ample line bundle on the projectivized bundle P (E). Similarly, E is numerically effective (nef) if O P (E) (1) is so. We could find that if E = L is a line bundle,then P (E) = X and O P (E) (1) = L, 7

15 so that this definition generalizes the rank one case. The ample vector bundles have the similar cohomological characterization criterion. 8

16 Chapter 3 Some results about mod p reduction 3.1 Some early statement for mod p reduction When studying special vector bundles, Hartshorne began to be interested in the ample vector bundles. Then he studied ample vector bundles on algebraic varieties. He paid attention to ample tangent vector bundles on non-singular projective variety. He put forward the famous Hartshorne s conjecture, for discussing the structure of the non-singular variety with ample tangent vector bundle. Hartshorne s Conjecture. (H n) If X is an irreducible n-dimensional non-singular projective algebraic variety with ample tangent vector bundle defined over an algebraically closed field k, then X is isomorphic to P n over k in the sense of an algebraic isomorphism. If we assume that k = C(the complex number field), People know that this conjecture has a strong connection with the following famous conjecture of Frankel in 9

17 complex differential geometry. Frankel s conjecture. (F n) A compact Kahler manifold X of dimension n with positive sectional curvature is biholomorphic to the complex projective space P n (C). In the case n = 1, it is easy to check that X is necessarily isomorphic to the projective space. In the case n = 2, they are solved by using classification of algebraic surfaces, by R. Harthorne and by Frankel and Andreotti respectively. Later on, T. Mabuchi proved the conjecture in the case n = 3, when k is of characteristic zero and the second Betti number of X equal to 1. After that, S. Mori removed the assumption on the second Betti number, and proved that (H 3) is true. S. Mori proved a criterion for P ic(x) = Z, which is that Let X be a nonsingular projective algebraic variety with ample anticanonical divisor K X, and then P ic(x) is isomorphic to Z if and only if every effective divisor on X is ample. He used this to prove that the Picard number ρ(x) of X with ample tangent vector bundle T X is 1. Next, he proved a characterization of projective spaces. Assume that X holds a nonzero global vector bundle which vanishes on an ample irreducible effective divisor D, i.e., H 0 (X, T X OX ( D)) 0, and then X is isomorphic to P n and D corresponds to a hyperplane in P n. In next subsection, we will state a critical part of Mori s proof of Hartshorne s conjecture in the case of every n-dimensional nonsingular projective variety. Introduce the bend-and-break lemma which is also called mod p reduction. 10

18 3.2 The bend-and-break lemma Now, let us turn to take a look at Mori s proof of Hartshorne s conjecture in the case of every n-dimensional nonsingular projective variety. In this proof, he researched the morphisms from a rational curve C to X, and used the technique of reduction modulo p and the construction of Frobenius morphism on the curve C. The use of Frobenius morphism is to make sure the lower bound of the dimension of the morphism scheme Mor(C, X; j {p} ) at the point i f is bigger than 1. At here, Mor(C, X; j {p} ) = {f : C X f is a morphism and f {p} = j {p} }. In his Spectacular proof, he proved the existence of rational curves in a nonsingular projective variety over an arbitrary field k, and this curve must break into an effective 1-cycle with a rational component passing through the fixed point p. Therefore, they are called bend-and-break lemmas. By using the fact of existence of rational curves, professor Mori proved the Hartshorne s conjecture. Lemma 3.1. {rigidity lemma} Let X, Y and Y be varieties and let ϕ : X Y and ϕ : X Y be proper morphisms. Assume ϕ O X O Y. (a) If ϕ contracts one fiber ϕ 1 (y 0 ), there is an open neighborhood Y 0 of y 0 in Y and a factorization ϕ ϕ 1 (Y 0 ) : ϕ 1 (Y 0 ) Y 0 Y (b) If ϕ contracts each fiber of ϕ, it factors through ϕ. Proposition 3.2. We have a non-singular projective variety X over an algebraically closed field k of characteristic p > 0. Then a rational curve exists in X if the canonical divisor K X is not numerically effective. 11

19 Proof. Since we know that K X is not numerically effective, we could find an irreducible curve Z in X satisfying that (K X Z) < 0. Assume that i : C 1 Z X is the normalization of Z. Assume that t is a sufficient big positive integer which satisfies that dimx g(c 1 ) p t (K X Z) 1. Here, we can increase the positive number p t (K X Z) to make sure that it is large enough by using composition of the Frobenius morphism. Let us suppose that q is a power of p. We define a 1/qth power endomorphism ϕ : Spec k Spec k, and let C q Spec k be the base change of C 1 Spec k of ϕ. Because ϕ is flat, H 1 (C q, O Cq ) = ϕ H 1 (C 1, O C1 ). Hence we have g(c q ) = g(c 1 ). Next, let q = p t, C = C q, and define the morphism h : C C 1 the qth power Frobenius k-morphism. Now we take a k-rational point Q on C, and j : Q (i h)(q) X. Since we know q =deg h, we have dim [i h] Mor(C, X; j) χ(c, (i h) T X O C ( Q)) = dimxg(c) q(k x Z) 1 Here, χ is the Euler number of the sheaf. Now we could obtain a non-singular curve E and a finite morphism f : E Mor k (C, X; j) with [i h] f(e). We want to show that E is not a complete curve. Suppose that the contrary is true, and define F : C E X the morphism induced by f. From the definition of Mor k (C, X; j), F (Q E) = (i h)(q). By the rigidity lemma, we have that F (u E) = (i h)(u) for every closed point u on C. Then we have that f(e) is a point [i h], which is a contradiction, since f(e) is an infinite set. Therefore we prove our claim. Let E E be a non-singular compactification of E. Because of the fact that f can not extend to E, we can not extend the morphism F : C E X to C E X. So this rational map F : C E X is not a morphism. After that we do the final blowing-up to eliminate the indeterminacy of F. We can 12

20 have a surjective morphism θ from the final blowing-up variety S to X, θ : S X. The image of θ is of dimension 2. On the other hand, there is a point t 0 in E satisfying that F is not defined on (Q, t 0 ). The fibre of t 0 under the projection S E is the union of the strict transform of C {t 0 } and a connected exceptional rational 1-cycle H. And H can not be entirely contracted by θ and it meets the strict transform of {Q} E. Because the strict transform of {Q} E is contracted by θ to the single point (i h)(q). We obtain a rational 1-cycle which is mapped to a rational curve in X. This rational 1-cycle θ H in X is what we want. Remark 3.3. Let X be a scheme. All of its local rings are of characteristic p. We define the Frobenius morphism F : X X as follows: F is the identity map on the topological space of X, and the induced morphism F : O X O X is the pth power map. F induces a local homomorphism on every local ring by the fact that the local rings are of characteristic p, so actually F is a scheme morphism. Now, we want to state a result in arbitrary characteristic 0. Proposition 3.4. We have a non-singular projective variety X over an algebraically closed field k with the ample anti-canonical divisor K X. Then a rational curve exists in X. Proof. First, we want to construct an integral domain R in k which is a finitely generated ring over Z. Assume that X is defined by these homogeneous polynomials {f 1, f 2,, f m }. Let x be a point of X. R is the smallest ring that contains all of the coefficients of these homogeneous polynomials, the coordinates of x, and the integer ring Z. Then we can obtain a projective smooth morphism ϕ : W S = SpecR. At here W = SpecR[f 1, f 2, f m ]. Then we have W k = W SpecR Speck X, and the dual relative anticanonical divisor K W/S is ϕ-ample. We choose every closed point in S, which is treated as a geometric point p in S. Next, we have that the residue field 13

21 k(p) of p is of characteristic p > 0, since k(p) contains Z p. p is a prime integer. And by proposition 3.2 W k(p) contains a rational curve, and ( K Wk(p) C) dimx + 1. Assume that T Mor X (PS 1, W ) is the open and closed subscheme containing all the morphisms g : P 1 s W (s S is the point dominated by g) satisfying that 0 < deg P 1g ( K W/S ) dimx + 1. We know that T is quasi-projective by the fact that K W/S is ϕ-ample. So T contains at least one closed point for each closed point on S. Hence T k also contains one closed point over k. Then the image of one closed point of T k over P 1 k is a rational curve in X. Now we know the existence of a rational curve on X. Then we want it to break up into an effective 1-cycle with rational components under some conditions. Proposition 3.5. We have a projective variety X over k and a rational curve f : P 1 X. If dim [f] (Mor(P 1, X; f {0, } )) 2, the 1-cycle f P 1 is numerically equivalent to a connected non-integral effective rational 1-cycle passing through f(0), and f( ). Proof. By the deformation theorem, the lower bound of dim [f] (Mor(P 1, X; f {0, } )) is K X f P 1 dim(x). First, we know the group of automorphisms of P 1 fixing 2 points is the multiplicative group variety G m = k {(0)}. Assume that D is the normalization of one 1-dimensional subvariety of Mor(P 1, X; f {0, } ) containing [f] but not contained in G m -orbit. The corresponding map F : P 1 D X D is finite. Next, let D be a smooth compactification of D and let S be the normalization in the rational field K(P 1 D) of the closure of F (P 1 D) in P 1 D. Then we obtain F : S X D the standard normalization finite morphism. Since P 1 D is normal, P 1 D = F 1 (X D). 14

22 S is of dimension 2, and it may not be non-singular. We also know that no component of a fiber of π is contracted by e, since it could be contracted by F. Because D is a non-singular curve and S is an integral surface, π is a flat morphism. Then every fiber E is a 1-dimensional projective scheme without embedded component, whose genus is a constant, which is 0. In particular, every component E 1 of E red is non-singular and rational, since O E1 is a quotient of O E. Thus H 1 (E 1, O E1 ) is a quotient of H 1 (E, O E ), and it is 0 module. Particularly, if E is integral, it is a non-singular rational curve. We suppose that each fiber of π is integral, the contrary, and we want to obtain a contradiction. Each fiber is an integral curve whose genus is 0, so it is P 1. Then S is a ruled surface. We assume that D 0 is the closure of {0} D in S and let D be the closure of { } D. These sections are contracted by e. If H is ample divisor on e(s), which is a surface by the construction, then we can have (e H) 2 > 0 and e H D 0 = e H D = 0; Thus D0 2 and D 2 are negative by Hodge index theorem. Nevertheless, since D 0 and D are both sections of π, their difference is linearly equivalent to the pull-back by π of a divisor on D(see [Ha1], V, prop. 2.3). Particularly, 0 = (D 0 D ) 2 = D D 2 2D 0 D < 0, which is a contradiction. Then we can have that at least one fibre of π is not integral. By the fact that none of its components is contracted by e, its direct image on X is the required 1-cycle. By using the bend-and-break lemma, professor S. Mori proved the famous Hartshorne s conjecture. The theorem is that every irreducible n-dimensional nonsingular projective algebraic variety whose tangent vector bundle is ample, is isomorphic to P n. So, we have known that non-singular projective variety with ample 15

23 tangent bundle is just projective space. 16

24 Chapter 4 Some results about projective varieties with numerically effective tangent bundles and nef anticanonical bundles 4.1 Some background As we know, S. Mori proved the Hartshorne-Frankel conjecture, which tells us the structure of non-singular projective varieties with ample tangent bundles. Hence, it is very natural for us to explore the structure of non-singular projective varieties X whose tangent bundles T X are in a degenerate condition of ampleness, which is numerical effectivity (simplified by nef ). It means that the tautological quotient line bundle O P (TX )(1) on P (T X ) is numerically effective, i.e. (O P (TX )(1) C) 0 for all curves C P (T X ). Particularly, if T X is generated by global sections, then T X is numerically effective. Generally, the surfaces X whose tangent bundles are nef, 17

25 can be described clearly. This surface is either abelian surface, or P 2, P 1 P 1 or a special ruled surface over an elliptic curve. In the following sections, we want to give a classification of the projective surfaces whose tangent bundles are numerically effective and state the results of the classification of 3-dimensional non-singular projective varieties with numerically effective tangent bundles. 18

26 4.2 Some results about numerical effective vector bundles We assume that X is a projective variety and E is a rank r-vector bundle on X. Let π : P (E) X be the associated projective space bundle and let O P (E) (1) be the natural quotient line bundle on P (E) with π (O P (E) (1)) E. On a projective variety, let ρ(x) = dim(p ic(x) R/numerical equivalence) be the Picard number of X. For a subvariety Z X, N Z X = Hom OZ (I/I 2, O Z ) means its normal sheaf of Z in X. Definition 4.1. We have a Q-divisor D, and then the corresponding locally free sheaf E O X (D) is a nef bundle if and only if the Q-divisor O P (E) (1) π (D) is nef. Next, we state some useful properties of the nef vector bundles in the following proposition and lemma. Proposition 4.2. X is a projective variety, and E is a vector bundle on X of rank r. (1) E is nef if and only if E C is nef for every curve C. (2) E is nef if and only if E O X (D) is ample for every ample Q-divisor D on X ( by definition, it means that the Q-divisor O P (E) (1) π (O X (D)) is ample. (3) If E is nef and F is a quotient bundle of E, then F is nef. (4) Let 0 F E G 0 be an exact vector bundle sequence. If F and G are nef, then E is nef. (5) If f : X X is a finite map of smooth curves, E is nef iff f E is nef. (6) Let X be a smooth curve. Then E is nef iff every quotient bundle of E has nonnegative degree. (7) Let 0 F E G 0 be an exact vector bundle sequence. Suppose that c 1 (G) = 0 and E is nef, and then F is nef. 19

27 (8) Suppose c 1 (E) = 0. Then E is nef iff the dual sheaf E is nef. (9) If E and F are nef, then E F, S m E, k E are nef (m, k N). (10) If E is nef, then c i (E) 0, i.e. (c i (E) Z) 0 for all irreducible closed subset Z X of dimension i. Proof. (1). This can be proved by the definition of nef vector bundle. (2). First, we suppose that the vector bundle E O X (D) is ample for every ample Q-divisor D on X. Then we choose one ample divisor D 0 and let α > 0. Then (c 1 (O P (E) (1) + π (O X (αd)) C) > 0, Thus, (c 1 (O P (E) (1)) C) 0. Conversely, by using Nakai-Moishezon criterion, for every curve C, we have (c 1 (O P (E) (1)) C) 0, and π (D)) C > 0. Hence, (c 1 (O P (E) (1) + π (O X (D)) C) > 0. So E O X (D) is ample. (3). We can construct an injective closed embedding P (F ) as a subspace of P (E). And we have O P (F ) (1) O P (E) (1) P (F ). By the definition of nef vector bundle, F is nef, since O P (F ) (1) is nef. (4). Assume that D is an ample Q-divisor. We take a finite covering g : X X such that g (D) is a divisor D. Since g (F ) O( D) and g (G) O( D) are ample, 20

28 and F O(D) and G O(D) are ample by (2), g (E) O( D) = g (E O(D)) is ample. Then by using (2), E is nef. (5),(6). We prove (5), (6) together. Define that E is q-nef iff every quotient of E has non-negative degree. First, we prove (5) by using (5) for q-nef bundles. After that, we will prove (6) by using (6) for q-nef bundles. Therefore nef and q-nef are the same notions and (5) is proved in its original form. Suppose that f (E) is q-nef. Let F be a quotient of E. Then f F is a quotient of f E, and then c 1 (f F ) 0. Since c 1 (f F ) = degf c 1 (F ), E is q-nef. Suppose that E is q-nef. From the first part of (5), we may assume that f is a Galois covering. If f E is semi-stable, f E is q-nef. Otherwise let G f E be the maximal destabilizing subsheaf. By [Mi, Lemma3.2] we can assume G f (Q) with some subbundle Q E. Now c 1 (E/Q) 0, and then c 1 (f E/F ) 0. Since G is maximal, c 1 (F ) 0 for every quotient f F of f E. So f E is q-nef. To prove (6), we could suppose that X is a curve. One direction can be proved by (3). In the other direction let E be q-nef, i.e. all quotients have non-negative degree. Let C P (E) be an irreducible curve. Let h : C C be the normalization and f = h π : C X. C induces a section C of P (f E C). Let F be the corresponding quotient line bundle of f E. Since f E is q-nef by (5), we can have c 1 ( F ) 0. On the other hand, c 1 ( F ) = (c 1 (O P (E) (1)) C). Hence O P (E) (1) is nef. (7). We suppose that F is not a nef vector bundle. Hence, we can find a quotient bundle L of F with c 1 (L) < 0. Also, we suppose that X is a smooth curve. Then we find a subbundle J of F with c 1 (J) > c 1 (F ). Since c 1 (G) = 0, J is also a subbundle 21

29 of E with c 1 (J) > c 1 (E). So E has a quotient of negative degree, which is impossible. (8). We have the fact that E r 1 E dete and r 1 E is nef. Then we can get the conclusion. (9). We prove that S m E is nef. Assume that X is a smooth curve. Choose an ample Q-divisor D. We take the covering g : X X such that g (D) = D is a divisor. Since E is nef, g E O( D m ) is ample, and thus Sm (g E O( D m )) is ample. So Sm E O(D) is ample. By (2), S m (E) is nef. (10). Assume that E (αd) is ample. We have (c i (E (αo X (D))) Z) > 0. Let α go to 0. Then (c i (E)) Z) 0. Lemma 4.3. If E is nef on X and w H 0 (X, E ) and w 0, then w has no zero points. Proof. If w has a zero point p Z(w), we can find a curve C and p C. Then we have E C = O C (mp) G. 0 O C (mp) E C. Then we have E C O C ( mp) 0. We have the fact that vector bundle E is nef iff in the exact sequence E L 0, for all quotient line bundles L, L is nef. By the fact that O C ( mp) is not nef, E C is not nef on C. It is a contradiction. So w has no zero point. Remark 4.4. Assume that X is a projective manifold and E is a rank r-vector bundle. We recall that E is called semi-stable with respect to a polarization(= ample 22

30 line bundle) H if and only if δ H (F ) c 1(F ) H rankf c 1(E) H r = δ H (E) for all subsheaves F E of positive rank. If the strict inequality always holds, E is called stable. We call that E is stable(semistable) iff E is stable(semi-stable) with respect to any polarization. A maximal destabilizing subsheaf F for E with respect to H is a proper subsheaf F having the following properties: (1) δ H (F ) δ H (E). (2) F E is a subsheaf with δ H (F ) δ H (E) and F F and then F = F. If X is an irreducible reduced curve, then a vector bundle E on X is said to (semi- )stable iff the pull-back of E to the normalization of X is (semi-)stable. Next we will have a result for H-destabilizing subsheaf. Proposition 4.5. Assume that X is a smooth surface, and that E is a rank 2 bundle on X. Assume E is nef and c 1 (E) = 0. Let F E be a maximal H-destabilizing subsheaf. Then we have that F is a subbundle of E. Proof. We suppose F 0. Because F is also a subsheaf of E and since F F (F is torsion free), F must be reflexive (of rank 1). X is smooth and F is locally free. The inclusion F E does not vanish in codimension 1, otherwise there is an effective divisor D 0 such that F O(D) E. Since F F O(D), δ H (F O(D)) > δ H (F ); this contradicts the maximality of F. Thus Q = E/F is locally free outside a finite set. Q is also torsion free, otherwise F would not be maximal (consider Ker(E 23

31 Q/torsion)). Because E is nef. we can get c 1 (Q) 0, hence c 1 (F ) = c 1 (Q) = 0. So c 2 (E) = c 2 (Q) = 0. In order to calculate c 2 (Q), we consider the exact sequence 0 Q Q R 0. Here R is the quotient sheaf. c 2 (Q ) = c 2 (Q) + c 2 (R). Q being a line bundle, we can obtain c 2 (Q) = c 2 (R). Since suppr is concentrated on points, c 2 (R) 0 and c 2 (R) = 0 if and only if Q is locally free. Since c 2 (E) 0, we can have c 2 (Q) 0 and thus c 2 (R) = 0. So Q is locally free and therefore F E is a subbundle. 24

32 4.3 Some general results Proposition 4.6. We have a projective manifold X whose tangent bundle T X is nef. C 0 X is a rational curve. Then the deformations of C 0 can cover X. Proof. If C 0 is a smooth curve, by a theorem of Kodaira, it is true. Now assume that C 0 is not smooth. We consider the graph of i v : Ĉ 0 X where i : C 0 X is the embedding into X and v is the normalization. Then we can obtain the result by using Kodaira-Griffiths result. Corollary 4.7. κ(x) = if and only if There is a rational curve in X. Otherwise we have κ(x) = 0. Proposition 4.8. Let X be a projective manifold. If the tangent bundle T X is nef, then the Albanese map f : X Alb(X) = Y is smooth. Proof. f is smooth iff T X df f T Y 0 is surjective. Then it is equal to 0 f Ω Y Ω X. Let {ω i } 1 i n be a basis of H 0 (X, Ω X ). We have f : x ( x x 0 w 1,, x x 0 w n ) = (z 1,, z n ), and f : Ω Y Ω X, f dz n = w n. Here x 0 is a fixed base point. f ( a i dz i ) = a i w i = 0. Since {w i } 1 i n, we have a 1 = = a n = 0. Then ai dz i = 0. So f is injective. Thus, we have that df : T X f T Y is surjective. So, the Albanese map from a projective manifold with nef tangent bundle is smooth. Theorem 4.9. Suppose T X is nef. (1) If K X is nef, K X 0, and there is an etale covering T X, T is an Abelian variety. This kind of projective manifold X is called hyperelliptic. 25

33 (2) If K X is not nef, κ(x) = and every extremal ray defined a fibration ϕ : X Y (dimy < dimx). Proof. (1) Since K X is also nef, we have K X 0. By Yau[Y], we just need to prove c 2 (X) = 0 in H 4 (X, R). Since T X is nef, we have c 2 1(X) c 2 (X), i.e. c 2 1(X)H 1... H n 2 c 2 (X)H 1... H n 2 0 for all ample H i. Thus c 2 (X) = 0, and c 2 (X) is represented by a semi-positive form [Y]. (2) we can obtain it from Mori theory. Proposition Assume that T X is nef, A(X) is the Albanese torus of X, and φ : X A(X) is the Albanese map. Then φ is a surjective, smooth, and connected map. All fibers F have nef tangent bundles. Proof. If not, we could have a non-zero section of Ω 1 X having zero points by taking φ (w), and a general section w H 0 (A(X), Ω 1 ). It contradicts lemma 4.3. Smoothness is from lemma 4.3. By applying proposition 4.2(6), we could have the last fact. Definition For any projective manifold X, define q(x) = max{q( X) an etale map f : X X, X compact connected}. q(x) = h 1 (X, O X ). Corollary Assume that X is projective, and T X is nef. Then dimx q(x). Proof. Otherwise we can find an etale covering f : X X with dim X = dimx < q( X). Since T X is also nef, we get a contradiction to Theorem(4.9). 26

34 Proposition Assume that X is n-dimensional projective manifold, and let T X be nef. Then we have, (1) h 0 (X, Ω p ) ( n p). (2) X (O X ) 2 n 1. (3) If q(x) > 0, then X (O X ) = 0 and K n X = 0. Proof. (1) Since rankω p = ( n p), otherwise we can have a section of Ω p with zero points. Also, Λ p T X is nef, so this contradticts (4.2, 4.3). (2) it can be proved from (1) by the fact of h p (X, O) = h 0 (X, Ω p ). (3) If q(x) > 0, then q( X) > 0 for certain etale cover X of X. Then X has etale coverings X k of any order k. Since X (O Xk ) = kx (O X), we get a contradiction to (2). The last fact comes from (4.10). Proposition Suppose that X is a projective manifold with nef tangent bundle. Suppose that q(x) = q(x). φ : X A(X) is the Albanese map. Then we have q(f ) = 0 for every fiber F of φ. Proof. From (4.10) all fibers of φ are smooth and connected. Suppose that q(f ) 0 and let f : F F be a finite etale covering such that q(f ) = q( F ). Therefore, we can find a compact irreducible analytic subset Ḡ of C(X/A(X)), the relative cycle space of X over A(X), with a natural map φ : Ḡ A(X) such that every connected component of φ 1 (φ(f )) consists of a family G for a map f : F F as above. Furthermore, the Stein factorisation ϕ : Ā(X) A(X) of φ is unramified over A(X), because its fibers are orbits of π 1 ( F ) under the natural mondromy representation of π 1 (A(X)) on π 1 (F ). Suppose that X Ḡ A(X) X is the graph of the universal family of cycles of X parametrized by Ḡ: it is unramified over X. Let φ : X Ā(X) be the composition of φ : Ḡ Ā(X) with the natural projection from X to Ḡ. Let γ : F G be the Albanese map of F, and we know it is surjective and connected. Let F g := γ 1 (g), for g G. Let F g := f ( F g ). 27

35 So (F g ) g G is an analytic family of cycles of F parametrized by G. Let G C(F ) be the image of the natural map ψ : G C(F ) associated to it, where C(F ) is the Chow scheme of F. Let Γ G F be the graph of the universal family of cycles parametrized by G. It is unramified over F, since it is covered by F. This construction can now be made in a relative setting over A(X), because the map f : F F is defined by some finite index subgroup of π 1 (F ), and it is a topological invariant of F. The map φ : X A(X), η = φ v, is a differentiable bundle with typical fiber F. So we have the following commutative maps, β : X A( X/ Ā( X)) = B, η : A( X/Ā( X)) Ā( X) φ = η β : X Ā(X) So it is the relative Albanese reduction of φ. Then η 1 (z) is the Albanese torus of φ 1 (x) for every x Ā(X). But the map η is a locally trivial fiber bundle with fiber an abelian variety F, since the universal covering of Ā(X) does not admit any non-constant holomorphic map to any Siegel domain parametrizing polarized marked abelian varieties by using Liouville s theorem. B is projective. There is a finite etale covering c : Ã Ā(X) such that c (η) : B = c (B) Ã is a principal bundle with fiber F and group Aut 0 (A). Then we can deduce that q( B) = q(ã) + q(f ) = dim B. Thus B is an abelian variety. Now we see that c ( φ) : X 0 = c ( X) Ã. Then we can have a finite etale covering g : X 0 X as well as an induced map β 0 : X 0 B. Finally, we have q(x) q(x 0 ) q( B) q( X) + q( F ) > q( X), which is a contradiction. 28

36 Remark The following generalization of (4.13) should be true. Assume X is a projective n-dimensional manifold with nef T X. Then the following statements are equivalent. (1) X is Fano. (2) ( K X ) n > 0. (3) q(x) = 0. (4) X (O X ) 0. (5) X (O X ) = 1. (6) H 0 (X, Ω p ) = H p (X, O X ) = 0 for all p > 0. It is easy to know that (1) (6), (6) (5) (4) (3), (1) (2). By using the Kawamata-Shokurov base point free theorem and the fact that if K X is not ample, mk X defines a modification X Z contracting some rational curve, (2) implies (1). It contradicts T X nef. So (2) (1). Let us take a look at the fibrations. Proposition Assume that X is a n-dimensional projective manifold with nef tangent bundle T X. Let f : X Y be a surjective map with connected fibers, dimx > dimy. Then we have (1) Assume F is a smooth fiber of f. Then T F is nef. (2) If f is smooth, T Y is nef. Proof. (1) can be proved from the exact sequence 0 T F T X F N F X 0, and the triviality of N F X and (4.2). (2) can be proved from the exact sequence 0 T X Y T X f (T Y ) 0. 29

37 Proposition Assume that X is a projective manifold with nef T X. Let D = ni D i be an effective divisor on X. Then D is nef. Proof. First suppose that D is an irreducible normal hypersurface. Suppose that there is a curve C with D C < 0. Then we have C D. We can get an exact sequence N D X C φ Ω 1 X C Ω 1 D C 0. Let f : C C be the normalization. By the assumption, φ is non-zero, and hence we could get the exact sequence 0 f (N D X C) φ f (Ω 1 X C) f (Ω 1 D C) 0. Thus f (N D X C) is nef, i.e. (D C) 0. Now, if D is not normal, consider m N such that the canonical map N D S m Ω 1 X does not vanish identically along C. Proposition Assume that X is a n-dimensional projective manifold with nef T X. Let f : X Z with dimz 2 be a surjective map with connencted fibers on a projective subvariety Y of X. Y X is an irreducible hypersurface. Then dimf(y ) > 0. Proof. Suppose dimf(y ) = 0. Then we can find a very ample divisor L on Z such that f (L) = L + Y. Next, choose a general curve C Y, and then (L C) > 0. Since (f L C) = 0, we get Y C < 0, which contradicts remark(4.15). 30

38 4.4 Projective surfaces with nef tangent bundles In this subsection, we will give an easy classification of projective surfaces whose tangent bundles are nef. Theorem Assume that X is a smooth projective surface and let T X be nef. Then X is minimal and exactly one of the following surfaces in the following list. (1) X is an abelian surface. (2) X is hyperelliptic. (3) X = P 2. (4) X = P 1 P 1. (5) X = P (E), E is a rank 2-vector bundle on the elliptic curve C with either (a) E = O C L, L P ic 0 (C), or (b) E is given by a non-split extension 0 O C E L 0 with L = O C, or degl = 1. (These are exactly the semi-stable 2-bundles on C). Proof. By using (4.6), X is minimal. If κ(x) = 0, we need to apply (4.13). Then X is abelian surface or hyperelliptic. So let κ(x) =. Hence X is rational or ruled surface. X P 2, X is a ruled surface over a curve C. By using (4.16), g(c) 1. Let X = P (E) with a 2-bundle E. If C = P 1, X P 1 P 1, X contains an exceptional rational curve, so T X is not nef. Now suppose that C is elliptic. By using the sequence 0 T X C T X p (T C ) = O 0 31

39 we would see that T X is nef iff T X C is nef by (4.2). Since T X C O P (E) (2) p (dete ), and p : P (E) C is the projection, this is true iff E dete 2 is nef, which means that E is semi-stable. Now the semi-stable bundles on C are exactly those as described in (5) [At1]. Also, we can use the elliptic ruled surfaces theory to obtain them. Remark If X is a smooth curve, the vector bundle E of rank r is semi-stable iff E dete r is nef. 32

40 4.5 Projective 3-folds with nef tangent bundles In this subsection we want to give a classification of projective 3-folds with nef tangent bundle T X. By using (4.7), we could assume κ(x) = and that X contains an extremal ray R. We can fix R and use the associated contraction φ : X Y to prove this classification of 3-folds. Due to the complexity of this classification, we just state the classification theorem. Theorem Assume that X is a projective 3-fold. Then T X is nef iff X is up to finite etale cover one of the projective varieties in the following list: (1) X = P 3 ; (2) X = Q 3, the 3-dimensional quadric; (3) X = P 1 P 2 ; (4) X = P 1 P 1 P 1 ; (5) X = P (T P2 ) (6) X = P (E), with E a numerically flat rank 3 bundle over an elliptic curve C; (7) X = P (E) C P (F ), with E, F numerically flat rank 2 bundles over an elliptic curve C; (8) X = P (E), with E a numerically flat rank 2 bundle over an abelian surface; (9) X = abelian variety. 33

41 4.6 Structure of the Albanese map In 1996, Professor Qi Zhang solved a conjecture of D ly, Peternell and Schneider. Conjecture. Assume that X is a compact Kahler manifold with nef anticanonical bundle K X. Then the Albanese map α : X A(X) is surjective. By Professor Qi Zhang s proof in his paper On projective manifolds with nef anticanonical bundles, he proved the following theorem. Theorem Let X be a smooth projective variety over the field of complex numbers C such that K X is nef. Then the Albanese map α : X A(X) is surjective and has connected fibers. Proposition Let X be a Kähler manifold with nef T X. Then the Albanese map α is a submersion onto the Albanese torus A(X) and the relative tangent bundle to α is nef. The fibres are connected and have nef tangent bundles. Proof. d(α) = (u 1,..., u q ), and (u 1,..., u q ) is a basis of H 0 (X, O X ). If we had rank (dα(x)) < q at some point x X, then there exists a non zero linear combination u = λ 1 u λ q u q with u(x) = 0. Since T X is nef, u H 0 (X, TX ) do not have zero points, contradiction. Thus α is a submersion. Then we have the exact sequence 0 T X/A(X) T X dα α T A(X) 0 where T A(X) is trivial means that T X/A(X) is nef. It implies particularly that the fibres have nef tangent bundles. If the fibres were not connected, we can get a factorization α : X Y A(X) from their connected components, in which X Y has connected fibres and Y A(X) is a finite etale cover. Thus Y is also a complex torus, and then the universal property of the Albanese map gives a factorization α : X A(X) Y in which A(X) Y is also etale. But the fibres of X Y will not be connected, contradiction. 34

42 Some important propositions without proof. Proposition Assume that X is a Kähler n-fold such that T X is nef. If X admits a nef bundle H such that c 1 (H) n 0, then X is a projective algebraic manifold and H is ample. Proposition Let X be a Kähler manifold with nef T X. One of the two following situations is true: (1) c 1 (X) n > 0. Then X is a Fano manifold (that is a projective manifold with ample divisor K X ) and X (X, O X ) = 1. (2) c 1 (X) n = 0. Then X (X, O X ) = 0 and X contains a non-trivial holomorphic p-form of odd degree. Moreover, there exists a finite etale cover X of X such that q( X) := h 0 (X, Ω 1 X) > 0. Proposition Let X, Y be compact Kähler manifolds and let g : X Y be a smooth fibration with connected fibres. If F denotes any fibre of g, then (1) q(x) q(y ) + q(f ), (2) q(x) q(y ) + q(f ). (3) Assume that Y is a complex torus and that the fibres F have the following properties: π 1 (F ) contains an abelian subgroup of finite index, and for every finite etale cover F of F the Albanese map F A( F ) has constant rank. Then q(x) = q(y ) + q(f ). Theorem Let X be a compact Kähler manifold with nef tangent bundle T X. Let X be a finite etale cover of maximum irregularity q = q( X) = q(x). Then, we have (1) π 1 ( X) Z 2q. 35

43 (2) The Albanese map α : X A( X) is a smooth fibration over a q-dimensional torus with nef relative tangent bundle. (3) The fibres F of α are Fano manifolds with nef tangent bundles(i.e. K F is ample and T F is nef). Proof. We use the induction on n = dimx to prove this theorem. Assume that the conclusion is true in dimension < n. Then X A( X) has the property theorem(4.26, 2) by applying proposition(4.22). Particularly, these fibres F have nef tangent bundle, and π 1 (F ) contains an abelian subgroup of finite index. Then we can apply proposition (4.25, 3) to get q(f ) = 0. Thus the alternative proposition (4.24, 2) can not occur for F and thus F is a Fano manifold. Then property (3) is proved. Meanwhile, we know that Fano manifolds are simply connected by [Ko1]. The homotopy exact sequence of the Albanese fibration means that π 1 ( X) π 1 (A( X)) Z 2q. There exists a composition of finite etale covers X X X such that X is a Galois cover of X (take π 1 ( X) to be the normal subgroup of finite index in π 1 (X) obtained by taking the intersection of all conjugates of π 1 ( X)). Then π 1 ( X) Z 2q as a subgroup of finite index in π 1 (X) Z 2q, and the fundamental group π 1 (X) is an extension of a finite group by Z 2q. 36

44 4.7 Projective surfaces with nef anticanonical bundles In this subsection, we shall give a simple classification of projective surfaces over a field of characteristic 0 whose anticanonical divisors are nef. Lemma Assume that X is a terminal n-dimensional projective variety and K X is almost nef (there are at most finitely many rational curves which have negative intersection number with K X.). Then we have κ(x) 0. Besides the following three properties are equivalent. (a) κ(x) = 0. (b) K X 0. (c) K X is nef. Proof. Since K X is almost nef, if K X is not numerically equivalent to 0, it can not have any non zero global section. Then κ(x) 0. If κ(x) = 0 and if K X is not numerically equivalent to 0, then there is a non-zero section F tk X for some positive number t. Therefore, K X can not be nef. (b) implies (c) is obvious. If K X is nef, then K X D = 0 for all but finitely many curves. Then we choose some ample divisors A i on X to satisfy that K X A 1... A n 1 = 0. Therefore K X 0. Theorem Assume that X is a smooth projective surface over a field of characteristic 0 and let the anticanonical divisor K X be nef. Then X is one surface of the following surfaces in the following list. (1) X is P 2 or P 2 blown up in at most 9 points in sufficiently general position. (2) X are rational ruled surfaces (Hirzebruch surfaces) F 0, F 1, F 2. Assume that F n = P P1 (O P1 O P1 (n)). 37

45 (3) X is a K3 surface, which is defined as a surface with K X = 0 and irregularity q = 0 and p a = p g = 1. (4) X is a ruled surface over an elliptic curve C. E is the corresponding normalized locally free sheaf of rank 2 on C and X = P C (E). Here normalized sheaf is a sheaf with the property that H 0 (X, E) 0 but for all invertible sheaves G on C with degg < 0, and we have H 0 (X, E G) = 0. Assume e = dege. We have (a) If E is a decomposable sheaf which means a direct sum of two invertible sheaves, then E = O C L. degl = 0. (b) if E is an indecomposable sheaf, then e is 0 or 1. (5) X is a hyperelliptic surface. (6) X is an Abelian surface. Proof. The anticanonical divisor K X of X is nef. Let q = dim(h 1 (X, O X )) = dim(h 0 (X, Ω 1 X )). Let b m = dim(h 0 (X, O X (mk X ))). Case (1). If q = 0 and κ(x) =, then q = 0 and b 2 = 0. By Castelnuovo s Rationality Criterion, X is a rational surface. Since κ(x) =, X is a ruled surface. Let X = P C (E). If X is the blow-up of P 2 from r points, we have r 9 by KX 2 0. Hence, X must be P 2, or surfaces blown up from P 2 on 9 points in sufficiently general position and Hirzebruch surfaces F n. Case (2). If X is a Hirzebruch surface, by ruled surface theory, K X 2C 0 +(2g 2 e)f. Here, C 0 is the image of a section from C to X, such that O X (C 0 ) = O X (1). C0 2 = dege = e. g is the genus of the curve C. K X 2C 0 (2g 2 e)f is nef if and only if K X C 0 0 and K X f 0. K X C 0 = 2C (2g 2 e)c 0 f = 2e + (e + 2) 0. 38

46 Therefore e 2. K X f = 2C 0 f + (e + 2) f 2 = 2 0. Hence, X are F 0, F 1, F 2. Case (3). If q = 0 and κ(x) = 0, then K X = 0, since K X is nef. So X is a K3 surface, which is defined as a surface with K X = 0 and irregularity q = 0 and p a = p g = 1. Case (4). If q 1 and κ(x) =, X is a ruled surface. We can easily have that q 2. Assume that X is minimal. By ruled surface theory, q = g(c). Thus g(c) 1. Since K 2 X = 8(1 g(c)) 0, g(c) 1. Then we have g(c) = 1. So C is an elliptic curve and X is a ruled surface over the curve C. Assume that X = P C (E) and E is a normalized rank-2 locally free sheaf on C and e = dege. Using the similar formula as above, we can have K X 2C 0 + (2g(C) 2 e) = 2C 0 + ( e) f. K X C 0 = 2C 0 C 0 e(f C 0 ) = 2C 2 0 e 1 = 2 ( e) e = 3e 0. So e 0. K X f = 2C 0 f + ( e)f f = 2 e 0 = 2 0. (a). By elliptic ruled surface theory, if E is decomposable, E O C L, and L has the property degl 0. In this case, e 0. Also we have e 0, by the above calculation. So e = 0 and degl = 0. (b). If E is indecomposable sheaf, by ruled surface theory, e has the property that 2g(C) e 2g(C) 2. Since g(c) = 1, we can have e = 0, 1, 2. If X = P C (E) 39