ON A CLASS OF ALGEBRAIC SURFACES WITH NUMERICALLY EFFECTIVE COTANGENT BUNDLES A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Hongyuan Wang, B.S., M.A., M.A.S. ***** The Ohio State University 2006 Doctoral Examination Committee: Fangyang Zheng, Advisor Approved by Zbigniew Fiedorowicz Andrzej J. Derdzinski Advisor Department of Mathematics ABSTRACT In this dissertation, we will present two new results in complex geometry. The first one is for a borderline class of surfaces with nef cotangent bundle. The statement of the main theorem on this topic is as follows, suppose M is a general type surface, 2 0 such that the two Chern numbers satisfy c1 = c2, and P ic(M)/Pic (M) is generated by KM , then every nontrivial extension of the holomorphic tangent bundle TM by KM will be Hermitian flat. As a corollary, we obtain that every surface with the 1 above described properties and with h (T K∗ ) > 0 will be a generalized theta M ⊗ M divisor. The second result is an extended version of Hard Lefschetz theorem for the cohomology with values in Nakano semipositive vector bundles on a compact K¨ahler manifolds. ii ACKNOWLEDGMENTS I am indebted to my advisor, Prof. Fangyang Zheng, for his patience, expert guidance and constant encouragement which made this work possible. I am grateful for the numerous enlightening discussions and suggestions as well as the intellectual support he has provided. I would like to thank my committee members Prof. Andrzej J. Derdzinski and Prof. Zbigniew Fiedorowicz for their invaluable time and professional insight. Special thanks go to their inspiring classes, Ph.D candidate general test and enlightening discussions. I would like to thank Prof. Herbert Clemens and his students for the group study and invaluable discussions. I am thankful for the Department of Mathematics for their support expressed through special graduate assignments and fellowships that I enjoyed. Finally, I wish to express my deep gratitude to my family, especially my wife, for all their love, understanding and support. Special gratitude goes to my older uncle for his education to me through the years, without him this degree will be impossible. iii VITA October, 1978 ......................... Born - Wuxi, Jiangsu, P.R.China. 1993-1998 ............................. B.S.,Department of Special Class of Gifted Young, University of Science and Tech- nology of China. Hefei Anhui, China. 2002 - present ......................... Graduate Teaching and Research Asso- ciate, Department of Mathematics, The Ohio State University. FIELDS OF STUDY Major field: Mathematics Specialization: Complex Geometry, Compact Complex Surfaces iv TABLE OF CONTENTS Abstract....................................... ii Acknowledgments.................................. iii Vita ......................................... iv ListofFigures ................................... vi CHAPTER PAGE 1 Introduction................................. 1 2 Stabilityofvectorbundles ......................... 11 2.1 Coherentsheaftheory.................... 11 2.2 Semistable and stable sheaves . 18 3 Positivity of Hermitian vector bundles . .. 22 3.1 Curvatureofvectorbundles. 22 3.2 Nakano positivity, Griffiths positivity and Numerically ef- fectiveness .......................... 27 3.3 Amplevectorbundles .................... 32 4 Inequalities for Chern classes . 37 5 MaintheoremspartI............................ 40 6 MaintheoremspartII ........................... 60 Bibliography .................................... 69 v LIST OF FIGURES FIGURE PAGE 1.1 TheGeographyofChernNumbers . 5 vi CHAPTER 1 INTRODUCTION A major goal in complex geometry is to classify compact complex manifolds in a given dimension up to birational equivalence classes. In early 19th century, Castel- nuovo, Enriques and many others had succeeded in creating an impressive geomet- ric theory of birational classification for smooth algebraic surfaces. It was discov- ered that main birational invariants for an algebraic surface M are the first Betti 1 number b1(M) = h (M, C) and the plurigenera Pn(M) of M, which is defined by 0 n Pn(M)= h (M,KM⊗ ). After sheaf theory had been developed, and applied by Hodge, Serre, Hirzebruch, Grothendieck and others to analytic and algebraic geometry, the classification ques- tions got a decisive progress since 1950’s. In 1960’s and 1970’s Kodaira and others carried out the classification project for projective surfaces, and also extended it to 1 cover all compact complex surfaces. It is now referred to in the literature as Enrique- Kodaira classification theory. Given a compact complex surface M, there are four possibilities for its plurigenera: 1), all Pn(M) vanish; 2), not all Pn(M) vanish, but all are either 0 or 1; 3), P (M) n; n ∼ 4), P (M) n2. n ∼ Nowadays this fact is expressed by saying that the Kodaira dimension kod(M) of M is respectively , 0, 1, or 2. The algebraic surfaces in class 1) are formed by all −∞ the birationally ruled surfaces, namely, those M that is birational to P1 C, for any × compact complex curve C. The non-algebraic surfaces in class 1) are so called V II0- class, which are defined as compact complex surfaces with first Betti number b1 =1 and all Pn = 0. For instance, all Hopf surfaces (i.e. compact complex surface with universal cover C2 0 ) belong to this set. There are other examples that belongs to \{ } this set as well. However, an exhaustive list is yet to be established. For case 2), there are four types of K¨ahlerian surfaces distinguished by their first Betti numbers and plurigenera, namely the birational classes of tori, hyperelliptic surfaces, K-3 surfaces and Enriques surfaces. The non-K¨ahler ones in case 2) consists of Kodaira surfaces 2 which are all affine quotients of C2. The case 3) are all elliptic surfaces, i.e. there is a smooth curve B and a surjective morphism p : M B whose generic fibre is an → ellpitic curve. Kodaira had studied all possible types of singular fibres, there are 7 of them, and gave a complete classification of this type of surface. Finally, surfaces in class 4), are simply called surfaces of general type. This is of course means 99.99% of the surfaces, and not much is known about this class except some very basic results. It is known that, for any surfaces with non-negative Kodaira dimension, there is a unique surface in each birational equivalent class that is minimal, i.e. can not be blown down any further. This surface is called the minimal model of the birational class. By Gieseker’s thoerem [Gies], p236, we know that there exists a quasi-projective coarse module scheme for minimal surfaces of general type with fixed Chern numbers 2 c1 and c2. However, little is known about the structure of the Gieseker scheme in 2 general. Even the very basic questions such as, for which pair (c2,c1) is the Gieseker 2 scheme non-empty, are not completely answered. The known restrictions on (c2,c1) are expressed by 1.1. Theorem If M is any minimal surface of general type, then 1), c2(M)+ c (M) 0(12), 1 2 ≡ 3 2 2), c1(M) > 0 and c2(M) > 0, 3), c2(M) 3c (M), 1 ≤ 2 4), 5c2(M) c (M)+36 0 (if c2(M) is even), 5c2(M) c (M)+30 0 (if c2(M) 1 − 2 ≥ 1 1 − 2 ≥ 1 is odd). 2 Let c1 = n and c2 = m. This theorem tells us that no integer pair (m, n) with n> 3m or 5n m + 36 < 0, can be the two Chern numbers of any minimal general − type surface. In light of this, the question now is following: If D Z2 := (m, n) m Z, n Z is given by the inequalities 0 < n 3m, ⊂ { | ∈ ∈ } ≤ m 5n + 36 and 12 (n + m), then for which pair (m, n) D, does there exist a ≤ | ∈ 2 minimal surface of general type M with c1 = n and c2 = m? One can divide D into two parts D := (m, n) D n 2m and D := 1 { ∈ | ≤ } 2 (m, n) D n> 2m . D is the region where τ = 1 p = 1 (c2 2c ) 0. Here τ is { ∈ | } 1 3 1 3 1 − 2 ≤ the signature of M and p1 is the first Pontryagin number. Similarly, D2 is the region for those M with positive signature. The simple examples such as, the complete intersections or the theta divisors, which are the smooth ample divisors of the three dimensional abelian varieties, usually lie in the region D1. In particular, the theta 2 divisors have c1 = c2, so they are on the line n = m. Indeed for a long time only 4 very few examples with Chern pairs in D2 were known. The geography of the Chern numbers is plot as Figure 1.1. 2 n =3m n =2m n = m n = c1 D2 D D1 5n m + 36 0 − ≥ m = c2 Figure 1.1: The Geography of Chern Numbers Although we can easily find lots of pairs in D, but it is difficult to show that all pairs in D can be represented. The combined efforts of Persson ([Pers]) for region D1 and Chen ([Che1] [Che2]) for the regions D1 and D2 gave rise to the following results 5 1.2. Theorem Given any pair (m, n) D, there exists a minimal general type surface ∈ 2 M with c1 = n and c2 = m, except maybe for points on the finitely many lines n 3m +4k = 0 with 0 k 347. In fact, for at most finitely many exceptions − ≤ ≤ on these lines all pairs in D1 occurs as the Chern numbers of minimal general type surfaces. This thoerem does not give surfaces on the borderline n = 3m. By Yau’s deep theorem [Yau1] [Yau2] on K¨ahler-Einstein metrics 1.3. Theorem (Yau) Let M be a compact K¨ahler manifold of dimension n, (a) if c > 0 (resp. c < 0), i.e. the (1, 1) cohomology class c (resp. c ) is 1 1 1 − 1 represented by some positive (1, 1)-form.