CHERN FORMS OF POSITIVE VECTOR BUNDLES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Dincer GULER ***** The Ohio State University 2006 Dissertation Committee: Approved by Prof. Fangyang Zheng, Advisor Prof. Jeffery McNeal Advisor Prof. Andrzej Derdzinski Graduate Program in Mathematics ABSTRACT In this thesis we consider the problem posed by Griffiths, which asks to determine which polynomials of Chern forms will be always positive for any Griffiths positive vector bundle E. Following an idea used in the work of Yau and Zheng, we explore the induced metric on the projectivized bundle, the Grothendieck equation and the push forward of forms and we are able to prove that the signed Segre forms are always positive. ii Dedicated to my wife and my parents iii ACKNOWLEDGMENTS I would like to thank my advisor, Professor Fangyang Zheng, for his patience, guidance and constant encouragement which made this work possible. I am grateful for the enlightening discussions and suggestions as well as the intellectual support he has provided. I would like to thank Professor Jeffery McNeal for his enlightening discussions on L2 methods in Complex Differential Geometry that has given me new insights in the theory. I would like to thank Professor Andrzej Derdzinski for his invaluable time and professional insight. I am thankful for the Department of Mathematics for their support expressed through several teaching and research fellowships that I enjoyed. Last but not the least I wish to thank my family and my wife for their constant support and encouragement. iv VITA 1975 . Born in Razgrad, Bulgaria 1998 . B.Sc. in Mathematics, Middle East Technical University, TURKEY 1998-Present . Graduate Teaching Associate, The Ohio State University PUBLICATIONS Guler, Dincer; Zheng, Fangyang On Ricci rank of Cartan-Hadamard manifolds, Internat. J. Math. 13 (2002), no. 6, 557–578 FIELDS OF STUDY Major Field: Mathematics Specialization: Complex Differential Geometry v TABLE OF CONTENTS Abstract . ii Dedication . ii Acknowledgments . iv Vita......................................... v CHAPTER PAGE 1 Introduction . 1 2 Chern Forms, Segre Forms and Schur Polynomials . 8 2.1 Preliminaries . 8 2.2 Chern and Segre Forms . 10 2.3 Positive Polynomials . 13 3 Positive Differential Forms, Push Forward and Positive Vector Bundles 21 3.1 Positive Differential Forms . 21 3.2 Push Forward of Differential Forms . 24 3.3 Positive Vector Bundles . 26 4 Positivity of the Segre Forms . 33 4.1 Proof of the Main Theorem . 33 4.2 Semipositivity . 40 4.3 Some Remarks For Further Research . 43 Bibliography . 44 vi CHAPTER 1 INTRODUCTION In complex geometry the concept of positivity has played a central role. Many peo- ple contributed to the development of it, most noticably Grauert, Hartshorne and Griffiths. In Griffiths paper [G] there were five notions of positivity, all closely related. In the end it seems that two of them become the most important. One is the ampleness of holomorphic vector bundles over complex projective varieties, coined by Hartshorne. For a line bundle E over Xn, one of the many equivalent definitions of E being ample is that there exists an integer m and a holomorphic embedding f : X → PN of X ∗ ⊗m N N into a complex projective space such that f OP (1) = E , where OP (1) is the hyperplane line bundle. When the rank of E is greater than one, one may consider −1 ∗ the projectivized bundle π : P(E) → X, where for any x ∈ X, π (x) = P(Ex) is the space of the hyperplanes in the fibre Ex. Following Grothendieck and Hartshorne, ∗ ∗ denote by OP(E)(1) the dual of the tautological line subbundle of π E , then E is defined to be ample (over X) if OP(E)(1) is ample over P(E). OP(E)(1) is also called the associate (hyperplane) line bundle over P(E). 1 The other positivity notion for E is the so called Griffiths positive: If E admits a Hermitian metric h whose curvature is Griffiths positive. To be precise, that means 1,0 that for any x ∈ X and any 0 6= v ∈ Ex and any 0 6= V ∈ Tx X, the directional curvature r ¯ X k ¯ Θvv¯(V, V ) = viv¯jhk¯jΘi (V, V ) i,j,k=1 P is positive, where r is the rank of E, v = viei, hi¯j = h(ei, ej), with {e1, ..., er} k being a local frame of E near x and (Θj ) the curvature of E under this frame, which is an r × r matrix of (1, 1) forms. In other words, the curvature Θvv¯ is a positive ¯ (1, 1) form. Note that Θvv¯(V, V ) is also called in some literature Griffiths curvature or bisectional curvature . For line bundles, these two positivity notions coincide. One side is clear, the N standard metric on OP (1) has Griffiths positive curvature, so if E is ample, the ⊗m ∗ N pull back metric on E = f OP (1) naturally induces a Hermitian metric on E (by taking the m-th root) which is still positively curved. So ample implies Griffiths positive for line bundles. The other direction is known as the Kodaira embedding theorem, a fundamental result in complex geometry. For vector bundles of rank higher than one, Griffiths positivity implies ampleness. This is because any Hermitian metric h on E naturally induces a Hermitian metric ˆ ˆ h on OP(E)(1) (see section 3.3 for details), and h has Griffiths positive curvature if h does. The other direction is still an open question: Is any ample vector bundle Griffiths positive? The only known case is when the base manifold is a curve, in this case the answer 2 is yes, proved by Unemura in [U]. In general case E being ample means that the line bundle OP(E)(1) admits a positively curved metric g. However g in general may not be the induced metric hˆ from a Hermitian metric h on E. Instead g only gives a so called Finsler metric on E. The complex Finsler geometry is still largely uncharted territory, and the answer to the above question remained illusive, despite the general belief that the answer should be an affirmative one. One of the most fundamental properties associated to the positivity notion of E is the positivity of its characteristic classes/forms. Let ck(E) denote the k-th Chern class of E. It is a cohomology class in Hk,k(E)∩H2k(X, Z). When E is equipped with a Hermitian metric h, the curvature form gives rise to the Chern forms Ck(E), which are real global (k, k) forms on X representing the k-th cohomology classes ck(E). Let P(c1, ..., cr) be the polynomial ring over Q of the Chern classes of E and denote by Pk the Q-vector space of weighted homogeneous polynomials of degree k, with weight assignement deg(ci) = i. In [G] Griffiths raised the question ”What kind of polynomials in Pk will be always positive for any ample vector bundle E over X?”, where both the rank of E and the dimension of X are no less than k. Here a (k, k) cohomoly class is considered positive if its integral over any k dimensional subvariety of X is positive. In fact this question was raised at both the cohomology class level and the form level. At the cohomology level, this question was answered by the celebrated work of Fulton and Lazarsfeld in [FL], expanding the earlier result of Bloch-Gieseker [BG]. To describe their result let us first fix some notations. Let E be a holomorphic vector 3 bundle of rank r over a projective manifold X of dimension n. Denote by Λ(k) the set of all partitions of k by nonnegative integers. Thus an element λ ∈ Λ(k) is of the form λ = (λ1, ..., λk) where X k ≥ λ1 ≥ λ1 ≥ ... ≥ λk ≥ 0, λj = k Each λ ∈ Λ(k) gives rise to a Schur polynomial Pλ of degree k defined as the k × k determinant c c . c λ1 λ1+1 λ1+k−1 c c . c λ2−1 λ2 λ2+k−2 Pλ = . ... .. c c . c λk−k+1 λk−k+2 λk where by convention c0 = 1 and ci = 0 if i does not belong to [0, k]. It turns out that {Pλ : λ ∈ Λ(k)} forms a basis of the vector space Pk, the weighted homogeneous polynomials of degree k of the Chern classes of E. So any polynomial of the Chern P classes in Pk can be expressed as P = aλPλ where each aλ ∈ Q. For k = 2 the two 2 Schur polynomials are c2 and c1 − c2. For k = 3 there are three Schur polynimials c3, 3 c1c2 − c3 and c1 − 2c1c2 + c3. For k = 4, there are 5 Schur polynomials: 2 2 2 4 2 2 c4, c1c3 − c4, c2 − c1c3, c1c2 − c1c3 − c2 + c4 , c1 − 3c1c2 + 2c1c3 + c2 − c4 corresponding to the partitions (4, 0, 0, 0), (3, 1, 0, 0), (2, 2, 0, 0), (2, 1, 1, 0) and (1, 1, 1, 1) k respecyively. Note that for λ = (1, 1, ..., 1), pλ is just (−1) sk(E), where sk is the Segre class of E determined by the inverse Chern classes, namely (1 + s1(E) + s2(E) + ...)(1 + c1(E) + c2(E) + ...) = 1 4 for any vector bundle E. Recall that a cohomology class α ∈ Hk,k(X) is said to be positive if Z α > 0 Z for any k-dimensional subvariety Z of X. 1.0.1. Theorem([FL]): If E is ample then for any k ≤ min{n, r}, Pλ is positive for P each λ ∈ Λ(k).