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Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties*

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Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties* Inventiones math. 28, 1 - t06 (1975) by Springer-Verlag 1975 Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties* Maurizio Cornalba (Pisa) and Phillip Griffiths (Cambridge, Mass.) Table of Contents wo. Introduction A. Orders of Growth on Algebraic Varieties ........... 4 w 1. Review of the Classical Theory ............. 4 w2. Generalization to Algebraic Varieties ........... 6 w3. Exhaustion Functions and K~ihler Metrics on Special Affine Varieties ....................... 9 84. Order of Growth of Analytic Sets ............ 13 w5. Order of Growth of Holomorphic Mappings; the First Main Theorem of Nevanlinna Theory ............. 15 w6. Orders of Growth of Holomorphic Vector Bundles ..... 17 B. Holomorphic Vector Bundles with Growth Conditions ..... 22 w 7. Statement of Main Results about Algebraicity of Vector Bundles ....................... 22 w 8. Statement of Main Results about Transcendental Vector Bundles ....................... 26 w 9. The Kodaira Identity and Consequences ......... 28 w 10. Applications of Nevantinna Theory ........... 31 w l 1. Proof of Theorems I and III(ii) for Affine Varieties ..... 35 w 12. Proof of Theorem [I .................. 37 w 13, Proof of Theorem ltI ................. 42 w 14. Proof of Theorem IV ................. 47 w 15. Proof of Theorem V .................. 50 C. Applications to Algebraic Geometry ............. 52 w 16. G.A.G,A. and Serre's Theorems A and B ......... 52 w 17. An ApproximationTheorem .............. 53 w 18. The Algebraic de Rham Theorem ............ 56 D. Grauert's Theorem and the Oka Principle .......... 61 w 19, Complex Structures on Vector Bundles and Holomorphic Homotopies of Complex Vector Bundles ......... 61 w20. Statement and Proof of Grauert's Theorem ........ 64 E. A Lower Bound on the Growth of Analytic Subvarieties Represent- ing a Given Homology Class ................ 69 w21. Statement of Theorem VII ............... 69 822. Proof of Theorem VII ................. 73 " This research was partially supported by NSF grant GP 38886. I Inventionesmath,, Vol. 28 2 M. Cornalba and P. Griffiths F. Discussion of Some Open Problems ............. 75 w23. Runge Problem for Analytic Sets ............ 75 w24. Extension Problem for Analytic Sets ........... 78 w25. A Characterization of Algebraic Varieties ........ 79 w26. On the Growth of Holomorphic Vector Bundles and Related Matters ....................... 80 w27. Order of Growth and Hodge Theory ........... 82 Appendix 1 : Chern Classes of Holomorphic Bundles and the Funda- mental Class of an Analytic Subvariety ............ 86 Appendix 2: Vector Bundles and Chern Classes/ll) ........ 9l Appendix 3: Solving the c5-Equation with Dependence on Parameters ................... 99 w0. Introduction a) In this pap6r we shall use function theory with growth conditions to study some questions on algebraic varieties which have analytic, but not necessarily algebraic, solutions. The main such question arises from the observation that, on a smooth affine variety A, the cohomology H2k(A, Q) is generated by fundamental classes of analytic subvarieties 1. These subvarieties cannot in general be algebraic, and one of our main discoveries is that the transcendental level of the subvariety may be related to the Hodge type of its fundamental class. We are able to prove a lower bound for this (Theorem VII), and to show that the lower bound is sharp for divisors (Theorem V). However, we could not find good upper bounds for higher codimension cycles, and so must be content to state some conjectures whose solution would complete our program (w16726 and 27). Once we began the study of cycles from a "transcendental point of view, we were soon led to the conclusion that an extremely large number of naturally posed geometric problems on affine algebraic varieties admit analytic but not necessarily algebraic solutions. To properly understand the nature of these solutions, it seems necessary to prove the Oka principle with growth conditions on such a variety z. The primary example of the Oka principle is the theorem of Grauert [4], which states that on a Stein manifold M the natural mapping (0.1) Vectho ~(M) ~ Vect,o p (M) is a bijection of sets. One of our main problems was to prove (0.1) with growth conditions in case M is an affine variety. Although unable to do this with what presumably are the correct growth conditions (except A complete proof of this fact is given in the appendices to this paper. 2 Roughly speaking, the Oka principle states that, on a Stein manifold M, an analytic problem has an analytic solution if, and only if, it has a topological solution. Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties 3 in the case of line bundles), we are able to give a new proof of Grauert's theorem based on the LZ-methods for the 0-operator and a certain lineari- zation trick (w16719 and 20), a proof which does give some growth conditions and which may suggest a better method for treating the problem (cf. w26). In addition to an estimate on the growth of cycles in a given cohomol- ogy class and proof of Grauert's theorem, a third main part of this paper is a systematic discussion of the order of growth of holomorphic vector bundles on algebraic varieties. Included are i) a characterization of alge- braic vector bundles and their cohomology (Theorems I and II), and ii) a proof of the equivalence of several definitions of growth for line bundles (Theorem III; we do not know the analogous result for vector bundles of rank > 1). Numerous examples ([11]) together with our work here and further heuristic reasoning, also given below, indicate that the finite order category is the correct one in which to prove the Oka principle for affine varieties. Moreover, the precise order of growth should, in many cases, be an integer which can be calculated by Hodge theory. It is this principle which has been our working hypothesis throughout. As a fourth part of this paper (Section C) we have in w16716-18 used the general techniques to prove some linear results in algebraic geometry. These include Serre's G.A.G.A and special cases of his Theorems A and B for locally free algebraic sheaves, Grothendieck's algebraic de Rham theorem, and an approximation theorem for sections of an algebraic vector bundle over an affine variety. A final part of our work (Section F) is devoted to formulating a few of the many open questions which arose in our study. We are able to prove special cases of all these problems and to give additional heuristic arguments for most, but the general questions eluded us, for reasons due primarily to a lack of techniques for handling global non-linear geometric problems in codimension > 1. b) We want to discuss briefly the two main techniques, the Nevanlinna theory and the LZ-estimates for the O-operator, which we have used in this paper. The latter gives a good method for handling linear problems with growth conditions. All that is really necessary beyond the techniques in Chapter IV of [20] is to put the basic estimate in in- trinsic geometric form, which is accomplished by the Kodaira identity (w9), and then to carefully construct suitable plurisubharmonic exhaustion functions and complete Kiihler metrics with "bounded curvature" on a non-compact algebraic variety (w3), and find a suitable "partition of unity" argument for the Zariski topology (w 13). The Nevanlinna theory is used to relate the order of growth of analytic sets (w4) to the growth of the defining functions, and is reviewed in w5. Due to the Cornalba-Shiffman example (cf. [5]), this theory works 1" 4 M. Cornalba and P. Griffiths well only in codimension one, and this is perhaps the basic reason why we are unable to do very much in the general case. When combined, these two techniques give, as far as they go, a very nice method for study- ing non-compact algebraic varieties. However, because of the failure of the analytic Bezout theorem, more subtle techniques seem necessary for higher codimension questions. c) To conclude this introduction, we wish to give a few remarks concerning the organization and contents of this paper, i) In w1671 and 2, we have recorded some historical and informal remarks concerning growth conditions on algebraic varieties. This area does not seem to be widely known among algebraic geometers, and it therefore seemed a good idea to establish a frame of reference for our later work. ii) In the appendices we have given a proof of the fact that, on a Stein manifold M, the even cohomology H2k(M, (~) is generated by analytic subvarieties. This result follows" formally from a) Grauert's theorem (a proof of which is given in w 19), b) standard results concerning the fundamental class of an analytic subvariety and Chern classes of holomorphic vector bundles (Appendix I), and, c) the Atiyah-Hirzebruch spectral sequence in K-theory (our reference for this is [8]). The latter is based on Bott periodicity, a deep result on the 2g-homo- topy of the unitary groups. However, all we really need is a Q-version of Bott periodicity, and it seemed a good idea to give a complete proof of this; one intended for non-topologists. Based on ideas of John Morgan and Dennis Sullivan, we have given such a proof in Appendix II. A. Orders of Growth on Algebraic Varieties w 1. Review of the Classical Theory In classical complex function theory, the concept of the order of growth is of fundamental importance. In order to establish a frame of reference for this paper, it is useful to review what for our purposes are the essential aspects of this theory: a) Associated to an entire meromorphic function f(~), there is an order function [28] TO (f, r), a convex function of log r which measures the growth off In particular, (1.t) f(~) is rational <=~ To(f, r)= O(log r).
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