COMPLEX AND ALMOST-COMPLEX STRUCTURES ON SIX DIMENSIONAL MANIFOLDS A Dissertation presented to the Faculty of the Graduate School University of Missouri-Columbia In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by JAMES RYAN BROWN Dr. Jan Segert, Dissertation Supervisor MAY 2006 c Copyright by James Ryan Brown 2006 All Rights Reserved The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled COMPLEX AND ALMOST-COMPLEX STRUCTURES ON SIX DIMENSIONAL MANIFOLDS presented by James Ryan Brown a candidate for the degree of Doctor of Philosophy and hereby certify that in their opinion it is worthy of acceptance. Jan Segert Bahram Mashhoon Dan Edidin Carmen Chicone Yuri Latushkin Acknowledgments As a student in the mathematics department at the University of Missouri-Columbia I have been influenced by many excellent teachers. My first thanks goes to the staff of the mathematics department, both past and present. Without their efforts I would have undoubtedly spent much more time on the administrative aspects of my position. I would like to thank the department and university for financial support to travel to conferences and for making it possible for me to be a part of the larger mathematical and academic communities. Of the professors I have had at the University of Missouri, Shuguang Wang and Zhenbo Qin are among those who have most influenced my development. The members of my committee have been particularly helpful over the time I have spent here. I would like to thank Bahram Mashhoon for several helpful comments on earlier work which have contributed to the writing of this manuscript. I would like to thank Carmen Chicone for introducing me to the exciting work of Kazdan and Warner during my comprehensive examination. It has been in this context that I learned about elliptic partial differential equations. I would like to thank Yuri Latushkin for a number of conversations in which he stimulated me intellectually and ii offered many encouragements and suggestions. Dan Edidin has influenced me greatly both in the several classes I have taken with him as well as the many conversations we have had. I am very appreciative for his helping me to tie together many of the ideas common to both differential geometry and algebraic geometry. I would especially like to express my appreciation to my advisor, Jan Segert. Without his guidance and careful reading of this and many other writings I would likely not have completed this degree. It was a rare conversation that I did not leave with more insight. I am deeply indebted to my family and friends for all of their moral and logistical support. There are too many to enumerate, but among the many colleagues I wish to thank are Geoff, Chris, Jared, Jaewon, Joseph, Yevgen, Krishna, Georgiy, Samar, Sabine, and Raj. I would like to thank Eskil, Lorentz, Mark, Stacie, Jordan, Zak, Scott, Jim, Chris, and David for all of their grounding conversations. Karen, Car- olyn, Eva, Charles, Aaron and Julianne have been a great help over the past several years. My mom Donna and my dad Walt have both helped more than I can express. Amanda, Shannon, and Luciane have supported me in many ways and have provided excellent examples of how to complete projects. Finally I am most grateful to HP and to my wife Ardith. Without their support and love I would certainly not have completed this manuscript. iii Contents Acknowledgments ii Abstract vi 1 Preliminaries 1 1.1 Almost-complex structures and torsion . 2 1.2 Summary of known results on S6 . 6 1.3 Almost-complex structures on CP3 . 10 1.4 Summary of known results on projective space . 13 1.5 Statement of results . 14 2 Properties of hypothetical complex structures on CP3 16 2.1 Dolbeault cohomology and the Fr¨olicher spectral sequence . 17 2.2 Cohomology relations for hypothetical exotic complex structures . 23 2.3 Fr¨olicher Spectral Sequence Computations . 27 2.3.1 The Fr¨olicher Spectral Sequence for a hypothetical complex S6 28 2.3.2 The Fr¨olicher Spectral Sequence for hypothetical exotic structures 30 iv 2.3.3 General descriptions of the terms of the Fr¨olicher spectral se- quence . 36 3 Symmetric complex structures 39 3.1 Index theorems . 40 3.2 Actions of C∗ . 42 3.3 Future directions . 53 A Octonions and almost-complex structures 55 B Index theorems 59 B.1 Characteristic Classes . 59 B.2 The Atiyah-Singer Index Theorem . 63 B.3 The Atiyah-Bott Fixed Point Formula . 65 B.4 The Atiyah-Segal G-Index Theorem . 67 Bibliography 69 Vita 79 v COMPLEX AND ALMOST-COMPLEX STRUCTURES ON SIX DIMENSIONAL MANIFOLDS James Ryan Brown Dr. Jan Segert ABSTRACT We investigate the properties of hypothetical exotic complex structures on three dimensional complex projective space CP3. This is motivated by the long stand- ing question in differential geometry of whether or not the six sphere S6 admits an integrable almost-complex structure. An affirmative answer to this question would imply the existence of many exotic complex structures on CP3. It is known that CP3 admits many topologically different almost-complex structures, but it is unknown whether or not CP3 admits an integrable almost-complex structure other that the standard K¨ahler structure. In this manuscript we give lower bounds on the Hodge numbers of hypothetical exotic structures on CP3 and a necessary condition for the Fr¨olicher spectral sequence to degenerate at the second level. We also give topolog- ical constraints on the classes of hypothetical exotic complex structures which are C∗-symmetric. We give restrictions on the fixed point sets of such C∗ actions. vi Was sich ub¨ erhaupt sagen l¨asst, l¨asst die Worte fassen; und wovon man nicht reden kann, darub¨ er mus man schweigen. -Ludwig Wittenstein, Tractatus Logico-Philosophicus What can be said at all can be said clearly; and whereof one cannot speak thereof one must be silent. -C.K. Odgen, English translation vii Chapter 1 Preliminaries A famous long-standing question in differential geometry is whether or not the six sphere S6 admits a complex structure. Hirzebruch and Yau have each listed this prob- lem among the fundamental open problems in differential geometry in [36, Problem 13] and [72, Problem 52], respectively. Borel and Serre [14] have shown that the only spheres which admit almost-complex structures are the two sphere S2 and the six sphere S6, see appendix A for constructions of almost-complex structures on S2 and S6 modeled on the sets of quaternions and octonions, respectively. By deforming a given almost-complex structure one obtains many almost-complex structures on S 2 and S6. It is well-known that S2 admits a unique complex structure. It is unknown, however, whether or not S6 admits any integrable almost-complex structure. Over the past fifty years none of the assertions of resolution of this question has withstood close scrutiny. There have been several published and unpublished manuscripts claiming to show that S6 does not admit a complex structure, e.g. [2] and [39], but none has gained acceptance among experts. On the other hand there 1 is also a recent preprint [28] which asserts that S6 is indeed a complex manifold, but this was found to have a gap in the proof of the main theorem and was quickly withdrawn by its author, leaving the question unanswered still. In addition to having no definitive answer to this existence question, there are relatively few results which give an indication of what the actual answer is. In this chapter we list and summarize the prerequisite material and previous results in this field. In section 1.1 we define complex and almost-complex structures on manifolds and give a classical condition which is both necessary and sufficient for an almost-complex structure to be complex. In section 1.2 we list some of the advances on the six sphere question. In section 1.3 we describe how this question may be reframed in the context of projective space followed by a summary of results on projective space in section 1.4. We conclude in section 1.5 with an announcement of the results to be found in this dissertation. 1.1 Almost-complex structures and torsion In this section we follow [45] and [46] to describe briefly complex and almost-complex structures on manifolds, and we state a classical theorem giving a necessary and sufficient condition for an almost-complex structure to be integrable. Recall that a smooth n-dimensional manifold X is a second countable Hausdorff space which admits a smooth atlas U ; ' , i.e., U is an open covering of X, f i ig f ig ' : U ' (U ) Rn is a homeomorphism, and ' '−1 : ' (U U ) ' (U U ) i i ! i i ⊂ i ◦ j j i \ j ! i i \ j 2 is smooth. A complex manifold X of complex dimension n is a smooth 2n-dimensional n manifold with the additional conditions 'i(Ui) is homeomorphic to a subset of C and ' '−1 : ' (U U ) ' (U U ) is holomorphic. i ◦ j j i \ j ! i i \ j We would like to study manifolds X which are not necessarily complex, but retain a particularly important feature of complex manifolds, that is, the tangent bundle T X of X is a complex vector bundle. We digress for a moment into a discussion of complex vector spaces and complex structures on real vector spaces. Definition 1.1.1. A complex structure J on a real vector space V is an endomor- phism J : V V with the property J 2 = I where I is the identity map.