COMPLEX AND ALMOST-COMPLEX STRUCTURES ON SIX DIMENSIONAL MANIFOLDS A ...

88 Pages·2012·0.3 MB·English

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 Ful(cid:12)llment
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
(cid:13)
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 in(cid:13)uenced by many excellent teachers. My (cid:12)rst thanks goes to the sta(cid:11)
of the mathematics department, both past and present. Without their e(cid:11)orts I would
haveundoubtedlyspentmuchmoretimeontheadministrative aspectsofmyposition.
I would like to thank the department and university for (cid:12)nancial support to travel to
conferences and for making it possible for me to be a part of the larger mathematical
andacademiccommunities. OftheprofessorsIhavehadattheUniversity ofMissouri,
Shuguang Wang and Zhenbo Qin are among those who have most in(cid:13)uenced 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 di(cid:11)erential equations. I would like to thank Yuri
Latushkin for a number of conversations in which he stimulated me intellectually and
ii
o(cid:11)ered many encouragements and suggestions. Dan Edidin has in(cid:13)uenced 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 di(cid:11)erential 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 Geo(cid:11), 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 Fro(cid:127)licher spectral sequence . . . . . . 17
2.2 Cohomology relations for hypothetical exotic complex structures . . . 23
2.3 Fro(cid:127)licher Spectral Sequence Computations . . . . . . . . . . . . . . . 27
2.3.1 The Fro(cid:127)licher Spectral Sequence for a hypothetical complex S6 28
2.3.2 TheFro(cid:127)licherSpectralSequence forhypotheticalexoticstructures 30
iv
2.3.3 General descriptions of the terms of the Fro(cid:127)licher spectral se-
quence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Symmetric complex structures 39
3.1 Index theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Actions of C(cid:3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 di(cid:11)erential geometry of whether or not the six sphere S6 admits an
integrable almost-complex structure. An a(cid:14)rmative answer to this question would
imply the existence of many exotic complex structures on CP3. It is known that CP3
admits many topologically di(cid:11)erent almost-complex structures, but it is unknown
whether or not CP3 admits an integrable almost-complex structure other that the
standard Ka(cid:127)hler structure. In this manuscript we give lower bounds on the Hodge
numbers of hypothetical exotic structures on CP3 and a necessary condition for the
Fro(cid:127)licher 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(cid:3)-symmetric. We give restrictions on the (cid:12)xed point sets of such C(cid:3) actions.
vi
Was sich u(cid:127)berhaupt sagen la(cid:127)sst, la(cid:127)sst die Worte fassen; und wovon man
nicht reden kann, daru(cid:127)ber 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 di(cid:11)erential geometry is whether or not the six
sphere S6 admitsacomplex structure. Hirzebruch andYauhave each listed thisprob-
lem among the fundamental open problems in di(cid:11)erential 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 S2
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 (cid:12)fty 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