Complex Structures| Complex Structures |
Complex Vector Space > s.a. Polar
Decomposition; Complex Numbers; i.
* Idea: A real vector
space V with a tensor J which
effectively converts it into
a complex one, by allowing us to define the product of x 
V and
= a +
ib C by
x: = ax + bJx;
In other words, it is as if we defined the imaginary unit by ix:= Jx,
etc.
$ Def: A complex structure on a real even-dimensional vector space V is
a map J: V → V, with J2 = –I.
Almost Complex Manifold
$ Weakly almost complex:
An n-manifold M such that
[TM]–[M × Rn]
is the image of the map KC(M) → KR(M)
defined by taking the underlying real
vector bundle.
$ Almost complex: An n-manifold M with
a (smooth assignment of a) complex structure on the tangent space at each point,
or, equivalently, such that its frame
bundle
is reducible to a GL(n/2,C)-bundle.
* Necessary conditions:
The dimension n is even, M is orientable.
* Necessary and sufficient
condition:
There is a lift r: BU(n) → BO(n) of f : M →
BO(n)
to g: M → BU(n) with r 
g = f.
Complex Manifold
$ Complex structure:
An integrable almost complex structure on a manifold.
* Idea: The integrability
condition allows to introduce local complex coordinates on M such
that the transition functions between different patches are holomorphic.
* Isomorphism: A map which preserves
the complex structure of a manifold is a biholomorphic map (f and
f –1 both holomorphic).
* Necessary conditions: n even, M orientable.
And Other Structures > s.a. symplectic
structure [Kähler]; Hyperkähler.
$ Kähler structure:
A triple (M, 
, J),
with and J strongly
compatible, and J integrable
(a complex structure J is
strongly compatible with a symplectic structure if
J is
symmetric and positive
definite).
$ And Hilbert space:
On a real 
with
(weakly nondegenerate) symplectic form ,
define (i) A complex structure J; (ii) A new real
inner product s(x,y):= –(Jx, y),
or s(Jx, y)
= B(x, y);
(iii) A Hermitian
inner product h(x, y):= s(x, y)
+ i (x, y),
i.e., is
the imaginary part of a Hermitian inner product.
@ Kähler: Hashimoto et al JMP(97) [hyperkähler metrics from
Ashtekar variables]; Pedersen et al LMP(99)
[quasi-Einstein Kähler metrics]; Chen JDG(00)
[space of Kähler metrics]; Cortes m.DG/01-ln
[special Kähler
manifolds]; Ross &
Thomas JDG(06)
[contant curvature, obstructions].
@ Generalizations: Varsaie JMP(99) [on supermanifolds].
Mathematical Examples
* On spheres: Of all
S2n,
only S2 and S6 have
an almost complex structure, and only S2 has
a complex structure [@ in Steenrod].
* On projective spaces: RP2n does
not have one, it is not even orientable; RP2n–1 is
weakly almost complex; HPn is
not even weakly almost complex for n 
2;
HP1 is w.a.c.; \tilde{Gn,k}: & P
Sankaran.
Examples and Applications > s.a. phase
space; {& P Sankaran: Calgary seminar 4.10.1990}.
* Idea: For a field theory, a choice of complex structure on the
phase space is equivalent to a choice of decomposition into positive and
negative
frequency modes.
* On the phase space of
a linear field theory: If the background spacetime has a timelike Killing
vector field ta,
then a complex structure which is compatible with the symplectic one is J:= –(–
t2)–1/2t,
acting on solutions of the field equations.
* On the Klein-Gordon phase
space: The general prescription translates into J(
,
):=
(|
|–1/2,
–||1/2 ).
* On the Maxwell phase
space:
The general prescription translates into J(Ai, Ei):=
(||–1/2Ei,
–||1/2Ai).
@ General references: Esposito in(93)gq/95 [complex
spacetime with torsion]; Marshakov & Niemi MPLA(05)ht [examples,
gauge theory].
@ And quantum mechanics: Isidro IJGMP(05)ht/04 [complex
structure on phase space]; Marmo et al IJGMP(05)ht [and
classical limit]; Isidro IJMPA(06)
[complex geometry and Planck cone]; > s.a. Complex
Numbers, geometric quantization.
@ For field theory: Gibbons & Pohle NPB(93)gq [on space of solutions in curved background, and quantization].
@ Complex techniques in general relativity: Boyer & Plebanski JMP(77)
[heavens]; Esposito gq/99 [long
introduction]; Martina et al JPA(01)mp [solutions
of heavenly equation]; Esposito IJGMP(05)ht/05 [intro];
Maran gq/05 [and
reality constraints]; > s.a. BF theory.
References > s.a. Riemann
Surface.
@ General: Yano 65; in Chern 79; Flaherty 76; in Griffiths & Harris 78; Kodaira
86; in Willmore 93.
@ Complex manifolds: Newman ln; Wells 80 [analysis].
@ Generalizations: Bandelloni & Lazzarini JMP(98)
[Kodaira-Spencer
deformation theory]; Sobczyk a0710 [geometry of moving planes].
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send feedback and suggestions to bombelli at olemiss.edu – modified
15 mar 2009