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 *λ* = *a* + i*b* ∈ \(\mathbb C\) by
*λx*: = *a**x* + *b**Jx*; In other words,
it is as if we defined the imaginary unit by i*x*:= *Jx*, etc.

$ __Def__: A complex structure on a real
even-dimensional vector space *V* is a map *J*: *V* →
*V*, with *J*^{2} = −I.

**Almost Complex Manifold**

$ __Weakly almost complex__: An *n*-manifold *M*
such that [T*M*]−[*M* × \(\mathbb R\)^{n}]
is the image of the map *K*_{C}(*M*)
→ *K*_{R}(*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, \(\mathbb C\))-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* \(\circ\) *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__:
The dimension *n* is even, *M* is orientable.

@ __References__: Fernández de Córdoba & Isidro a1505 [generalised complex geometry, and thermodynamical fluctuation theory].

**And Other Structures**
> s.a. symplectic structure [Kähler structure].

$ __And Hilbert space__:
On a real \({\cal H}\) with (weakly non-degenerate) symplectic form Ω,
define (i) A complex structure *J*; (ii) A *new* real inner product
*s*(*x*,* y*):= −Ω(*Jx*,* y*),
or *s*(*Jx*,* y*) = *B*(*x*, *y*);
(iii) A Hermitian inner product *h*(*x*,* y*):=
*s*(*x*, *y*) + i Ω(*x*, *y*),
i.e., Ω is the imaginary part of a Hermitian inner product.

@ __Generalizations__:
Varsaie JMP(99) [on supermanifolds].

**Mathematical Examples**

* __On spheres__:
Of all S^{2n}, only
S^{2} and S^{6} have
an almost complex structure, and only S^{2} has
a complex structure [@ in Steenrod 51].

* __On projective spaces__:
\(\mathbb R\)P^{2n} does not have one, it is not
even orientable; \(\mathbb R\)P^{2n−1} is
weakly almost complex; \(\mathbb H\)P^{n} is not even
weakly almost complex for *n* ≥ 2; \(\mathbb H\)P^{1}
is w.a.c.; \(\tilde G_{n,k}\): & P Sankaran.

**Examples and Applications**
> s.a. extensions of general relativity;
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 *t*^{a},
then a complex structure which is compatible with the symplectic one is *J*:=
\(-(-{\cal L}_t{}^2)^{-1/2}{\cal L}_t\), acting on solutions of the field equations.

* __On the Klein-Gordon phase space__:
The general prescription translates into *J*(*φ*,* π*):=
(|Δ|^{−1/2}*π*,
−|Δ|^{1/2} *φ*).

* __On the Maxwell phase space__:
The general prescription translates into
*J*(*A*_{i},* E*_{i}):=
(|Δ|^{−1/2}*E*_{i},
−|Δ|^{1/2}*A*_{i}).

@ __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];
Gibbons a1111-conf
[quantum mechanics as a real theory with a multitude of complex structures];
Aste a1905;
> s.a. Complex Numbers; geometric quantization.

@ __For field theory__: Gibbons & Pohle NPB(93)gq
[on space of solutions in curved background, and quantization];
Much & Oeckl a1812
[for Klein-Gordon theory in globally hyperbolic spacetimes].

**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 in(13)-a0710 [geometry of moving planes].

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