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]; > s.a. Complex
Numbers; geometric quantization.

@ __For field theory__: Gibbons & Pohle NPB(93)gq [on space of solutions in curved background, and quantization].

**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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9 nov 2016