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 xV and λ = a + ib ∈ \(\mathbb C\) by λx: = 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: VV, with J2 = −I.

Almost Complex Manifold
$ Weakly almost complex: An n-manifold M such that [TM]−[M × \(\mathbb R\)n] 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, \(\mathbb C\))-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 S2n, only S2 and S6 have an almost complex structure, and only S2 has a complex structure [@ in Steenrod 51].
* On projective spaces: \(\mathbb R\)P2n does not have one, it is not even orientable; \(\mathbb R\)P2n−1 is weakly almost complex; \(\mathbb H\)Pn is not even weakly almost complex for n ≥ 2; \(\mathbb H\)P1 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 ta, 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|−1/2π, −|Δ|1/2 φ).
* On the Maxwell phase space: The general prescription translates into J(Ai, Ei):= (|Δ|−1/2Ei, −|Δ|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]; 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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