#18786: Implementation of Almost-Complex structures for manifolds through Hodge
structures
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Reporter: bpillet | Owner: bpillet
Type: | Status: new
enhancement | Milestone: sage-6.8
Priority: major | Keywords: Almost-complex,
Component: geometry | Hodge_structure, differential, geometry,
Merged in: | manifolds
Reviewers: | Authors:
Work issues: | Report Upstream: N/A
Commit: | Branch:
Stopgaps: | Dependencies: #18528
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= /!\ This ticket is under construction =
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This ticket is about enhancing [ticket:18528 SageManifold] toward complex
geometry. It deals mainly with implementation of almost-complex structures
on real differentiable manifolds.
This ticket only expresses my own point of view on the subject but I hope
it will spark a fruitful discussion on the question. Moreover I only deal
with mathematics here but any comment related to actual implementation is
very welcome.
== Content ==
* Some definitions
* Almost-complex structure
* Splitting of the tangent space
* Hodge structure of weight ''m''
* Why Hodge structures ?
* Heritage on tensors
* Other uses
* How to encode Hodge structure
* Sub-modules
* Filtrations
* Representations of '''''S'''''
== Some definitions ==
=== Almost-Complex structure ===
Let ''M'' be a real smooth manifold of even dimension ''2n'' and ''TM'' be
its tangent bundle. An almost-complex structure on ''M'' is the datum of
an anti-idempotent endomorphism of the tangent bundle of ''M''. That is :
* For all point ''x'' in ''M'' a '''''R'''''-linear map ''J,,x,, : T,,x,,
M -> T,,x,, M''
* ''J,,x,,'' depends smoothly on ''x''
* ''J,,x,, J,,x,, = -Id'' where ''Id'' is the identity endomorphism on
''T,,x,, M''
The manifold ''M'' together with ''J'' is called ''almost-complex
manifold''.
Example : On the tangent space to '''''C''''' seen as the manifold
'''''R'''''^2^, the multiplication by ''i = sqrt(-1)'' is an almost-
complex struture.
{{{#!comment
=== From Complex Structure ===
Given a complex manifold ''X'', ''X'' is locally isomorphic to a product
of copies of '''''C''''' so there is a natural action of ''i'' on its real
tangent space. Asking the change of charts of ''X'' to be holomorphic
amounts to the ''i'' gluing together and yielding an almost-complex
structure on the real tangent bundle.
Hence complex manifolds yield almost-complex manifolds. The converse is
false in general (There is some examples of almost-complex structures on
the sphere ''S^6^'' (related to octonions) which comes from no actual
complex structure).
}}}
=== What happens to the tangent space ? ===
Let call ''T'' the tangent bundle and ''T^C^'' its complexification (we
consider complex linear combinations of tangent vectors to ''M''). Then
the endomorphism ''J'' extended to ''T^C^'' is diagonalisable (with
eigenvalues ''+/-i'') and induces a splitting
''T^C^ = T^1,0^ + T^0,1^''
...
== Why Hodge structures ? ==
* Heritage on tensors
* Other use of Hodge structures
== How to encode Hodge structure ==
* Submodules
* Filtration
* Representation of **S**
== References : ==
1. Milne, ''Introduction to Shimura varieties''.
1. Daniel Huybrechts, ''Complex Geometry''.
1. Claire Voisin, ''Hodge structures''.
--
Ticket URL: <http://trac.sagemath.org/ticket/18786>
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