Opened 7 years ago
Last modified 10 months ago
#18786 new enhancement
Implementation of Almost-Complex structures for manifolds through Hodge structures
Reported by: | bpillet | Owned by: | bpillet |
---|---|---|---|
Priority: | major | Milestone: | sage-6.8 |
Component: | geometry | Keywords: | Almost-complex, Hodge_structure, differential, geometry, manifolds |
Cc: | egourgoulhon | Merged in: | |
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | #18528 | Stopgaps: |
Description
/!\ This ticket is under construction
This ticket is about enhancing 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.
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 :
- Milne, Introduction to Shimura varieties.
- Daniel Huybrechts, Complex Geometry.
- Claire Voisin, Hodge structures.
Change History (3)
comment:1 Changed 7 years ago by
- Cc egourgoulhon added
comment:2 Changed 10 months ago by
comment:3 Changed 10 months ago by
To that end, it might also be interesting to establish a connection between complex manifolds and their corresponding real structure.
May I ask what the status on this ticket is?