Implementation of Almost-Complex structures for manifolds through Hodge structures

[sagetrac-bpillet]

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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 R², the multiplication by i = sqrt(-1) is an almost-complex struture.

=== 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.
2. Daniel Huybrechts, Complex Geometry.
3. Claire Voisin, Hodge structures.

Depends on #18528

CC: @egourgoulhon

Component: geometry

Keywords: Almost-complex, Hodge_structure, differential

Issue created by migration from https://trac.sagemath.org/ticket/18786

[mjungmath]

comment:2

May I ask what the status on this ticket is?

[mjungmath]

comment:3

To that end, it might also be interesting to establish a connection between complex manifolds and their corresponding real structure.