Implementation of Almost-Complex structures for manifolds through Hodge structures

[bpillet]

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/!\ This ticket is under construction

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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 R², 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 :

1. Milne, Introduction to Shimura varieties.
2. Daniel Huybrechts, Complex Geometry.
3. Claire Voisin, Hodge structures.