Opened 2 years ago

Last modified 2 years 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 Jx : Tx M -> Tx M
    • Jx depends smoothly on x
    • Jx Jx = -Id where Id is the identity endomorphism on Tx 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 TC its complexification (we consider complex linear combinations of tangent vectors to M). Then the endomorphism J extended to TC is diagonalisable (with eigenvalues +/-i) and induces a splitting

TC = T1,0 + T0,1

...

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.

Change History (1)

comment:1 Changed 2 years ago by egourgoulhon

  • Cc egourgoulhon added
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