Opened 7 years ago

# Implementation of Almost-Complex structures for manifolds through Hodge structures

Reported by: Owned by: Basile Pillet Basile Pillet major sage-6.8 geometry Almost-complex, Hodge_structure, differential, geometry, manifolds Eric Gourgoulhon N/A #18528

# /!\ 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.