Opened 3 years ago

Last modified 3 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-linear map*R**J*_{x}: T_{x}M -> T_{x}M*J*depends smoothly on_{x}*x**J*where_{x}J_{x}= -Id*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*is diagonalisable (with eigenvalues

^{C}*+/-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*.

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