[ToricVarieties] Introduce Chern classes of tangent bundle

[HereAround]

This PR is inspired by developments in FTheoryTools. It aims to provide functionality to compute the Chern classes of the tangent bundle of a simplicial and complete toric variety.

[HereAround]

@lkastner As of right now, I am not happy with the performance of this method.

I have a particular toric variety in mind for which i would like to be able to compute the 4-th Chern class of the tangent bundle. As of right now, this seems almost hopeless with the current implementation. Suggestions/advice appreciated.

Sadly, I cannot attach the corresponding .mrdi file here as github says "We don't support that file type". But I will of course be happy to share said file e.g. via slack with anyone interested.

[codecov]

Codecov Report

Attention: Patch coverage is 85.36585% with 6 lines in your changes missing coverage. Please review.

Project coverage is 84.62%. Comparing base (38cc0a9) to head (3889afd).
Report is 9 commits behind head on master.

Files	Patch %	Lines
...ies/NormalToricVarieties/standard_constructions.jl	0.00%	6 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #3996      +/-   ##
==========================================
+ Coverage   84.56%   84.62%   +0.05%     
==========================================
  Files         597      600       +3     
  Lines       82190    82596     +406     
==========================================
+ Hits        69506    69894     +388     
- Misses      12684    12702      +18     

Files	Coverage Δ
...ental/IntersectionTheory/src/IntersectionTheory.jl	100.00% <ø> (ø)
...cVarieties/CohomologyClasses/special_attributes.jl	100.00% <100.00%> (ø)
...ies/NormalToricVarieties/standard_constructions.jl	95.10% <0.00%> (-4.17%)	⬇️

... and 28 files with indirect coverage changes

[lgoettgens]

I have some comments about inferability and thus making the code a bit faster. Please have a look and replace the placeholder types by the actual ones (if possible)

[HereAround]

I have some comments about inferability and thus making the code a bit faster. Please have a look and replace the placeholder types by the actual ones (if possible)

Thank you @lgoettgens . Let me work those changes in.

[HereAround]

I have some comments about inferability and thus making the code a bit faster. Please have a look and replace the placeholder types by the actual ones (if possible)

Thank you @lgoettgens . Let me work those changes in.

Done.

[micjoswig]

As long as hirzebruch_surface can only return NormalToricVariety, this argument is redundant.

Just in case you were asking: no, it does not make things globally more consistent.

[HereAround]

I have just added simplified constructors in 3889afd. This should (hopefully) fix this issue.

[lgoettgens]

It is not redundant here, as adding a default will keep this default necessary in all of Oscar 1.x.

[HereAround]

Ok. I removed it again. To be discussed at a convenient time/setting.

[lgoettgens]

The changes in 3889afd (#3996) are now in conflict with what was decided upon in #922.

In particular, there should only be a default added for all of these, if this is discussed with people from all areas that already have (or could add) another implementation of this.

[HereAround]

The changes in 3889afd (#3996) are now in conflict with what was decided upon in #922.

In particular, there should only be a default added for all of these, if this is discussed with people from all areas that already have (or could add) another implementation of this.

Fine @lgoettgens . Removed again and to be discussed at a convenient time/setting then.

The concerning commit has been dropped.

[micjoswig]

The changes in 3889afd (#3996) are now in conflict with what was decided upon in #922.

I do not see this as a conflict, unless someone has a precise plan to implement Chern classes for arbitrary varieties or schemes. The example back then were projective spaces and projective planes which pop up everywhere (and already exits in Oscar).

It is not so easy to derive a design from a few general principles; sometimes it is good to keeping things lean.

[lgoettgens]

The changes in 3889afd (#3996) are now in conflict with what was decided upon in #922.

I do not see this as a conflict, unless someone has a precise plan to implement Chern classes for arbitrary varieties or schemes. The example back then were projective spaces and projective planes which pop up everywhere (and already exits in Oscar).

It is not so easy to derive a design from a few general principles; sometimes it is good to keeping things lean.

Since the (now again removed) commit doesn't seem to be available on github anymore, let me summarize its changes: it did not only add a shorthand for hirzebruch_surface (which would have been fine for me), but as well for affine_space, projective_space and `weighted_projective space. So it indeed was relevant for the liked issue about projective spaces.

[micjoswig]

Since the (now again removed) commit doesn't seem to be available on github anymore, let me summarize its changes: it did not only add a shorthand for hirzebruch_surface (which would have been fine for me), but as well for affine_space, projective_space and `weighted_projective space. So it indeed was relevant for the liked issue about projective spaces.

Sorry for creating a mess. My comment only refered to hirzebruch.

[HereAround]

I opened a new PR #4051 regarding the above discussion on hirzebruch_surface(NormalToricVariety, 3) and alike.