[ToricVarieties] Support computation of basis of H^(2,2) and H^4

[HereAround]

cc @lkastner

Motivated from F-theory applications.

(You might find this entertaining @apturner @emikelsons .)

[apturner]

At a glance, it looks like this is only computing the vertical part of the middle cohomology, is that right? Of course the others will be trickier, but I’m wondering based on the name of you are planning to compute the horizontal part as well.

[codecov]

Codecov Report

Attention: Patch coverage is 91.46341% with 7 lines in your changes missing coverage. Please review.

Project coverage is 84.61%. Comparing base (852b6d1) to head (7f67902).
Report is 10 commits behind head on master.

Files with missing lines	Patch %	Lines
...al/FTheoryTools/src/G4Fluxes/special_attributes.jl	92.85%	5 Missing ⚠️
...cVarieties/CohomologyClasses/special_attributes.jl	83.33%	2 Missing ⚠️

Additional details and impacted files

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##           master    #4243   +/-   ##
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  Coverage   84.60%   84.61%           
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Files with missing lines	Coverage Δ
...cVarieties/CohomologyClasses/special_attributes.jl	97.64% <83.33%> (-2.36%)	⬇️
...al/FTheoryTools/src/G4Fluxes/special_attributes.jl	92.85% <92.85%> (ø)

... and 4 files with indirect coverage changes

[HereAround]

At a glance, it looks like this is only computing the vertical part of the middle cohomology, is that right? Of course the others will be trickier, but I’m wondering based on the name of you are planning to compute the horizontal part as well.

I only want to get the vertical part.

[apturner]

Glad to see this is working now. Thanks @HereAround, looks good!

[HereAround]

@lkastner and @apturner I will merge this PR now, as I need this so that I can proceed with the draft of a paper on the FTheoryTools. If you have feedback, please let me know, and I will be happy to update/improve the methods.

Please note that I placed the method basis_of_h22 in experimental (FTheoryTools), as it is relevant for and inspired by the related physics discussions. Still, from a math/algorithm perspective, it achieves the exact same (for simplicial, complete toric varieties) as basis_of_h4. As such, it would also seem reasonable to have both methods be called basis_of_h4, both be placed in the toric part of OSCAR, and call them with an optional "algorithm = ..." argument. The default algorithm could be the current method basis_of_h4, which relies on monomial_basis. Let me know your thoughts.

In case you wonder @lkastner , the algorithm with monomial_basis is painfully slow for big toric varieties, as it triggers a Groebner basis computation. One F-theory application that we are currently working on involves a simplicial, complete toric variety, whose cohomology ring is the quotient of a multivariate polynomial ring with 310 indeterminates and the ideal divided out is generated by about 47k generators. In this case, a GB computation is hopeless, which triggered the other algorithm basis_of_h22. For this involved variety, basis_of_h22 completes in about 10 seconds on my computer.

(@wdecker Based on our conversation, you might find this entertaining. In particular, you were right - this problem could be cast into a linear equation system.)