#9326: Add cohomology of toric varieties
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Reporter: vbraun | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.5
Component: algebraic geometry | Keywords:
Author: Volker Braun | Upstream: N/A
Reviewer: Andrey Novoseltsev | Merged:
Work_issues: |
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Old description:
> This patch adds a cohomology ring to toric varieties. This is then used
> for characteristic classes.
>
> I'm using the usual description of the rational cohomology ring via the
> Stanley-Reisner ideal of the fan. This does not work for singular
> varities with worse than orbifold singularities. I don't have any good
> candidate for the "space where the Chern classes live" in that case.
>
> Prerequisite: #9245: Add library of toric varieties.
New description:
This patch adds a cohomology ring to toric varieties. This is then used
for characteristic classes.
I'm using the usual description of the rational cohomology ring via the
Stanley-Reisner ideal of the fan. This does not work for singular varities
with worse than orbifold singularities. I don't have any good candidate
for the "space where the Chern classes live" in that case.
Prerequisites:
* #9245: Add library of toric varieties.
* #9429: Undesirable behaviour when deriving from `QuotientRingElement`
--
Comment(by vbraun):
As for the cones, once I get to the Chow group it would also be nice if
`cone.A()` would return the Chow cycle associated to the cone. And a cone
should be able to return the orbit closure as a toric variety. Really, in
toric geometry there is lots of stuff associated to a cone and it would be
nice if we could write `cone.stuff()`.
As for the cohomology ring, I did the name changes. The new patch also
defines classes `CohomologyRing(QuotientRing)` and
`CohomologyClass(QuotientRingElement)`, so we have something to attach
`exp()` and `truncate_to_degree()` to.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9326#comment:6>
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