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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2012
    • (edited Feb 27th 2012)

    prompted by this G+ post by David Roberts, I have started an entry arithmetic Chow group .

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 27th 2012

    So is a general search for differential cohomological situations likely to yield new examples of cohesion? How about this?

    We report on a joint project with Mike Hopkins and introduce Hodge filtered complex cobordism groups for complex manifolds. These groups combine integral topological and complex geometric information in analogy to differential cohomology groups for smooth manifolds. By taking the Hodge filtration into account these groups form a natural generalization of Deligne cohomology and become especially interesting for Kahler manifolds.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2012

    So is a general search for differential cohomological situations likely to yield new examples of cohesion?

    In this generality I have to reply like this: just as I cannot prevent anyone from calling “cohomology” something that is not computed by homs in an oo-topos, so I cannot prevent anyone from calling “differential cohomology” something that is not computed in a cohesive oo-topos. But in both cases I expect that a “good” definition will make this true.

    But more concretely, in the case that you point to:

    establishing cohesion for complex geometry should be pretty straightforward. If I had a single second to spare (which I don’t), I would try to work that out in more detail. Alternatively, that would be the kind of topic I might hand to a student (success on immediate questions guaranteed, with a good supply of arbitrarily advanced questions already visible).