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nLab > Latest Changes: generalized Calabi-Yau manifold

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 27th 2012
    • (edited May 27th 2012)

    stub for generalized Calabi-Yau manifold, (for the moment just to record some references)

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeMay 27th 2012
    • (edited May 27th 2012)

    I do not like the terminology. It is like calling a functor among nn-categories generalized functor, while it is just THE appropriate notion of the functor in that context. On the other hand, there are some generalizations of Calabi-Yau manifolds within usual varieties (or, sometimes, within orbifolds/algebraic stacks) which better deserve the name (as they indeed generalize the notion, within more or less the standard context, rather than transfer it mutatis mutandis to a new context). Kontsevich and Soibelman have one notion like that (which they also call generalized CY manifolds) within smooth compact complex varieties where one requires CY-like behaviour for certain Hodge numbers and some cohomological condition.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 27th 2012

    I don’t even like the whole “generalized”-terminology of generalized complex geometry that this comes from, since it is so very unspecific and so ignorant of all the other sensible generalizatons that there are.

    Unfortunately, however, it has become very much standard terminology. Therefore I’d opt for keeping the entry names this way and maybe add a discussion of terminology problems in the entry.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMay 27th 2012
    • (edited May 27th 2012)

    Maybe it is standard in some community. The algebraic geometry community, which has the main contributions to the study of Calabi-Yau has other preferences for what is generalized Calabi-Yau; similarly with some other “generalized” notions.

    I consider the terminology “generalized complex geometry” entirely standard (if “complex” is included, absolutely not otherwise), but not assuming that prefixing generalized to any notion in geometry makes it referring toward there. Instead one can postfix the terminology by “in generalized complex geometry” and then it is clear to anybody what is meant (like saying “functor among weak nn-categories”). So why not adopting “Calabi-Yau manifold in generalized complex geometry” ? I mean what will we do when generalized Calabi-Yau of algebraic geometry start more massively appearing on the net ? Ignore and not making an entry, just because a more shallow notion in other field has been introduced ?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2012

    I agree, renaming the entry to “Calabi-Yau in generalized complex geometry” might be a good idea. I can do so later (will have to fight the Cache bug once I do, no leisure to do so right now).

    Meanwhile, I have added a note on the definition of “Calabi-Yau in generalized complex geometry” in terms of reduction of the structure group.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2020
    • (edited Dec 18th 2020)

    added pointer to

    for the relation to (non-integrable) G-structure for G=G = SU(n).

    And added cross-link with MSU.

    diff, v7, current

1 to 6 of 6

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