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nLab > Latest Changes: homotopy of rational maps

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2021
    • (edited Jul 19th 2021)

    starting something, on the kind of theorems originating with

    Nothing to be seen here yet, but I need to save. (Am not sold on the entry title, except that “topology” is not really the right term here.)

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2021

    added statement (here) of Segal’s original theorem from 1979.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2021

    added a remark (here) highlighting the similarity/difference to the statement of the homotopical Oka principle for non-compact domain surfaces.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 19th 2021
    • (edited Jul 19th 2021)

    added a brief remark cross-linking with twistor string theory (here) – to be expanded

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2021

    added statement of the analogous theorem (here) for maps between projective spaces, from

    diff, v5, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2021

    I have added the qualifier “regular” to “rational map”, but still need to say this better – the terminology here is confusing.

    diff, v8, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2021

    I have added more words to the Idea-section and a remark on the regularity of the rational functions.

    diff, v9, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2021
    • (edited Jul 29th 2021)

    I have added a remark (here) that the space of rational maps ℂP 1→ℂP n\mathbb{C}P^1 \to \mathbb{C}P^n that is considered in Segal’s theorem is also considered in Gromov-Witten theory (after compactification and quotienting), as is nicely explicit in Bertram 02, p. 9.

    This confluence looks like it ought to have drawn attention, but I don’t find literature in this direction.

    diff, v9, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 1st 2021

    re #8:

    I see that it is this connection which the preprint

    • David Ayala, Homological Stability among Moduli Spaces of Holomorphic Curves in Complex Projective Space (arXiv:0811.2274)

    was after, before the author discovered the mistake highlighted in v2.

    Mistake or not, that’s the right question to ask. But it looks like it wasn’t followed up.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2021

    added pointer to:

    • Jacob Mostovoy, Spaces of Rational Loops on a Real Projective Space, Transactions of the American Mathematical Society, Vol. 353, No. 5 (May, 2001), pp. 1959-1970 (jstor:221802)

    diff, v16, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2021
    • (edited Aug 15th 2021)

    added (here) brief statement of the main theorem from

    identifying the full homotopy type of the space of pointed rational maps ℂP 1→ℂP n\mathbb{C}P^1 \to \mathbb{C}P^n with that of a configuration space of points. (Am making a similar addition now to this latter entry, too.)

    diff, v18, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2021

    Fixed the statement of Segal’s theorem (here), now pointing to n-equivalence (following Segal’s note on terminology on p. 44)

    diff, v19, current

1 to 12 of 12

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