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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2020

    starting something

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2020

    I gather that (in the complex-analytic case, hence in characteristic zero) the total number of degenerate points of an elliptic fibration, counted with multiplicity, is generally the Euler number. Is that right?

    And that, accordingly, the “24” in the number-counted-with-multiplicity of degenerate fibers of an elliptic fibration of a K3-surface “is” the 24 which is the Euler number of the topological manifold underlying K3. Is that right?

    Looking around, I see that this is what it does seem to say in Schütt-Shioda 09, Section 6.7, around Theorem 6.10. But the statement there is a little terse.

    They attribute the statement to a Prop. 5.16 in Cossec-Dolgachev 1989, but I haven’t yet managed to identify such a proposition in that book. (Which page is it??)

    v1, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2020
    • (edited Dec 4th 2020)

    I found a few more pertinent references and added them. If anyone has better suggestions on #2, please let me know.

    There is a striking analogy between:

    • the algebraic K3 with its 24 singular elliptic fibers (counted with mulitplicities)

    • the differential geometric K3 with its 24 vanishing loci of a smooth vector field (counted with multiplicities).

    Both 24s here get identified with the Euler number and hence with each other. But is there a more substantial relation?

    Removing 24 isolated singular points from the differential geometric K3 makes it a framed cobordism which witnesses the order of the third stable homotopy group of spheres (as indicated there).

    Does removing the 24 singular elliptic fibers from the algebraic K3 make it an algebraic cobordism of some relevance?

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