Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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?