In https://mathoverflow.net/a/44885/4177, Tilman points out that one can use a $K3$ surface minus the zeroes of a generic vector field to build a nullcobordism for $24[SU(2)]$. Given that a) this is a purely topological notion and b) we can write down defining equations for smooth projective varieties that are $K3$s, it may well be easy for experts to write down an explicit and, ideally, highly symmetric vector field on eg the Fermat quartic with 24 zeroes.

More generally: what's an example, algebraic or otherwise, of such a vector field on an explicitly given $K3$ surface?

  • $\begingroup$ Nice question. Sorry to pick a nit, but what does "affine K3" mean? $\endgroup$ – potentially dense Oct 15 '15 at 8:40
  • $\begingroup$ Erg, I meant projective variety, not affine! $\endgroup$ – David Roberts Oct 15 '15 at 9:45
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    $\begingroup$ Dear David, a smooth projective surface minus finitely many points cannot be affine, because functions which are regular outside a subset of codimension 2 are regular everywhere. (Sorry to keep being annoying.) $\endgroup$ – potentially dense Oct 15 '15 at 13:12
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    $\begingroup$ Are there $K3$s with enough algebraic vector fields to make this happen? $\endgroup$ – Allen Knutson Oct 15 '15 at 19:28
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    $\begingroup$ K3s have no (nontrivial and global) algebraic (or even analytic) vector fields, see page 15 of math.uni-bonn.de/people/huybrech/K3Global.pdf .The best I imagine you are going to get is something like the answer given by Danny Ruberman. $\endgroup$ – mkemeny Oct 17 '15 at 21:56

This is probably not as explicit as what you requested, but there are recipes for how to build a Morse function on any smooth hypersurface in $CP^3$ with no index 1 or 3 critical points. There will be exactly 24 critical points, and any gradient vector field for this Morse function will be of the sort you asked about.

Such Morse functions are due to Rudolph (Topology 14,301-303. 1975), Mandlebaum-Moishezon (Topology 14, 23-40. 1976) and Harer (Math Annalen 238, 51-158, 1978). The first two use a Morse function coming from the distance (squared) of the hypersurface to a point in the complement, in the fashion of the Morse theory proof of the Lefschetz hyperplane theorem (cf. Andreotti-Frankel, Annals of Math 69 713–717, 1959, Bott, Michigan Math Journal 6 (3): 211–216, 1959, or Milnor's book on Morse theory). The latter uses a Lefschetz pencil. In principle, all of these could be converted to explicit formulas, but I don't know how, or if anyone has written this down.

You can find pictures of the resulting handlebody structure, with only index 0,2 and 4 handles (so exactly 24 of them) in Harer-Kas-Kirby (Mem. AMS 1986). You can find similar pictures for all elliptic surfaces in the book of Gompf-Stipsicz.

  • $\begingroup$ This is fine for a generic K3, but what if we pick a really nice explicit example? $\endgroup$ – David Roberts Oct 15 '15 at 20:51
  • $\begingroup$ The only thought I had before I wrote the above answer was to look at a K3 obtained by the Kummer construction as a Z/2 quotient of the 4-torus with singularities resolved. It's pretty easy to describe vector fields on the torus, and my notion was to look for one that's equivariant and then think about what happens near fixed points. But I haven't given this any serious thought. $\endgroup$ – Danny Ruberman Oct 16 '15 at 1:02
  • $\begingroup$ What about the Fermat quartic :-) $\endgroup$ – David Roberts Oct 16 '15 at 5:21

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