# Vector field on a K3 surface with 24 zeroes

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?

• Nice question. Sorry to pick a nit, but what does "affine K3" mean? – potentially dense Oct 15 '15 at 8:40
• Erg, I meant projective variety, not affine! – David Roberts Oct 15 '15 at 9:45
• 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.) – potentially dense Oct 15 '15 at 13:12
• Are there $K3$s with enough algebraic vector fields to make this happen? – Allen Knutson Oct 15 '15 at 19:28
• 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. – 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.