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Oct 16, 2015 at 5:21 comment added David Roberts What about the Fermat quartic :-)
Oct 16, 2015 at 1:02 comment added Danny Ruberman 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.
Oct 15, 2015 at 20:51 comment added David Roberts This is fine for a generic K3, but what if we pick a really nice explicit example?
Oct 15, 2015 at 12:07 history answered Danny Ruberman CC BY-SA 3.0