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Nov 25, 2016 at 8:31 comment added Will Sawin @VivekShende I doubt I will be able to say much. In my example, $K$ is numerically trivial on $X$, but the necessary conditions are purely numerical, so I don't know how to distinguish $K$ numerically trivial from $K$ trivial.
Nov 24, 2016 at 22:04 comment added Vivek Shende Do your necessary conditions shed any light on the question of whether this can happen when X is calabi-yau, in the sense of having trivial canonical bundle?
Nov 23, 2016 at 16:36 history edited Will Sawin CC BY-SA 3.0
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Nov 22, 2016 at 23:52 vote accept Vivek Shende
Nov 22, 2016 at 20:48 history answered Will Sawin CC BY-SA 3.0