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For questions related to divisors in the sense of algebraic geometry (Cartier divisors, Weil divisors and so on). For question on divisors in the number theoretic sense please use the tag divisors-multiples.
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Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?
Interesting question. I think the answer is yes, let me try to prove it.
As you noticed, the ideal sheaf sequence shows that $h^1(D)=0$ is equivalent to the fact that $H^0({\mathcal O}_D)$ is 1-dimen …