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Ampleness and acyclicity are related; for example, large tensor powers of an ample sheaf are acyclic and the converse holds for line bundles. The tensor product of two ample sheaves is ample; this provokes the question of whether the tensor product of two acyclic sheaves is acyclic. If this is false, are there modest additional assumptions that could be required of the sheaves in the product that guarantee the resulting sheaf is acyclic?

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No. For example, on projective space $\mathbb{P}^k$ the line bundles $\mathcal{O}(-1),\ldots, \mathcal{O}(-k)$ are acyclic, but the line bundles $\mathcal{O}(-n)$ with $n>k$ are not. It is still possible for "negative" (or anti-ample) bundles to be acyclic, and their high tensor powers are not acyclic (as they have top cohomology by Serre duality).

To get a result you'll need to actually assume some sort of positivity condition instead of just cohomology vanishing. I'm not sure what the best known results are.

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