Noncommutative K3 surfaces over finite fields get zeta functions whose point counts can be negative, obstruct geometricity, and in one explicit example, perfectly mimic a K3 surface without being geometric.
On Hyperk\"ahler manifolds of K3$^{[n]}$-type with large Picard number
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Inspired by well-known examples of hyperk\"ahler manifolds, we show that any hyperk\"ahler manifold $X$ of K3$^{[n]}$-type with Picard number $\rho(X) \geq 4$ is always isomorphic to a moduli space of twisted stable sheaves on a K3 surface. Additionally, we provide explicit descriptions of hyperk\"ahler manifolds of K3$^{[n]}$-type with Picard ranks below this crucial value (e.g., $\rho(X)=3$) that are not birational to such moduli spaces.
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Zeta functions of K3 categories over finite fields
Noncommutative K3 surfaces over finite fields get zeta functions whose point counts can be negative, obstruct geometricity, and in one explicit example, perfectly mimic a K3 surface without being geometric.