Categorical Entropies of Hilbert Schemes of Points on Surfaces and Hyperkähler Manifolds

This paper studies the categorical entropy of autoequivalences of derived categories of Hilbert schemes of points on surfaces and hyperkähler manifolds. One of the central questions about categorical entropy is whether it satisfies a Gromov-Yomdin type formula $h_{\mathrm{cat}}(Φ) = \logρ(Φ)$. We say that $X$ has the Gromov-Yomdin (GY) property if this formula holds. We prove that if a surface $S$ fails to satisfy the (GY) property (e.g., K3 surfaces), then so does $\mathrm{Hilb}^n(S)$. Moreover, we show that no hyperkähler or Enriques manifold satisfies the (GY) property by constructing an explicit autoequivalence with positive categorical entropy but unipotent action on the cohomology ring.