Categorical Entropies of Hilbert Schemes of Points on Surfaces and Hyperk\"ahler Manifolds

Abstract

This paper studies the categorical entropy of autoequivalences of derived categories of Hilbert schemes of points on surfaces and hyperk\"ahler manifolds. One of the central questions about categorical entropy is whether it satisfies a Gromov--Yomdin type formula $h_{\mathrm{cat}}(\Phi) = \log\rho(\Phi)$. We say that $X$ has the Gromov--Yomdin (GY) property if this formula holds. We prove that if a surface $S$ with $q(S)=0$ fails to satisfy the (GY) property (e.g., K3 surfaces), then so does $\mathrm{Hilb}^n(S)$. We further prove that no projective hyperk\"ahler or Enriques manifold satisfies the (GY) property by constructing an explicit autoequivalence with positive categorical entropy but unipotent action on the cohomology ring. In fact, this discrepancy is arbitrarily large: categorical entropy is unbounded among autoequivalences with spectral radius one.