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

TL;DR

This paper investigates the categorical entropy of autoequivalences in derived categories of Hilbert schemes on surfaces and hyperk"ahler manifolds, revealing that the Gromov-Yomdin property often fails in these cases.

Contribution

It proves that if a surface lacks the Gromov-Yomdin property, then its Hilbert schemes also do, and shows hyperk"ahler and Enriques manifolds do not satisfy this property by explicit construction.

Findings

Surfaces failing Gromov-Yomdin property imply their Hilbert schemes also fail.

Hyperk"ahler and Enriques manifolds do not satisfy the Gromov-Yomdin property.

Explicit autoequivalences with positive entropy but unipotent cohomology action are constructed.

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 $S$ fails to satisfy the (GY) property (e.g., K3 surfaces), then so does $\mathrm{Hilb}^n(S)$. Moreover, we show that no 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.