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

Tomoki Yoshida

TL;DR

The paper investigates when a Gromov–Yomdin-type formula for categorical entropy holds for autoequivalences of derived categories of Hilbert schemes of points on surfaces and for hyperkähler and Enriques manifolds. Using equivariant category methods and the Bridgeland–King–Reid/Haiman framework, it proves that positive categorical entropy on a surface $S$ propagates to all Hilb$^{n}(S)$, while failing to satisfy the GY property transfers as well. It then constructs explicit non-standard autoequivalences on hyperkähler manifolds (via a $\mathbb{P}^{n}$-twist composed with tensoring by a line bundle) that have $h_{\mathrm{cat}}>0$ but whose action on cohomology is unipotent, hence $\log\rho(\Phi^{N})=0$, giving counterexamples to (GY) in the HK setting; a parallel descent argument shows Enriques manifolds also lack (GY). Overall, the work reveals rich categorical dynamics beyond topological entropy and delineates the limitations of a universal GY formula in higher-dimensional geometries.

Abstract

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.

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

TL;DR

The paper investigates when a Gromov–Yomdin-type formula for categorical entropy holds for autoequivalences of derived categories of Hilbert schemes of points on surfaces and for hyperkähler and Enriques manifolds. Using equivariant category methods and the Bridgeland–King–Reid/Haiman framework, it proves that positive categorical entropy on a surface propagates to all Hilb, while failing to satisfy the GY property transfers as well. It then constructs explicit non-standard autoequivalences on hyperkähler manifolds (via a -twist composed with tensoring by a line bundle) that have but whose action on cohomology is unipotent, hence , giving counterexamples to (GY) in the HK setting; a parallel descent argument shows Enriques manifolds also lack (GY). Overall, the work reveals rich categorical dynamics beyond topological entropy and delineates the limitations of a universal GY formula in higher-dimensional geometries.

Abstract

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 . We say that has the Gromov-Yomdin (GY) property if this formula holds. We prove that if a surface fails to satisfy the (GY) property (e.g., K3 surfaces), then so does . 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.
Paper Structure (10 sections, 27 theorems, 66 equations, 1 table)

This paper contains 10 sections, 27 theorems, 66 equations, 1 table.

Table of Contents

  1. Introduction
  2. Dynamical System on Derived Categories
  3. Results
  4. Entropies of Exact Endofunctors
  5. Non-Standard Autoequivalences of Derived Categories
  6. Categorical Entropies on Hilbert Schemes of points on surfaces
  7. Equivariant Categories
  8. Hilbert Schemes of points on Surfaces
  9. Categorical Entropy on Hyperkähler Manifolds
  10. Enriques Manifolds

Key Result

Theorem 1

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $f$ be a surjective endomorphism of $X$. Then, the following equality holds: where $\rho(-)$ is the spectral radius of the induced linear map on $\oplus_{p}H^{p,p}(X, \mathbb{Z})$.

Theorems & Definitions (52)

  • Theorem 1: gromov_1987_entropy_homology_and_semialgebraic_geometrygromov_2003_on_the_entropy_of_holomorphic_mapsyomdin_1987_volume_growth_and_entropy
  • Theorem A: \ref{['theorem: positivity inherits to hilbert scheme', 'theorem: GY property for Hilb']}
  • Definition 2
  • Theorem B: \ref{['theorem: GY property of hyperkahler manifold']}
  • Remark
  • Theorem C: \ref{['theorem: pos entr and counteg of GY for Enriques manifolds']}
  • Definition 1.1
  • Theorem 1.2: orlov_2009_remarks_on_generators_and_dimensions_of_triangulated_categories
  • Definition 1.3
  • Theorem 1.4: dimitrov_haiden_katzarkov_kontsevich_2014_dynamical_systems_and_categories
  • ...and 42 more