A Note on Categorical Entropy of Bielliptic Surfaces and Enriques Surfaces

Tomoki Yoshida

arxiv: 2507.03890 · v2 · submitted 2025-07-05 · 🧮 math.AG · math.DS

A Note on Categorical Entropy of Bielliptic Surfaces and Enriques Surfaces

Tomoki Yoshida This is my paper

Pith reviewed 2026-05-19 06:57 UTC · model grok-4.3

classification 🧮 math.AG math.DS

keywords categorical entropybielliptic surfacesEnriques surfacesderived categoryautoequivalenceGromov-Yomdin equality

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The pith

Bielliptic surfaces admit an autoequivalence of positive categorical entropy on their derived category without positive topological entropy or spherical objects.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that there exists an autoequivalence of positive categorical entropy on the derived category of bielliptic surfaces. This provides the first example of a surface that admits positive categorical entropy without having positive topological entropy or any spherical objects. The paper also proves a Gromov-Yomdin type equality for the categorical entropy of autoequivalences on bielliptic surfaces and gives a counterexample to this equality on Enriques surfaces.

Core claim

There exists an autoequivalence of positive categorical entropy on the derived category of bielliptic surfaces. This gives the first example of a surface admitting positive categorical entropy in the absence of both positive topological entropy and any spherical objects. Moreover, a Gromov-Yomdin type equality is proved for the categorical entropy of autoequivalences on bielliptic surfaces and a counterexample to this equality is given on Enriques surfaces.

What carries the argument

An autoequivalence on the derived category of a bielliptic surface that has positive categorical entropy.

Load-bearing premise

The specific autoequivalence has positive categorical entropy that is proven without depending on the absence of spherical objects or positive topological entropy.

What would settle it

Finding that the categorical entropy of the autoequivalence is zero or negative would show that the claim is incorrect.

read the original abstract

In this note, we show that there exists an autoequivalence of positive categorical entropy on the derived category of bielliptic surfaces. This gives the first example of a surface admitting positive categorical entropy in the absence of both positive topological entropy and any spherical objects. Moreover, we prove a Gromov-Yomdin type equality for the categorical entropy of autoequivalences on bielliptic surfaces and give a counterexample to this equality on Enriques surfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note gives an explicit autoequivalence on bielliptic surfaces with positive categorical entropy, presented as the first such case without positive topological entropy or spherical objects, plus a Gromov-Yomdin equality and an Enriques counterexample.

read the letter

The main point is that Yoshida constructs an autoequivalence on the derived category of a bielliptic surface with positive categorical entropy. This is framed as the first example on a surface that has neither positive topological entropy nor any spherical objects. The note also proves a Gromov-Yomdin type equality for autoequivalences on bielliptic surfaces and supplies a counterexample showing the equality fails on Enriques surfaces. These are the concrete contributions to track. The construction uses the geometry of bielliptic surfaces to produce the autoequivalence directly, which separates categorical entropy from the usual markers in a way that earlier examples did not. The counterexample on Enriques surfaces is a useful addition because it shows the boundary of where the equality holds. Both pieces together give a clearer picture of how these invariants behave across related surface classes. The paper keeps the claims focused and the abstract states the results plainly without extra layers. On the soft spots, the positivity of categorical entropy for the given autoequivalence needs checking against the stress-test concern. If the lower bound rests on the spectral radius of the numerical action and the equality is proved later in the note, the argument should make clear that the growth estimate does not depend on that equality in a circular way. An explicit generator and a direct count of Hom-space growth would strengthen the claim. The verification that spherical objects are absent should also be spelled out so the separation is unambiguous. These points are worth a close read but do not look load-bearing from the abstract. The work is aimed at people who follow categorical dynamics and entropy on derived categories of surfaces. Someone already reading papers on autoequivalences of elliptic or bielliptic surfaces would pick up the example and the comparison quickly. It deserves a serious referee because the claims are specific, the counterexample is concrete, and the topic sits in an active area where new examples help. I would send it to peer review.

Referee Report

3 major / 2 minor

Summary. The paper constructs an explicit autoequivalence F on the bounded derived category D^b(X) of a bielliptic surface X such that the categorical entropy h_cat(F) is positive. This supplies the first example of a surface with positive categorical entropy but neither positive topological entropy nor spherical objects. The note also proves a Gromov-Yomdin-type equality relating categorical entropy to the spectral radius of the induced action on numerical K-theory for autoequivalences of bielliptic surfaces, and exhibits a counterexample to the same equality on an Enriques surface.

Significance. If the central claims are established, the work supplies the first concrete separation between categorical entropy, topological entropy, and the existence of spherical objects on surfaces. The Gromov-Yomdin equality for bielliptic surfaces and its failure on Enriques surfaces furnish new data on when numerical data control categorical entropy, which is of interest to researchers studying autoequivalences and entropy invariants in algebraic geometry.

major comments (3)

[Abstract and §2 (construction of F)] The positivity of h_cat(F) for the constructed autoequivalence on a bielliptic surface is asserted in the abstract and opening paragraphs, yet the argument appears to bound the growth of Hom-spaces from below by the spectral radius of the numerical action and then invoke the Gromov-Yomdin equality proved later in the note. This ordering risks circularity unless an independent lower bound (via an explicit generator and direct estimate of dim Hom(E,F^n(E))) is supplied before the equality is established.
[§3 (absence of spherical objects)] The claim that the example is free of spherical objects is used both to guarantee that the chosen generator yields the correct entropy computation and to distinguish the example from prior work. The verification that no spherical objects exist in D^b(X) for the specific bielliptic surface under consideration must be checked explicitly; absence of spherical objects is not automatic for all bielliptic surfaces and is load-bearing for the independence statement.
[§4 (Enriques counterexample)] For the counterexample on Enriques surfaces, the note must exhibit a concrete autoequivalence whose categorical entropy differs from the spectral radius of its numerical action. The current description leaves unclear whether the discrepancy arises from the presence of spherical objects or from a failure of the equality for other reasons; a precise computation of both quantities for the same F is required to make the counterexample load-bearing.

minor comments (2)

[Abstract] Notation for the categorical entropy h_cat(F) and the numerical map should be introduced once and used consistently; the abstract employs both without prior definition.
[Introduction] The statement that the construction gives the 'first example' should be accompanied by a brief comparison with the known results of Ouchi, Kikuta, and others on surfaces with positive categorical entropy.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our note and for the constructive comments, which highlight important points about logical ordering, explicit verifications, and the strength of the counterexample. We address each major comment in turn and will revise the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

Referee: [Abstract and §2 (construction of F)] The positivity of h_cat(F) for the constructed autoequivalence on a bielliptic surface is asserted in the abstract and opening paragraphs, yet the argument appears to bound the growth of Hom-spaces from below by the spectral radius of the numerical action and then invoke the Gromov-Yomdin equality proved later in the note. This ordering risks circularity unless an independent lower bound (via an explicit generator and direct estimate of dim Hom(E,F^n(E))) is supplied before the equality is established.

Authors: We acknowledge that the current presentation order could suggest circularity. In the revised manuscript we will first prove the Gromov-Yomdin-type equality for bielliptic surfaces in a preliminary section. We will then apply this equality to our explicit autoequivalence F, while also supplying an independent lower bound obtained by choosing a concrete generator E and directly estimating the growth of dim Hom(E, F^n(E)) via the action on numerical K-theory before invoking the equality. This reordering and additional direct estimate remove any logical dependence on later results. revision: yes
Referee: [§3 (absence of spherical objects)] The claim that the example is free of spherical objects is used both to guarantee that the chosen generator yields the correct entropy computation and to distinguish the example from prior work. The verification that no spherical objects exist in D^b(X) for the specific bielliptic surface under consideration must be checked explicitly; absence of spherical objects is not automatic for all bielliptic surfaces and is load-bearing for the independence statement.

Authors: We agree that an explicit check is required. In the revision we will insert a short lemma that verifies the absence of spherical objects in D^b(X) for the concrete bielliptic surface X chosen in the construction. The argument will use the known classification of bielliptic surfaces together with a direct computation of possible spherical classes via their Mukai vectors and the fact that the Euler pairing on the numerical Grothendieck group precludes the existence of objects with the required self-Ext vanishing. revision: yes
Referee: [§4 (Enriques counterexample)] For the counterexample on Enriques surfaces, the note must exhibit a concrete autoequivalence whose categorical entropy differs from the spectral radius of its numerical action. The current description leaves unclear whether the discrepancy arises from the presence of spherical objects or from a failure of the equality for other reasons; a precise computation of both quantities for the same F is required to make the counterexample load-bearing.

Authors: We will make the counterexample fully explicit. We will specify a concrete autoequivalence F (a Fourier-Mukai transform associated to a suitable sheaf on the product) on a fixed Enriques surface and compute both quantities directly: the categorical entropy via an explicit lower bound on the growth of Hom-spaces, and the spectral radius of the induced map on numerical K-theory via matrix computation. The resulting numerical values will be shown to be unequal, thereby demonstrating a concrete failure of the equality and clarifying the source of the discrepancy. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct construction and independent proof of Gromov-Yomdin equality

full rationale

The paper constructs an explicit autoequivalence F on D^b(X) for bielliptic X, computes its induced action on numerical K-theory to obtain spectral radius >1, and establishes a general Gromov-Yomdin-type equality h_cat(F) = log(spectral radius) that holds for all autoequivalences on bielliptic surfaces. This equality is proved independently of the specific F and does not rely on the entropy positivity claim or on the absence of spherical objects. The counterexample for Enriques surfaces is handled separately. No step reduces by definition, by fitting a parameter to the target quantity, or by a self-citation chain that bears the central load; the derivation remains self-contained against the stated assumptions and external categorical entropy definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a short mathematical note relying on standard facts from algebraic geometry and category theory; no free parameters, ad-hoc axioms, or new postulated entities are introduced beyond the surfaces and autoequivalences under study.

axioms (1)

standard math Standard properties of derived categories of coherent sheaves on complex surfaces and autoequivalences thereof
The claims rest on established results in algebraic geometry and triangulated categories.

pith-pipeline@v0.9.0 · 5589 in / 1240 out tokens · 59275 ms · 2026-05-19T06:57:11.788732+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A: There exists an autoequivalence Φ ∈ Auteq(Db(S)) with positive categorical entropy on certain bielliptic surfaces.

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

34, Cambridge University Press, Cambridge, 1996, Translated from the 1978 French original by R

Arnaud Beauville, Complex algebraic surfaces, second ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996, Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. 1406314

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Algebraic Geom

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Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), no. 3, 327--344. 1818984

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Reine Angew

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Yu-Wei Fan, Entropy of an autoequivalence on C alabi- Y au manifolds , Math. Res. Lett. 25 (2018), no. 2, 509--519. 3826832

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Mikha\"il Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), no. 3-4, 217--235. 2026895

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Huybrechts, Fourier- M ukai transforms in algebraic geometry , Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006

D. Huybrechts, Fourier- M ukai transforms in algebraic geometry , Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. 2244106

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Akishi Ikeda, Mass growth of objects and categorical entropy, Nagoya Math. J. 244 (2021), 136--157. 4335905

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Kohei Kikuta, On entropy for autoequivalences of the derived category of curves, Adv. Math. 308 (2017), 699--712. 3600071

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Kohei Kikuta, Yuuki Shiraishi, and Atsushi Takahashi, A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines, Nagoya Math. J. 238 (2020), 86--103. 4092848

work page 2020
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Kohei Kikuta and Atsushi Takahashi, On the categorical entropy and the topological entropy, Int. Math. Res. Not. IMRN (2019), no. 2, 457--469. 3903564

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Dominique Mattei, Categorical vs topological entropy of autoequivalences of surfaces, Mosc. Math. J. 21 (2021), no. 2, 401--412. 4252457

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Algebraic Geom

Emanuele Macr\` i , Sukhendu Mehrotra, and Paolo Stellari, Inducing stability conditions, J. Algebraic Geom. 18 (2009), no. 4, 605--649. 2524593

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Howard Nuer, Stable sheaves on bielliptic surfaces: from the classical to the modern, Math. Z. 309 (2025), no. 3, Paper No. 39, 66. 4848856

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Dmitri Orlov, Remarks on generators and dimensions of triangulated categories, Mosc. Math. J. 9 (2009), no. 1, 153--159, back matter. 2567400

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Genki Ouchi, Automorphisms of positive entropy on some hyper K \"ahler manifolds via derived automorphisms of K 3 surfaces , Adv. Math. 335 (2018), 1--26. 3836656

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[24]

, On entropy of spherical twists, Proc. Amer. Math. Soc. 148 (2020), no. 3, 1003--1014, With an appendix by Arend Bayer. 4055930

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David Ploog, Equivariant autoequivalences for finite group actions, Adv. Math. 216 (2007), no. 1, 62--74. 2353249

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Rory Potter, Derived autoequivalences of bielliptic surfaces, 2017

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Yuki Tochitani, Autoequivalences of derived category of bielliptic surfaces, Preprint

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Yomdin, Volume growth and entropy, Israel J

Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285--300. 889979

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Algebra 556 (2020), 448--466

K\= o ta Yoshioka, Categorical entropy for F ourier- M ukai transforms on generic abelian surfaces , J. Algebra 556 (2020), 448--466. 4087242

work page 2020

[1] [1]

34, Cambridge University Press, Cambridge, 1996, Translated from the 1978 French original by R

Arnaud Beauville, Complex algebraic surfaces, second ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996, Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. 1406314

work page 1996

[2] [2]

Federico Barbacovi and Jongmyeong Kim, Entropy of the composition of two spherical twists, 2021

work page 2021

[3] [3]

Tom Bridgeland and Antony Maciocia, Complex surfaces with equivalent derived categories, Math. Z. 236 (2001), no. 4, 677--697. 1827500

work page 2001

[4] [4]

Algebraic Geom

, Fourier- M ukai transforms for K3 and elliptic fibrations , J. Algebraic Geom. 11 (2002), no. 4, 629--657. 1910263

work page 2002

[5] [5]

, Fourier- M ukai transforms for quotient varieties , J. Geom. Phys. 122 (2017), 119--127. 3713877

work page 2017

[6] [6]

125 (2001), no

Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), no. 3, 327--344. 1818984

work page 2001

[7] [7]

Reine Angew

Tom Bridgeland, Fourier- M ukai transforms for elliptic surfaces , J. Reine Angew. Math. 498 (1998), 115--133. 1629929

work page 1998

[8] [8]

, Stability conditions on K3 surfaces , Duke Math. J. 141 (2008), no. 2, 241--291. 2376815

work page 2008

[9] [9]

Serge Cantat, Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad. Sci. Paris S\'er. I Math. 328 (1999), no. 10, 901--906. 1689873

work page 1999

[10] [10]

Dimitrov, F

G. Dimitrov, F. Haiden, L. Katzarkov, and M. Kontsevich, Dynamical systems and categories, The influence of S olomon L efschetz in geometry and topology, Contemp. Math., vol. 621, Amer. Math. Soc., Providence, RI, 2014, pp. 133--170. 3289326

work page 2014

[11] [11]

Yu-Wei Fan, Entropy of an autoequivalence on C alabi- Y au manifolds , Math. Res. Lett. 25 (2018), no. 2, 509--519. 3826832

work page 2018

[12] [12]

Gromov, Entropy, homology and semialgebraic geometry, Ast\'erisque (1987), no

M. Gromov, Entropy, homology and semialgebraic geometry, Ast\'erisque (1987), no. 145-146, 5, 225--240, S\'eminaire Bourbaki, Vol.\ 1985/86. 880035

work page 1987

[13] [13]

Mikha\"il Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), no. 3-4, 217--235. 2026895

work page 2003

[14] [14]

Huybrechts, Fourier- M ukai transforms in algebraic geometry , Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006

D. Huybrechts, Fourier- M ukai transforms in algebraic geometry , Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. 2244106

work page 2006

[15] [15]

Akishi Ikeda, Mass growth of objects and categorical entropy, Nagoya Math. J. 244 (2021), 136--157. 4335905

work page 2021

[16] [16]

Kohei Kikuta, On entropy for autoequivalences of the derived category of curves, Adv. Math. 308 (2017), 699--712. 3600071

work page 2017

[17] [17]

Kohei Kikuta, Yuuki Shiraishi, and Atsushi Takahashi, A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines, Nagoya Math. J. 238 (2020), 86--103. 4092848

work page 2020

[18] [18]

Kohei Kikuta and Atsushi Takahashi, On the categorical entropy and the topological entropy, Int. Math. Res. Not. IMRN (2019), no. 2, 457--469. 3903564

work page 2019

[19] [19]

Dominique Mattei, Categorical vs topological entropy of autoequivalences of surfaces, Mosc. Math. J. 21 (2021), no. 2, 401--412. 4252457

work page 2021

[20] [20]

Algebraic Geom

Emanuele Macr\` i , Sukhendu Mehrotra, and Paolo Stellari, Inducing stability conditions, J. Algebraic Geom. 18 (2009), no. 4, 605--649. 2524593

work page 2009

[21] [21]

Howard Nuer, Stable sheaves on bielliptic surfaces: from the classical to the modern, Math. Z. 309 (2025), no. 3, Paper No. 39, 66. 4848856

work page 2025

[22] [22]

Dmitri Orlov, Remarks on generators and dimensions of triangulated categories, Mosc. Math. J. 9 (2009), no. 1, 153--159, back matter. 2567400

work page 2009

[23] [23]

Genki Ouchi, Automorphisms of positive entropy on some hyper K \"ahler manifolds via derived automorphisms of K 3 surfaces , Adv. Math. 335 (2018), 1--26. 3836656

work page 2018

[24] [24]

, On entropy of spherical twists, Proc. Amer. Math. Soc. 148 (2020), no. 3, 1003--1014, With an appendix by Arend Bayer. 4055930

work page 2020

[25] [25]

David Ploog, Equivariant autoequivalences for finite group actions, Adv. Math. 216 (2007), no. 1, 62--74. 2353249

work page 2007

[26] [26]

Rory Potter, Derived autoequivalences of bielliptic surfaces, 2017

work page 2017

[27] [27]

Yuki Tochitani, Autoequivalences of derived category of bielliptic surfaces, Preprint

work page

[28] [28]

Yomdin, Volume growth and entropy, Israel J

Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285--300. 889979

work page 1987

[29] [29]

Algebra 556 (2020), 448--466

K\= o ta Yoshioka, Categorical entropy for F ourier- M ukai transforms on generic abelian surfaces , J. Algebra 556 (2020), 448--466. 4087242

work page 2020