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Kikuta–Takahashi's categorical Gromov–Yomdin conjecture

Let XX be a smooth projective variety, let DbCoh⁡(X)D^b\operatorname{Coh}(X) be its bounded derived category, and let Φ:DbCoh⁡(X)→DbCoh⁡(X)\Phi: D^b\operatorname{Coh}(X) \to D^b\operatorname{Coh}(X) be an exact endofunctor. Write N(X)\mathcal{N}(X) for the numerical Grothendieck group of XX, and let [Φ][\Phi] be the induced endomorphism of N(X)\mathcal{N}(X). The categorical entropy of Φ\Phi at parameter 00 is denoted by h0(Φ)h_0(\Phi), and ρ\rho denotes spectral radius.

Kikuta–Takahashi's conjecture. One has

h0(Φ)=log⁡ρ([Φ]:N(X)→N(X)).h_0(\Phi) = \log\rho([\Phi]: \mathcal{N}(X) \to \mathcal{N}(X)).

The conjecture is a categorical analogue of the Gromov–Yomdin theorem, relating categorical entropy to the linear action on the numerical Grothendieck group. The source states that it is true in many cases but also has counterexamples, so it is refuted in the stated generality.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Kikuta–Takahashi's categorical Gromov–Yomdin conjecture

    Let XX be a smooth projective variety over C\mathbb{C}, and let ϕ∈Aut⁡(D⁡b(X))\phi\in\operatorname{Aut}(\operatorname{D}^b(X)) be an autoequivalence. Write HH∙(X)HH_\bullet(X) for the Hochschild homology of XX, let HH∙(ϕ)HH_\bullet(\phi) be the induced C\mathbb{C}-linear isomorphism, let ρ\rho denote spectral radius, and let h0h_0 denote categorical entropy at parameter 00. Kikuta–Takahashi's conjecture. For every such ϕ\phi,

    h0(ϕ)=log⁡ρ(HH∙(ϕ)).h_0(\phi)=\log\rho(HH_\bullet(\phi)).

    This is a categorical analogue of the Gromov–Yomdin theorem relating entropy to spectral radius. The paper's abstract states that the authors give a counterexample, so the conjecture is refuted.

    source: Dominique Mattei, “Categorical vs topological entropy of autoequivalences of surfaces”, arXiv:1909.02758 (2021).

Sources & referencesView supporting material

Primary source

Federico Barbacovi and Jongmyeong Kim, “On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity”, arXiv:2110.12597 (2021).

Additional references

2 papers in this index state this conjecture (2016–2021). The statement above is taken from the most recent of them; the others are arXiv:1608.05627.

categorical-dynamicscategorical-entropyderived-categoriesentropysurfaces

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