On entropy of $\mathbb{P}$-twists

TL;DR

The paper proves that the $\mathbb{P}$-twist $P_E$ associated to a $\mathbb{P}^d$-object $E$ on a smooth projective variety is not conjugate to any standard autoequivalence, by analyzing the categorical entropy function $h_t$ and showing a nontrivial splitting of shifting numbers, $\tau^{+}(P_E^k)\neq\tau^{-}(P_E^k)$ for $k\neq0$. It develops a framework using shifting numbers and hearts to compute $h_t$ without requiring the often difficult condition $E^{\perp}\neq\{0\}$, and obtains concrete entropy formulas: $h_t(P_E)=-2dt$ for $t\le0$ and $h_t(P_E)\le0$ for $t>0$ (with $h_t(P_E)=0$ if $E^{\perp}\neq\{0\}$); analogous results hold for spherical twists. In addition, the paper introduces categorical polynomial entropy $h_t^{\mathrm{pol}}$ and shows it vanishes for $t\neq0$ and equals $1$ at $t=0$ under a stability condition placing $E$ in the corresponding heart, thereby giving a refined invariant for twists. The work unifies and extends prior results on spherical twists to $\mathbb{P}$-twists and provides tools for distinguishing nonstandard autoequivalences in derived categories via dynamical invariants.

Abstract

We show that the $\mathbb{P}$-twist associated to any $\mathbb{P}$-object of a smooth project variety is not conjugate to a standard autoequivalence. This result is obtained by computing the categorical entropy functions of $\mathbb{P}$-twists. We also determine the categorical polynomial entropy of spherical twists and $\mathbb{P}$-twists, under an additional assumption.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2: ElaginLunts*Proposition 6.13, FanFilip*Theorem 2.1.7
• Remark 2.3
• Example 2.4
• Remark 2.5
• Proposition 2.7: FanShifting*Theorem 1.1
• Lemma 2.8
• Remark 2.9