Entropy of an autoequivalence on Calabi-Yau manifolds

Abstract

We prove that the categorical entropy of the autoequivalence $T_{\mathcal{O}}\circ(-\otimes\mathcal{O}(-1))$ on a Calabi-Yau manifold is the unique positive real number $\lambda$ satisfying $$ \sum_{k\geq 1}\frac{\chi(\mathcal{O}(k))}{e^{k\lambda}}=e^{(d-1)t}. $$ We then use this result to construct the first counterexamples of a conjecture on categorical entropy by Kikuta and Takahashi.