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Register a new userOn entropy of P-twists
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Yu-Wei Fan
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We compute the categorical entropy of autoequivalences given by P-twists, and show that these autoequivalences satisfy a Gromov-Yomdin type conjecture.
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We prove that for any $mathbb{P}^n$-functor all the convolutions (double cones) of the three-term complex $FHR xrightarrow{psi} FR xrightarrow{tr} Id$ defining its $mathbb{P}$-twist are isomorphic. We also introduce a new notion of a non-split $mathbb{P}^n$-functor.
Associated to a Mukai flop X ---> X is on the one hand a sequence of equivalences D(X) -> D(X), due to Kawamata and Namikawa, and on the other hand a sequence of autoequivalences of D(X), due to Huybrechts and Thomas. We work out a complete picture of the relationship between the two. We do the same for standard flops, relating Bondal and Orlovs derived equivalences to spherical twists, extending a well-known story for the Atiyah flop to higher dimensions.
We prove that the categorical entropy of the autoequivalence $T_{mathcal{O}}circ(-otimesmathcal{O}(-1))$ on a Calabi-Yau manifold is the unique positive real number $lambda$ satisfying $$ sum_{kgeq 1}frac{chi(mathcal{O}(k))}{e^{klambda}}=e^{(d-1)t}. $$ We then use this result to construct the first counterexamples of a conjecture on categorical entropy by Kikuta and Takahashi.
For classical dynamical systems, the polynomial entropy serves as a refined invariant of the topological entropy. In the setting of categorical dynamical systems, that is, triangulated categories endowed with an endofunctor, we develop the theory of categorical polynomial entropy, refining the categorical entropy defined by Dimitrov-Haiden-Katzarkov-Kontsevich. We justify this notion by showing that for an automorphism of a smooth projective variety, the categorical polynomial entropy of the pullback functor on the derived category coincides with the polynomial growth rate of the induced action on cohomology. We also establish in general a Yomdin-type lower bound for the categorical polynomial entropy of an endofunctor in terms of the induced endomorphism on the numerical Grothendieck group of the category. As examples, we compute the categorical polynomial entropy for some standard functors like shifts, Serre functors, tensoring line bundles, automorphisms, spherical twists, P-twists, and so on, illustrating clearly how categorical polynomial entropy refines the study of categorical entropy and enables us to study the phenomenon of categorical trichotomy. A parallel theory of polynomial mass growth rate is developed in the presence of Bridgeland stability conditions.
Given an extension of algebras $B/A$, when is $B$ generated by a single element $theta in B$ over $A$? We show there is a scheme $mathcal{M}_{B/A}$ parameterizing the choice of a generator $theta in B$, a moduli space of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples. A choice of a generator $theta$ is a point of the scheme $mathcal{M}_{B/A}$. This inspires a local-to-global study of monogeneity, piecing together monogenerators over points, completions, open sets, and so on. Local generators may not come from global ones, but they often glue to twisted monogenerators that we define. We show a number ring has class number one if and only if each twisted monogenerator is in fact a global generator $theta$. The moduli spaces of various twisted monogenerators are either a Proj or stack quotient of $mathcal{M}_{B/A}$ by natural symmetries. The various moduli spaces defined can be used to apply cohomological tools and other geometric methods for finding rational points to the classical problem of monogenic algebra extensions.
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