Serre functor and $\mathbb{P}$-objects for perverse sheaves on $\mathbb{P}^n$

TL;DR

This paper identifies the inverse Serre functor for the constructible derived category on projective space as a specific $P$-twist and classifies indecomposable perverse sheaves as $P$-like objects, providing explicit morphism constructions.

Contribution

It explicitly describes the inverse Serre functor as a $P$-twist and classifies all indecomposable perverse sheaves on $P^n$ as $P$-like objects with constructed morphisms.

Findings

Inverse Serre functor is a $P$-twist at a simple perverse sheaf.

All indecomposable perverse sheaves are $P$-like objects.

Constructed morphisms span endomorphism spaces.

Abstract

We show that the inverse Serre functor for the constructible derived category $\mathbf{D}^\mathrm{b}_\mathrm{c}(\mathbb{P}^n)$ is given by the $\mathbb{P}$-twist at the simple perverse sheaf corresponding to the open stratum. Moreover, we show that all indecomposable perverse sheaves on $\mathbb{P}^n$ are $\mathbb{P}$-like objects, and explicitly construct morphisms spanning their total endomorphism spaces.