# A trichotomy for the autoequivalence groups on smooth projective surfaces

## TL;DR

This paper classifies autoequivalence groups of derived categories on smooth projective surfaces into three types based on the maximal dimension of Fourier--Mukai kernels, providing a new trichotomy and conjectures for the case of dimension 2.

## Contribution

It introduces a novel trichotomy classification for autoequivalence groups based on Fourier--Mukai kernel dimensions and proposes a conjecture for the dimension 2 case, with partial proofs.

## Key findings

- Maximal Fourier--Mukai kernel dimension is 2, 3, or 4.
- Conjecture formulated for the dimension 2 case.
- Partial proofs provided for special cases.

## Abstract

We study autoequivalence groups of the derived categories on smooth projective surfaces, and show a trichotomy of types according to the maximal dimension of Fourier--Mukai kernels for autoequivalences. This number is $2$, $3$ or $4$, and we also pose a conjecture on the description of autoequivalence groups if it is $2$, and prove it in some special cases.