Derived categories of Burniat surfaces and exceptional collections

TL;DR

This paper constructs a maximal-length exceptional collection on Burniat surfaces with specific invariants, analyzes its endomorphism algebra, and studies the properties of the associated semiorthogonal complement category.

Contribution

It introduces a maximal exceptional collection on Burniat surfaces and characterizes its endomorphism algebra and semiorthogonal complement category.

Findings

Constructed a length 6 exceptional collection on Burniat surfaces with $K_X^2=6$

Calculated the DG algebra of endomorphisms of the collection

Identified the semiorthogonal complement as an almost phantom category

Abstract

We construct an exceptional collection $\Upsilon$ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$, a 4-dimensional family of surfaces of general type with $p_g=q=0$. We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $\mathcal A$ of $\Upsilon$ is an "almost phantom" category: it has trivial Hochschild homology, and $K_0(\mathcal A)=\bZ_2^6$.