Derived Partners of Enriques Surfaces
Lev Borisov, Vernon Chan, Chengxi Wang
TL;DR
The paper investigates derived equivalences for Enriques surfaces arising as quotients of $(2,2,2)$-complete intersections in $\mathbb{P}^5$ by a fixed-point-free involution. It constructs noncommutative Deligne--Mumford stacks with Azumaya algebras, establishing derived equivalences $\mathcal{D}^b(\operatorname{Coh}-\mathscr{Y},\mathcal{A}) \simeq \mathcal{D}^b(\operatorname{Coh}-X)$ and $\mathcal{D}^b(\operatorname{Coh}-\widehat{\mathscr{Y}},\widehat{\mathcal{A}}) \simeq \mathcal{D}^b(\operatorname{Coh}-X/\sigma)$, thereby making the Homological Projective Duality (HPD) perspective explicit in this setting. The construction uses full Clifford algebras and their centers to produce Azumaya algebras on stacky resolutions, together with Veronese-type comparisons linking to Kuznetsov's framework, and provides geometric realizations of the Brauer classes via Severi--Brauer varieties. These results offer a concrete NC-model for Enriques surfaces, extend HPD beyond commutative varieties, and supply tools for analyzing Brauer classes in related geometric contexts with potential applications to derived-category geometry and mirror-symmetric constructions.
Abstract
Let $V$ be a $6$-dimensional complex vector space with an involution $σ$ of trace $0$, and let $W \subset \Sym^2 V^\vee$ be a generic $3$-dimensional subspace of $σ$-invariant quadratic forms. To these data we can associate an Enriques surface as the $σ$-quotient of the complete intersection of the quadratic forms in $W$. We exhibit noncommutative Deligne-Mumford stacks together with sheaves of Azumaya algebras on them whose derived categories are equivalent to those of the Enriques surfaces. This provides a more accessible treatment of of Theorem 6.16 in https://www.ams.org/journals/jams/2021-34-02/S0894-0347-2021-00963-3/ .. We also construct geometric realizations of the Brauer classes coming from these sheaves of Azumaya algebras which may be of independent interest.