Sheaves on quoric manifolds

Introduction

There were two motivations for this work. The first was to study quoric manifolds [15] from a sheaf-theoretic perspective and in particular, to understand how their equivariant topology is reflected in the associated categories of equivariant and constructible sheaves. The second was to investigate whether Lunts’ techniques for toric varieties [19] can be applied to quoric manifolds. In particular, we wanted to determine whether the equivariant derived category of constructible sheaves on quoric manifolds admits a description in terms of the geometry of the orbit space and the isotropy data of the underlying group action.

In this paper we show that the equivariant derived category of constructible sheaves on a quoric manifold is equivalent to the derived category of constructible DG-modules over a sheaf of DG-algebras on the face poset of the orbit polytope. We further show that this sheaf of DG-algebras is formal, that is, quasi-isomorphic to its cohomology sheaf.

Equivariant derived categories of sheaves

Sheaf theory was introduced by Leray in the 1940s and developed in the following decade. Its basic principle is to organise local data on a space $X$ into global invariants: sheaves on $X$ form an abelian category $\operatorname{Sh}(X)$, and the global sections functor assigns to each sheaf its global invariants. This functor, and more generally the direct image $f_*$ along a continuous map $f$, is only left exact, and its failure to preserve short exact sequences is measured by higher direct images, in particular by sheaf cohomology. These higher terms are organised in the derived category, obtained from complexes of sheaves by localising at quasi-isomorphisms. There they become single functors $f_*$, which together with $f^*$, $\otimes$, $f_!$, $f^!$ and Verdier duality form the six operations [18].

This formalism is especially fruitful when $X$ carries extra structure. If $X$ is stratified, one restricts to complexes whose cohomology sheaves are constructible, that is, locally constant of finite rank along the strata. For suitable stratifications this subcategory is stable under the six operations and relates the local topology of the strata to the global structure of $X$. If a group $G$ acts on $X$, one obtains the abelian category $\operatorname{Sh}_G(X)$ of $G$-equivariant sheaves. Its naive derived category does not retain equivariant cohomology: for connected $G$ and $X = \mathrm{pt}$ it gives $D(\operatorname{Vect})$, whereas the equivariant cohomology of a point is $H^*(BG)$. This was addressed by Bernstein and Lunts [4], who define $D_G(X)$ by resolving $X$ by the free $G$-spaces $X \times E$ and descending to the quotients $X \times_G E$, where $E$ approximates $EG$. The resulting category carries the functors $f^*$, $f_*$, $f^!$, $f_!$ and the usual adjunctions. Combining the two structures gives the equivariant derived category $D_{G,c}(X)$ of constructible sheaves.

Toric varieties are the model case in which the group action, the stratification and the equivariant derived category are all controlled by combinatorics. A normal toric variety $X = X_\Sigma$ is a complex algebraic variety with an action of the algebraic torus $T^n := (\mathbb{C}^*)^n$ and a dense open orbit. It has finitely many $T^n$-orbits $O_\sigma$, in bijection with the cones $\sigma$ of the fan $\Sigma$, with $\dim O_\sigma= n - \dim\sigma$. Lunts [19] described the bounded $T^n$-equivariant derived category $D^b_{T^n,c}(X)$ of sheaves constructible with respect to the orbit stratification algebraically. It is equivalent to the derived category of DG-modules over a differential graded algebra, which is the Ext-algebra of the direct sum of the simple equivariant perverse sheaves attached to the orbits, and which is formal. Lunts also conjectured that an analogous description holds for a complex reductive group acting on a projective variety with finitely many orbits; this conjecture was extended in a number of directions [14 Bal09, 23 Jos22a]

In this paper we take a first step towards a real-topological analogue of this conjecture. We replace the complex reductive group by the compact Lie group $Q^n$, a product of copies of the group $Q \cong\mathrm{SU}(2)$ of unit quaternions, and the projective variety by a quoric manifold $M^{4n}$, the quaternionic analogue of a quasitoric manifold, the latter being a topological analogue of a toric variety [8]. This is a $4n$-dimensional manifold with an action of $Q^n$ whose orbit space is a simple $n$-polytope $P^n$. As in the toric case, the faces of $P^n$ induce a finite stratification of $M^{4n}$. Our results are an equivalence of $D^b_{Q^n,c}(M^{4n})$ with the derived category of DG-modules over a sheaf of DG-algebras on the face poset of $P^n$ (Theorem 4.13), and the formality of this sheaf of DG-algebras (Theorem 5.6).

Quoric manifolds, stratifications and the Borel construction

Quoric manifolds were introduced by Hopkinson [15], building on earlier work of Scott [24], as quaternionic analogues of the quasitoric manifolds of Davis and Januszkiewicz [8], which are themselves topological analogues of toric varieties [10]. A quoric manifold is a smooth $4n$-dimensional manifold with a smooth locally regular action of $Q^n = (S^3)^n$ whose orbit space is a simple $n$-dimensional polytope (Definition 2.11). This contrasts with a $2n$-dimensional quasitoric manifold, which carries an action of the compact torus $T^n = (S^1)^n$. The identification of the orbit space with the polytope is understood to preserve the face structure.

As in toric topology [5], the face structure of the orbit polytope records the isotropy data of the action (§ 3.2). Since $Q^n$ is non-abelian, stabilisers are recorded up to conjugacy. Over the relative interior of each face their conjugacy class is constant, although the stabiliser subgroups need not be equal, and distinct faces may carry the same conjugacy class. Thus the partition by stabiliser conjugacy classes need not have connected pieces.

Let $\pi:M^{4n}\to P^n$ be the orbit map, let $\mathcal{F}(P^n)$ be the set of nonempty faces of $P^n$ including $P^n$, and for a face $F$ with relative interior $\mathring{F}$ put $M_F:=\pi^{-1}(\mathring{F})$. Then $M^{4n}=\bigcup_{F\in\mathcal{F}(P^n)}M_F$. We encode this decomposition by a finite topological space: the face poset $(\mathcal{F}(P^n),\subseteq)$, ordered by inclusion, with the Alexandrov topology whose open sets are the up-sets (§ 3.3). Denote it by $\mathcal{F}_{\leq}$. The map $\rho_P:P^n\to\mathcal{F}_{\leq}$, $\rho_P(x)=F for $x \in\mathring{F}$, is continuous and open, and so is

whose fibres are precisely the subsets $M_F$. Our first main result is the following.

The face stratification coincides with the connected components of the orbit-type Whitney stratification (Corollary 2.31) and extends to the Borel construction. Let $E_NQ^n = (S^{4N+3})^n$ and $EQ^n = \bigcup_N E_NQ^n$. In § 3.5 we study the Borel space construction of a quoric manifold

together with the projection $b_Q: M_Q \longrightarrow\mathcal{F}_{\preceq}$, $[e,m] \mapsto\pi_{\mathcal{F}}(m)$. For a face $F$ let $U_F = \{G \in\mathcal{F}(P^n) \mid F \preceq G\}$ be its minimal open neighborhood and $V_{F,Q} := b_Q^{-1}(U_F)$ the corresponding open star. Our second main result is the following.

This is the topological input for the sheaf of DG-algebras of § 5.2.

Sheaves on quoric manifolds

Sheaves that retain information about both the $Q^n$-action and the face stratification are our main interest. In § 4 we study $Q^n$-equivariant sheaves on $M^{4n}$, with the face stratification $\{M_F\}_{F\in\mathcal{F}(P^n)}$ as the organising structure. The aims are to compute the categories of sheaves and local systems on orbits and strata, and to use this to describe constructible and perverse sheaves.

Every stabiliser of the $Q^n$-action is connected, being isomorphic to $Q^k = \mathrm{SU}(2)^k$, and each stratum splits equivariantly as $M_F \cong\mathring{F} \times Q^n/\lambda^b(F)$. Since $\mathrm{SU}(2)$ is simply connected and $\mathring{F}$ is contractible, every stratum is simply connected. This gives two computations.

• Orbits. The equivariant category of an orbit $Q^n/H$ is $\operatorname{Rep}(\pi_0(H))$, and $\pi_0(H)$ is trivial for every stabiliser. Hence $\operatorname{Sh}_{Q^n}(Q^n \cdot m)$, $\operatorname{Loc}(Q^n \cdot m)$ and $\operatorname{Loc}_{Q^n}(Q^n \cdot m)$ are all equivalent to $\operatorname{Vect}_{\mathbb{C}}$, so an equivariant sheaf on an orbit is constant.

• Strata. $\operatorname{Sh}_{Q^n}(M_F) \simeq\operatorname{Sh}(\mathring{F})$, so the group action carries no extra data on a stratum, and $\operatorname{Loc}(M_F) \simeq\operatorname{Loc}_{Q^n}(M_F) \simeq\operatorname{Vect}_{\mathbb{C}}$. The proofs use the vanishing of $\pi_1$ of the stratum and of its Borel construction $EQ^n \times_{Q^n} M_F \simeq B\lambda^b(F)$.

All of the above have been well studied in general. Our aim is to carry out explicit calculations in the case of quoric manifolds. Our third main result is the following.