1.2.1 Motivation

In classical algebraic geometry, one is typically interested in spaces (varieties, schemes, stacks, etc.) which are locally modeled on the spectrum of a commutative ring. According to the philosophy of non-commutative geometry, it is often profitable to regard $A_\infty$ categories as ‘non-commutative’ spaces. One can then attempt to generalize classical algebraic geometry to the setting of $A_\infty$ category theory.

For example, there are standard notions smoothness, properness, dimension, etc. in classical algebraic geometry. It turns out that there also exist non-commutative analogs of some of these notions. Namely, an $A_\infty$ category is defined to be smooth if its diagonal bimodule is perfect; it is defined to be proper if its diagonal bimodule is proper.Footnote ¹ The dimension of an $A_\infty$ category is simply its Rouquier dimension.Footnote ²

If $Y$ is an ordinary algebraic variety, then one can associate to it the $A_\infty$ category $D^b\operatorname {Coh}(Y)$. One typically regards $D^b\operatorname {Coh}(Y)$ as the non-commutative partner of the classical object $Y$. It is then natural to ask: do the non-commutative notions of smoothness, properness, and dimension coincide with their classical counterparts?

As one might hope, it is known that $D^b\operatorname {Coh}(Y)$ is smooth and proper if and only if $Y$ is smooth and proper in the classical sense. Concerning dimension, there is the following well-known conjecture of Orlov.

Conjecture 1.1 (Orlov [Reference OrlovOrl09a, Conjecture 10])

Let $Y$ be a smooth quasi-projective scheme of dimension $n$. Then

(1.1)\begin{equation} \operatorname{Rdim}D^{b} \operatorname{Coh}(Y) = n. \end{equation}

The assumption that $Y$ is smooth is necessary; see Example 2.17. Orlov's conjecture is, in general, wide open. It has only been established for the following rather restrictive classes of examples, thanks to the combined work of many authors (to the best of the authors’ knowledge, this list is exhaustive).

1. (i) Smooth affine varieties, projective spaces, and smooth projective quadrics [Reference RouquierRou08, Proposition 7.18, Examples 7.7, 7.8].

2. (ii) All smooth projective curves [Reference OrlovOrl09a, Theorem 6].

3. (iii) The following schemes defined over an algebraically closed field of characteristic $0$: del Pezzo surfaces with $\operatorname {rk} \operatorname {Pic}(Y) \leqslant 7$, Fano threefolds of type $V_5$ and $V_{22}$, toric surfaces with nef anti-canonical divisor, toric Deligne–Mumford stacks defined over $\mathbb {C}$ of dimension no more than $2$ or Picard number no more than $2$, Hirzebruch surfaces [Reference Ballard and FaveroBF12].

4. (iv) The product of varieties from (iii) [Reference YangYan16].

5. (v) The product of two Fermat elliptic curves with the Fermat $K3$ surface [Reference Ballard, Favero and KatzarkovBFK14, Theorem 1.6].

6. (vi) Smooth toric Fano varieties of dimension $3$ or $4$ [Reference Ballard, Duncan and McFaddinBDM19, Propositions 2.15 and 2.16].

7. (vii) Certain blow-ups of the projective space [Reference PirozhkovPir19, Theorem 4.1].

8. (viii) Some weak del Pezzo surfaces [Reference Elagin, Xu and ZhangEXZ21, Corollary 7.5].

We remark that Rouquier proved that, for smooth $Y$,

(1.2)\begin{equation} n \leqslant \operatorname{Rdim}D^{b} \operatorname{Coh}(Y) \leqslant 2n, \end{equation}

so the content of Orlov's conjecture lies in improving the upper bound. See § 2.6 for further discussion and references.

1.2.3 Methods: symplectic flexibility

Although Orlov's conjecture is a problem in algebraic geometry, our proof of Theorem 1.2 is entirely symplectic. As mentioned in (1.2), it is already known that $\operatorname {Rdim}D^b\operatorname {Coh}(Y) \geqslant n$. Hence, we only need to prove the claimed upper bounds. To do this, we apply homological mirror symmetry. This transforms Theorem 1.2 into a statement about the Rouquier dimension of wrapped Fukaya categories of Weinstein pairs; namely, it is enough to prove that the Rouquier dimension of a (polarizable) Weinstein pair of real dimension $2n$ is bounded above by $n$ when $n \leqslant 3$ and by $2n-3$ when $n \geqslant 4$. This is now purely a problem in symplectic topology which can be approached using symplectic tools.

According to the recent work of Ganatra, Pardon, and Shende [Reference Ganatra, Pardon and ShendeGPS22], wrapped Fukaya categories of Liouville manifolds satisfy ‘Weinstein sectorial descent’. Essentially, this means that if we (suitably) cover a Liouville manifold $X$ by Weinstein sectors $\{ X_\alpha \}_{\alpha \in I}$, then $\mathcal {W}(X)$ can be expressed as the homotopy colimit of the wrapped Fukaya categories of the form $\mathcal {W}(\cap _{\alpha \in J} X_\alpha )$ (where $J$ ranges over all non-empty subsets of $I$ and the homotopy colimit is indexed by the obvious diagram of inclusions).

It is not hard to show that the Rouquier dimension of a homotopy colimit can be bounded from above by the Rouquier dimension of its individual pieces. This bound depends on (a) the ‘depth’ of the diagram (i.e. the longest chain of arrows) and (b) the Rouquier dimension of the individual pieces; see Lemma 2.15 for a precise statement. This suggests a possible strategy for upper bounding the Rouquier dimension of a Weinstein manifold: construct a sectorial cover of ‘small’ depth, such that each piece in the cover has ‘small’ Rouquier dimension.

Let us first implement this strategy in the special case where $X=T^*M$, for $M$ a closed $n$-manifold. We construct a sectorial cover as follows. First, triangulate $M$. Then set $X_\alpha := T^*\operatorname {star}(v_\alpha )$ where $v_\alpha$ ranges over the vertices of the triangulation. Observe that the intersection of any collection of stars in a triangulation is either empty, or is the star of some higher-dimensional cell. From this, it follows that the cover has depth $n+1$. It also follows (after appropriate smoothing) that $\bigcap _{\alpha _1, \ldots, \alpha _n} T^*\operatorname {star}(v_{\alpha _i}) \simeq T^*D^n$ and, hence, the Rouquier dimension of the overlaps is zero. Plugging these numbers into Lemma 2.15, one finds that $\operatorname {Rdim} \mathcal {W}(T^*M) \leqslant n$. This upper bound is sharp in general.Footnote ³

We would like to extend this argument to an arbitrary Weinstein manifold $X$. Unfortunately, the naive generalization does not work. If $X$ is an arbitrary Weinstein manifold, its skeleton may be a highly singular object. Even assuming the skeleton can be triangulated (which is not automatic), it is completely unclear how to control the Rouquier dimension of the resulting sectorial cover. It is also unclear how to convert a cover of the skeleton into a sectorial cover of its thickening (in the cotangent bundle case, there is a canonical Morse–Bott form which allows one to easily perform this conversion).

To overcome these difficulties, we take advantage of recent advances from the flexible side of symplectic geometry. Namely, the arborealization theorem of Álvarez-Gavela, Eliashberg, and Nadler [Reference Alvarez-Gavela, Eliashberg and NadlerAGEN21a, Reference Alvarez-Gavela, Eliashberg and NadlerAGEN21b, Reference Alvarez-Gavela, Eliashberg and NadlerAGEN20] provides an h-principle for simplifying the singularities of the skeleton. The upshot is that, under rather mild topological assumptions, one can homotope a Weinstein manifold so that the singularities of its skeleton are of ‘arboreal’ type. Such homotopies do not affect the wrapped Fukaya category.

Arboreal singularities are indexed by finite rooted trees. Morally, if $\mathcal {S}_\sigma$ is a Liouville sector obtained by thickening an arboreal singularity $\sigma$, one expects that $\mathcal {W}(\mathcal {S}_\sigma )$ is Morita equivalent to $\operatorname {Rep}(T_\sigma )$, the derived representation category of the corresponding tree $T_\sigma$. Moreover, it is known that $\operatorname {Rdim} \operatorname {Rep}(T_\sigma )=0$ if $T_\sigma$ is of Dynkin type (i.e. the underlying graph is a Dynkin diagram) and $\operatorname {Rdim} \operatorname {Rep}(T_\sigma )=1$ otherwise. This suggests that sectorial thickenings of arboreal singularities are a good substitute for $T^*D^n$.

Arboreal skeleta admit a canonical stratification by ‘singularity order’. We prove in § 4.2 that this stratification is Whitney. By a well-known theorem of Goresky, any Whitney stratification can be refined to a (possibly non-Whitney) triangulation. This suggests the following updated strategy: given a (polarizable) Weinstein manifold $X$ of dimension $2n$, apply a homotopy to make the skeleton arboreal. It therefore admits a triangulation. Now construct a sectorial cover by taking arboreal thickenings of the stars of the vertices, like for cotangent bundles.

Unfortunately, there are significant obstructions to making this strategy rigorous. To begin with, one would need to develop a good theory of arboreal sectors (which should be closed under intersections), as well as a procedure for constructing a ‘sectorial thickening’ of an arboreal singularity. Moreover, one would need a method for construct arboreal sectors from stars of (possibly non-Whitney) triangulations. Due to this difficulties, it is unclear to us whether the above strategy can feasibly be implemented in the framework of Fukaya categories.

Instead, we work in the framework of microlocal sheaf theory. The details of this approach are carried out in § 4. After reviewing some basic properties of microlocal sheaves, we study the stratified topology of arboreal skeleta, and use this to construct a suitable triangulation. We then carry through a microlocal sheaf analog of the above covering argument to obtain an upper bound on the Rouquier dimension of the category of microlocal sheaves. Finally, we apply the main result of [Reference Ganatra, Pardon and ShendeGPS20b] to pass back to wrapped Fukaya categories.

The precise upper bound which we obtain via this argument (and under the assumption that $X$ is stably polarizable) is $\operatorname {Rdim} \mathcal {W}(X^{2n}) \leqslant n$ for $n \leqslant 3$, and $\operatorname {Rdim} \mathcal {W}(X^{2n}) \leqslant 2n-3$ for $n>3$. We know no counterexample to the inequality $\operatorname {Rdim} \mathcal {W}(X^{2n}) \leqslant n$ for arbitrary Weinstein manifolds.

1.2.4 A digression on motives

As a byproduct of our method, we derive some consequences for the structure of non-commutative motives on partially wrapped Fukaya categories. The notion of a non-commutative motive is was introduced by Tabuada [Reference TabuadaTab15] and is briefly summarized in § 4.5. To give an illustration, suppose that $(X,A)$ is a polarizable Weinstein pair such that $\mathcal {W}(X,A)$ is proper (it is always smooth according to [Reference Ganatra, Pardon and ShendeGPS22, Corollary 1.19]). Then we find that:

1. (i) $HH_\bullet (\mathcal {W}(X,A))$ is concentrated in degree zero;

2. (ii) $\operatorname {Perf} \mathcal {W}(X,A)$ satisfies the non-commutative Weil conjecture introduced by Tabuada (following proposals of Kontsevich) in [Reference TabuadaTab22].

Both of these statements follow from a more general result about non-commutative motives, which is itself an easy consequence of the arborealization method described previously. We refer the reader to § 4.5 for details.

1.3.1 Motivation

As motivation for the geometrically inclined symplectic topologist, we state some quantitative questions of rather classical flavor.

If $(X, \lambda )=(T^*M, pdq)$ for a closed manifold $M$, then ${\frak c}_X= 0_M$ and Question 1.4 is classical. As is now well-known, it follows from the existence of Floer cohomology (or finite-dimensional analogs) that $|{\frak c}_X \cap \phi ({\frak c}_X)|$ is bounded from below by the sum of the Betti numbers of $M$.

In general, however, the skeleton of a Weinstein manifold is a highly singular object for which there is no available notion of Floer cohomology (and it seems rather unlikely that such a notion exists). Note that even for the cotangent bundle of a closed manifold, equipped with a Weinstein structure homotopic to the standard Morse–Bott one, the skeleton is typically singular.

To the best of the authors’ knowledge, the only systematic result in the literature addressing Question 1.4 is the lower bound $|{\frak c}_X \cap \phi ({\frak c}_X)| \geqslant 1$, which holds whenever the Rabinowitz–Floer homology $RFH(X)$ is non-zero [Reference Cieliebak and FrauenfelderCF09]. The nonvanishing of $RFH(-)$ for Weinstein manifolds is now known to be equivalent to the nonvanishing of $SH(-)$ and $\mathcal {W}(-)$; see [Reference RitterRit13, § 1.6].

Let us denote this number by $\operatorname {Lef}_w(X) \in \mathbb {N} \cup \{\infty \}$. According to work of Giroux and Pardon [Reference Giroux and PardonGP17], any Weinstein manifold admits a Lefschetz fibration with Weinstein fibers, so $\operatorname {Lef}_w(X)<\infty$ if and only if $X$ is Weinstein (up to homotopy). It is also clear that $\operatorname {Lef}_w(X)=0$ if and only if $X$ is subcritical (by a result of Cieliebak [Reference CieliebakCie02], this means equivalently that $X$ admits a Weinstein presentation with only handles of index strictly less than the middle dimension, or that $X$ is Weinstein deformation equivalent to the product of a lower-dimensional Weinstein manifold with $\mathbb {C}$). For topological reasons, one has the lower bound $\operatorname {Lef}_w(X) \geqslant \operatorname {rk} H_n(X; \mathbb {Z})$ when $X$ has real dimension $2n$. These are the only lower bounds in the literature that we are aware of.

An analog of $\operatorname {Lef}_w(X)$ in real Morse theory was considered by Abouzaid and Seidel [Reference Abouzaid and SeidelAbSe10b] under the name ‘complexity’. By definition, the complexity of a Weinstein manifold $X$, which we denote by $\operatorname {WMor}(X)$, is the minimal number of critical points of a Weinstein Morse function on $X$, where $X$ is considered up to Weinstein homotopy.Footnote ⁵ Surprisingly, Lazarev [Reference LazarevLaz20] showed that for Weinstein manifolds having non-zero middle homology, one has $\operatorname {WMor}(X)= \operatorname {Mor}(X)$, where $\operatorname {Mor}(X)$ denotes the minimal number of critical points of a smooth Morse function on $X$. Does a similar phenomenon hold for $\operatorname {Lef}_w(X)$?

1.3.3 Methods: bounds on the Rouquier dimension coming from geometry

It turns out that the Rouquier dimension of wrapped Fukaya categories provides a lower bound for the quantities we wish to study.

Proposition 1.14 (= Proposition 6.9)

Let $(X, \lambda )$ be a Weinstein manifold with properly embedded cocores (this is a generic condition). If $\phi : X \to X$ is a generic compactly supported Hamiltonian symplectomorphism, then

(1.8)\begin{equation} | {\frak c}_X \cap \phi({\frak c}_X) | \geqslant \operatorname{Rdim} \mathcal{W}(X) +1. \end{equation}

(The result we prove in Proposition 6.9 is stated slightly differently and is more general.) Let us sketch the proof of Proposition 1.14. To begin with, it can be shown that the Rouquier dimension of any $A_\infty$ category is bounded above by the length of the shortest resolution of the diagonal bimodule by Yoneda bimodules (minus $1$). This is a purely algebraic fact which has nothing to do with Fukaya categories.

We now set $r=|{\frak c}_X \cap \phi ({\frak c}_X)|$. Our goal is to construct a resolution of the diagonal bimodule $\Delta _{\mathcal {W}(X)}$ of length $r$. We do this by the following geometric argument. First, observe that the intersection points of ${\frak c}_X$ with $\phi ({\frak c}_X)$ are in bijection with the intersection points of the skeleton of $(\overline {X} \times X, -\lambda \oplus \phi _*\lambda )$ with the diagonal Lagrangian. As ${\frak c}_X \pitchfork \phi ({\frak c}_X)$, the diagonal intersects the skeleton transversally in $r$ points. By applying deep work of Chantraine et al. [Reference Chantraine, Rizell, Gigghini and GolovkoCRGG17] or Ganatra, Pardon, and Shende [Reference Ganatra, Pardon and ShendeGPS22], one can then construct a resolution of the diagonal by $r$ cocores of $\overline {X} \times X$. Morally, the idea is that as one scales the diagonal Lagrangian by the Liouville flow, it converges to cocores at each point where it intersects the skeleton. The upshot is that we have resolved the diagonal Lagrangian by $r$ cocores in $\overline {X} \times X$, which are just products of cocores in $X$. Finally, we can turn this resolution of the diagonal Lagrangian into a resolution of the diagonal bimodule. This last step is carried out in the Appendix, following the methods of [Reference GanatraGan12, Reference Ganatra, Pardon and ShendeGPS22].

There is also an upper bound on the Rouquier dimension coming from Lefschetz fibrations.

Proposition 1.15 (= Proposition 6.6)

Let $X$ be a Liouville manifold. Then we have

(1.9)\begin{equation} \operatorname{Lef}_w(X) \geqslant \operatorname{Rdim} \mathcal{W}(X) +1. \end{equation}

The proof of Proposition 1.15 is straightforward and (at least implicitly) well-known to experts: the Fukaya–Seidel category associated to a Lefschetz fibration admits a semi-orthogonal decomposition of length equal to the number of critical points of the fibration, which immediately implies the corresponding upper bound for the Rouquier dimension.

1.3.4 Methods: commutative ring actions on wrapped Fukaya categories

In order to obtain conclusions about quantitative symplectic topology, we need a method for obtaining lower bounds on the Rouquier dimension of wrapped Fukaya categories which can then be combined with the upper bounds stated in Propositions 1.14 and 1.15. These lower bounds will ultimately arise from considering a canonical action of the Hochschild cohomology ring on the Fukaya category. To explain this, it is natural to start the discussion in a more general setting.

Let $\mathcal {C}$ be a pre-triangulated $A_\infty$ category and let $\mathcal {T}= H^0(\mathcal {C})$ be its homotopy category. Let $Z^\bullet (\mathcal {T})$ be the (graded-commutative) ring of graded natural transformations of the identity on $\mathcal {T}$. Given a graded-commutative ring $R$, a central action of $R$ on the triangulated category $\mathcal {T}$ is, by definition, a morphism of graded-commutative rings $R \to Z^\bullet (\mathcal {T})$. Concretely, this is the data of a morphism $R \to \hom _{H(\mathcal {C})}^\bullet (K, K)$ for each object $K \in \mathcal {C}$ such that the induced left and right module structures on $\hom _{H(\mathcal {C})}^\bullet (K, L)$ coincide up to sign.

The upshot is that the morphism spaces of objects of $\mathcal {T}$ define modules over $R$. One can then study the triangulated category $\mathcal {T}$ using tools from (graded-)commutative algebra. This perspective leads to a very rich theory; see, e.g., [Reference Benson, Iyengar and KrauseBIK08].

The workhorse result for lower-bounding the Rouquier dimension of wrapped Fukaya categories in this paper is the following:

Theorem 5.2 can be viewed as an ‘affine’ analog of a beautiful theorem of Bergh et al. [Reference Bergh, Iyengar, Krause and OppermannBIK10, Theorem 4.2]. In fact (as we show), one could directly apply [Reference Bergh, Iyengar, Krause and OppermannBIK10, Theorem 4.2] and still obtain useful lower bounds for certain examples. However, for our purposes, [Reference Bergh, Iyengar, Krause and OppermannBIK10, Theorem 4.2] suffers from two limitations. First, it is essentially never sharp (there is a pesky $-1$ which comes, ultimately, from working in the ‘projective’ rather than ‘affine’ setting). Second, it requires rather strong hypotheses on $R$ and $\operatorname {Hom}_{\mathcal {T}}^\bullet (G,G)$ which do not hold in many cases of interest to us (for instance, Examples 1.9 and 5.18 would fail).Footnote ⁷ In contrast, Theorem 5.2 is sharp for many examples (such as cotangent bundles of $n$-spheres and $n$-tori), and more widely applicable.

Our general approach to proving Theorem 5.2 via Koszul objects follows [Reference Bergh, Iyengar, Krause and OppermannBIK10]; however, the input from graded-commutative algebra is replaced by new arguments from (ungraded) commutative algebra. We crucially use the hypothesis that $R$ is of finite type. This hypothesis is rather severe from the representation theoretic perspective taken in [Reference Bergh, Iyengar, Krause and OppermannBIK10]. However, from the symplectic perspective which we adopt in this paper, this assumption is rather mild.

Of course, Theorem 1.16 is useless without a good supply of central actions. Happily, for any pre-triangulated $A_\infty$ category $\mathcal {C}$, there is a canonical central action of the Hochschild cohomology ring $HH^\bullet (\mathcal {C})$ on $\mathcal {T}= H^0(\mathcal {C})$ called the characteristic morphism. Given any object $K \in \mathcal {C}$, this is just the map $HH^\bullet (\mathcal {C}) \to \hom ^\bullet _{H(\mathcal {C})}(K, K)$ that projects a Hochschild cochain onto its length zero part. In general, the characteristic morphism is neither injective nor surjective.

In the case which is relevant to us, namely when $\mathcal {T}=H^0(\operatorname {Perf} \mathcal {W}(X))$ and $X$ is Weinstein, the characteristic morphism is known to admit a concrete description in terms of the familiar closed–open map. To explain this a little, recall that the closed–open map defines a ring morphism $SH^\bullet (X) \to HW^\bullet (L, L)$, for any object $L \in \mathcal {W}(X)$. On the other hand, if $X$ is Weinstein, there is an isomorphism of graded-commutative rings $HH^\bullet (\mathcal {W}(X))= SH^\bullet (X)$ [Reference Chantraine, Rizell, Gigghini and GolovkoCRGG17]. Now the point is simply that the obvious diagram formed by these maps commutes. As a result, we obtain the following corollary which is the desired lower bound.

Corollary 1.17 (= Corollary 5.17)

Suppose that $SH^0(X)$ admits a subring $R$ of finite-type over $k$, and suppose there exists a (split) generator $K \in \mathcal {W}(X)$ such that $HW^\bullet (K,K)$ is Noetherian over $R$, where $R$ acts via the closed–open map. Then

(1.10)\begin{equation} \operatorname{Rdim} \mathcal{W}(X) \geqslant \dim_R HW^\bullet(K,K). \end{equation}

Corollary 1.17 works with any choice of gradings, but $SH^\bullet (X)$ will typically be $\mathbb {Z}$-graded or $\mathbb {Z}/2$-graded, depending on the application. The flexibility to choose different gradings is convenient in practice, because of course $SH^0(X)$ will admit different subrings depending on the choice of grading.