Microlocal mirror symmetry on the torus

Dmitry Vaintrob

October 29, 2017

Abstract

Suppose k is an arbitrary field (of any characteristic) and U is an openvariety with compactification X with normal-crossings (or, more gener-ally, locally toric type) complement. To such an X we associate a “per-fectoid compactification” stack, X(orb-∞) compactifying U in a pro-sense,and define the log-perfectoid quasicoherent category of Qcohlog-perf(U,X)

of U as the category Qcohlog-perf(U) := QcohaX(orb-∞) of almost quasi-coherent sheaves in the sense of Faltings’ theory of almost-mathematics.We show that when U = Gnm is a complex torus and X is a toric com-pactification, the derived category Db Qcohlog-perf(T,X) is independent ofchoice of X as a symmetric monoidal dg category and is derived equiva-lent to the category of all topological sheaves on the Langlands dual realtorus (S1)n (with monoidal structure given by convolution), which can bethought of (using results of Nadler-Zaslow and others) as the microlocalFukaya category of the mirror symplectic variety T∨ = T ∗(S1)n. Ourproof works by using topos-theory models (in the sense of Grothendieck)to view Qcohlog-perf(U) as a special case of a new equivariant quasicoherenttheory of “Novikov toric varieties”, glued out of spectra of rings of Novikovpolynomials (polynomials in several variables with real exponents).

1 Introduction

1.1 Overview and motivationHomological Mirror Symmetry studies equivalences between derived cat-egories associated to algebraic and differential-geometric objects. Theoriginal formulation (due to Kontsevich, [6]) conjectures that to any (suit-ably nice) algebraic variety X one can associated a “mirror” symplecticvariety X∨ together with an equivalence of dg categories Db Qcoh(X) ∼=Fuk(X∨): hereDb Qcoh(X) is the derived ategory of quasicoherent sheaveson X and Fuk(X∨) is the (dg enhancement) of the A∞ (idempotent com-pleted) Fukaya category associated to X∨. In practice, known instancesof mirror symmetry are variations on this theme, with Db Qcoh(X) andFuk(X∨) modified in some way.

1

1.1.1 Mirror symmetry for the cylinder

Let C∗sym be the cylinder, viewed as a symplectic manifold with Kählerform ω := d|z| ∧ dθ, for |z|, θ the absolute value and angle coordinates.The mirror of C∗sym is the algebraic variety C∗alg (or, over more generalfields, Gm), with functoriality with respect to scalar multiplication mak-ing C∗sym,C∗alg naturally into dual tori. Mirror symmetry results takethe form of the statement that a Fukaya category on C∗sym is derivedequivalent to a coherent category on C∗alg. However, since C∗sym,C∗alg areopen, there is some ambiguity to which category should be taken on eitherside. Namely, on C∗sym there are two standard natural Fukaya categories.First, the compact Fukaya category FukC∗sym has objects exact compactLagrangians L ⊂ C∗sym together with a local system. The overwrappedFukaya category, Fukot C∗sym has objects all Lagrangians which are al-lowed to go to ∞, and must be deformed to wrapp around at ∞ in acertain asymptotically periodic way when composed. The known mirrorsymmetry statements for the cylinder take the following form:

Theorem 1. 1. The (idempotent-completed dg enhancement of) thecompact Fukaya category of the cylinder C∗sym is equivalent to the de-rived category of coherent sheaves on C∗ with bounded and compactly-supported cohomology.

2. The (idempotent-completed dg enhancement of) the overwrapped Fukayacategory of the cylinder C∗sym is equivalent to the derived category ofall coherent sheaves on C∗.

There are some problems with both categories. Namely, the com-pact Fukaya category does not remember as much geometric informationabout the symplectic variety C∗sym as one would like, and the overwrappedFukaya category loses some of the finite-dimensionality and symmetrythat the ordinary Fukaya category has. Now since C∗sym is naturally acotangent bundle, C∗sym ∼= T ∗S1, there is a third category, defined byNadler and Zaslow in [9], which interpolates between the compact andoverwrapped versions. We call this category the microlocal Fukaya cate-gory, Fukµ(C∗sym); its objects are Lagrangians with some conicity condi-tions at ∞. Conjecturally, such a category can be defined more generallyfor any symplectic manifold which admits a Lagrangian skeleton: for aninformal definition for Weinstein manifolds, see [8]. It is reasonable to asknow whether there is a new coherent category of sheaves on C∗alg which ismirror symmetric to Fukµ(C∗sym). In this paper, we define such a category,which we call the log-perfectoid category of coherent sheaves.

1.1.2 Coherent-constructible correspondence: the Nadler-Zaslow approach to the n-dimensional cylinder

One way to interpolate between the two different coherent categories above(compactly supported sheaves on Gm and all sheaves on Gm) is to studysheaves on the (unique) compactification P1 of Gm. It turns out that thederived category of sheaves on P1 is indeed related to a certain modifiedFukaya category for the cylinder. This approach becomes more interest-ing at higher dimensions, where the cylinder C∗sym = T ∗S1 is replaced by

2

(T ∗S1)n and the algebraic mirror is replaced by Gnm. Then the compact-ification is no longer unique, and the most well-studied compactificationsare toric varieties. This gives us a whole family of flavors of mirror sym-metry statements for (T ∗S1)n parametrized by toric varieties. As it turnsout, for each (smooth and compact) toric variety X compactifying Gnmthere is a fully faithful functor from the derived category Db(coh(X)) tothe microlocal Fukaya category, Fukµ(T ∗S1)n. The image of this functorshould then be cut out by a combinatorial boundary condition on La-grangians in (T ∗S1)n. One very fruitful point of view on the resultingtheory originates from a very old observation of Bondal, [1], that the de-rived category of coherent sheaves on an n-dimensional toric variety Xcan be faithfully embedded in the category of constructible sheaves onthe topological torus S := (S1)n (proved by [?]). Now it follows from[9], the category of constructible sheaves on S is equivalent to a certainFukaya category on T ∗S ∼= (T ∗S1)n, and boundary conditions on thisFukaya category translate into singular support conditions (a notion im-ported from microlocal analysis) on constructible sheaves on (S1)n. Thispoint of view results in a constellation of results and conjectures knownas the coherent-constructible correspondence conjectures (ccc conjecturesfor short), proved by Kuwagaki, [7].

1.1.3 Some motivation: coherent-constructible correspon-dence at the highly ramified limit

From now on we assume access to (possibly a variant of) the philosophy of[9] to identify the Fukaya category of the n-dimensional symplectic torusT ∗S with some category of topological sheaves on S itself. The fact thattoric compactifications of the torus Gnm give different full derived subcate-gories of Constr(S) is suggestive: perhaps there is some universal, or “limitcompactification” of Gnm which sees the entire category of constructible,or even all topological, sheaves on S. One candidate is to take the limitof all toric varieties (which naturally form a projective system by fanrefinement), but this is not quite good enough: for example, in the one-dimensional context, there is a unique compactification (P1) and it only“sees” constructible sheaves on S = S1 which are locally constant outsidethe basepoint 1 ∈ S1. We can do better by extending our world from toricvarieties to toric stacks. Namely, suppose (for now) that the basefield k isalgebraically closed of characteristic 0. Then for any positive integer N ,the group Z/N acts on both Gm and on P1 by multiplication by roots ofunity. Consider the unique étale N -fold cover G(N)

m of Gm. It is isomor-phic to Gm, but it will be convenient for us to differentiate it notationallyto think of it as a cover of the original Gm on which we want a mirrorsymmetry result. Now the map G(N)

m → Gm is the projection map inducedby factoring out the action of Z/N on G(N)

m . Let (P1)(N) (also isomorphicto P1 but thought of as fibering over it) be the unique smooth compacti-fication of G(N)

m . It maps to P1 with n-fold ramification at the boundarypoints, 0,∞. Now we can study the quotient stack (P1)(N)/(Z/N), whichcompactifies the free quotient stack G(N)

m /(Z/N) ∼= Gm. The categoryof coherent sheaves on this stack will simply be the category of Z/N -

3

equivariant sheaves on the ramified cover (P1)(N) (isomorphic to P1). Thegeneralization of the coherent-constructible correspondence in this con-text says that the derived category of sheaves on (P1)(N)/(Z/N) is derivedequivalent to the category of constructible sheaves on S1 which are locallyconstant outside the N roots of unity in S1. As N → ∞, the tower oforbifolds (P1)(N)/(Z/N) forms a projective system. One would naivelyexpect the category of coherent sheaves on this limit (a priori, a notionthat can be defined in many diffent ways) to be derived equivalent to thecategory of sheaves “with exceptional fibers at Q/Z ⊂ S1”, which from thepoint of view of constructible sheaves is a nonsense object. Miraculously,there is a way to make sense of such a construction to get a much morecomplete result.

1.1.4 Inverse systems, almost quasicoherent categories andthe main theorem

Note that given a family of surjective maps of affine varieties X0 ← X1 ←. . . , the projective limit can be understood as an honest affine variety,albeit not of finite type. Namely, the rings O(X0) → O(X1) → . . . area sequence of injective maps of commutative rings, and we can formallydefine lim

←Xi := Spec(lim

→O(Xi)) to be the spectrum of the direct limit

of these rings. A similar result holds for group schemes, and the inverselimit of the group schemes Z/N = Spec(k[Z/n]) is the group scheme

Z/∞ := Spec(k[Q/Z]).

This is the geometric version of the algebraic fundamental group of Gm,and acts on the universal cover of Gm, which is lim

←G(N)m := Spec(k[Q])

(note that we are taking finite polynomials with exponents in Q, notseries). Similarly, the inverse limit of ramified covers (A1)(N) has an affinelimit,

(A1)(∞) := Spec k[Q≥0].

Applying this construction locally (on the upper and lower hemispheres),we get a ramified limit (P1)(∞). Now the affine object (A1)(∞) (or, moreoften, a completion of its pro-p piece) comes up in number theory, whereone associates to it a new category of sheaves using an algebraic theory(originally studied by Faltings, [Faltings]) called “almost mathematics”.The map of rings k[Q≥0] → k taking a polynomial to its constant co-efficient gives a point 0 ∈ (A1)(∞). Given a vector space V , it givesa skyscraper sheaf δV over 0 on (A1)∞ (with k[Q≥0] action factoringthrough the unit coefficient). This is a Serre subcategory, Qcoh∂ (some-thing which is not true at any finite level), and in particular we have awell-defined quotient category, which is known as the “almost quasico-herent category” Qcoha((A1)(∞)) := Qcoh((A1)(∞))/Qcoh∂ . It was firstobserved by Faltings that, in the context of number theory, the categoryof almost quasicoherent sheaves Qcoha((A1)(∞)) is in many ways betterbehaved than the category of all quasicoherent sheaves (and, for someapplications, even better than the category of coherent sheaves at anyfinite stage, because the variety (A1)(∞) over a local field will be per-fectoid). A similar category makes sense for (P1)(∞) (quotient out the

4

Serre subcategory of skyscraper sheaves at 0 and ∞). This category hasaction by the geometric fundamental group Z/∞, and it makes sense toconsider the equivariant category, which we think of as the category ofalmost quasicoherent sheaves on the orbifold,

Qcoha(P1orb-∞) := Qcoha

((P1)(∞)/(Z/∞)

):

the “correct” inverse limit of the quasicoherent categories of (P1)(n)/(Z/n).Our main theorem in the one-dimensional case is now, in terms of this

notation, as follows.

Theorem 2. We have an equivalence of derived categories

Db Qcoha(P1orb-∞) ∼= Db Shv(S1),

where Shv(S1) is the category of all topological sheaves on S1.

It turns out that taking the inverse limit over all N is overkill, andwe can write down a similar statement for the tower of ramified covers(P1)(pn) for powers of p, or any other infinite tower of this sort. In thehigher-dimensional case, the result will turn out to be independent notonly on the tower of ramified covers, but also (on the level of derivedcategories) on the choice of toric compactification X of Gnm. Namely, forany toric variety X and any integer N there is a canonical variety X(N)

over X, which is isomorphic to X and extends the map t 7→ tn : T → Ton the open orbit. This map has action by the N -torsion group (Z/N)n

in the torus. In the limit, we get a variety X(∞) which has an action bythe geometric fundamental group (Z/∞)n of T = Gnm. It also has a well-defined category Qcoha(X(∞)) of almost-quasicoherent sheaves, which isthe Serre quotient by sheaves pushed forward from the boundary variety.We once again write down the quotient X(∞)/(Z/∞)n := Xorb−∞ whichcompactifies T , and define the category Db Qcoha(Xorb-∞) as the categoryof equivariant objects in Db Qcoha(X(∞)). Now in the general case, wehave the following theorem.

Theorem 3. We have an equivalence of derived categories

Db Qcoha(Xorb-∞) ∼= Db Shv(S),

where Shv(S) is the category of all topological sheaves on S.

(See Theorem 5 for a more precise statement).In particular, the derived category Db Qcoha(Xorb-∞) does not depend

on the choice of compactification X of the torus. (Note however that theAbelian category Qcoha(Xorb-∞) does depend on the specific compactifi-cation X.)

Our proof of the coherent-constructible-correspondence in this con-text uses the toric structure on X. But the set-up for the categoryQcoha(Xorb-∞) does not. Indeed, say U ⊂ X is any subvariety withnormal crossings (or, more generally, locally toric) boundary and withgeometric fundamental group surjecting onto the local (monodromy) fun-damental group near any point of the boundary. Then given a cofiltrationΓ1 ← Γ2 ← . . . of the geometric fundamental group πg1(U) by finite quo-tients, we get a sequence of covers U (1) ← U (2) ← . . . . It is not hard to

5

see that there is a unique extension of U (i) to a maximally ramified coverX(i) locally of toric type. We now obtain a quasicoherent category by theperfectoid-inspired procedure:

1. Take an inverse limit X(∞) of the X(N), viewed as a (non-finite type)scheme and similarly view the inverse limit Γ∞ of the ΓN as a groupscheme.

2. Consider the category of “almost coherent” sheaves, defined as a quo-tient of Qcoh(X(∞)) by sheaves pushed forward from the boundary.

3. Consider the category of Γ∞-equivariant objects in Qcoha(X(∞)).

The output of this procedure is a category Qcohlog-perf(U,X) which wecall the “log-perfectoid” category. We conjecture that this is the rightobject to consider on the algebraic side in more general microlocal mirrorsymmetry conjectures (when the B side involves an open variety and theA side has a symplectic skeleton).

1.2 Topos-theoretic interpretation of coherent cat-egoriesThe guiding principle of the present paper is to replace algebraic cate-gories and functors with topos-theoretic ones using (Abelian) categoriesof sheaves on Grothendieck sites. (The fact that constructible sheaves ona toric variety is a Grothendieck topos was first observed without proofin the note [2], and interestingly, implies exotic tensor product structureson the categories we consider.) While many of the arguments can be in-terpreted using modern derived theory with derived glueing of categoriesreplacing topoi (similarly to [7]), the point of view of sheaves on sitesseems essential (or at least very convenient) for certain “analysis-style”limit arguments: especially, for comparing categories in the “Novikov”and “perfectoid” contexts where the relevant equivalences of Grothendiecktopologies simply falls out of the fact that Q is dense in R (see Lemma39).

Recall that a Grothendieck topology (also known as a “site”) is acategory-theoretic extension of the notion of topological space with justthe right axioms to be able to define a sheaf condition. Grothendieckinitially introduced Grothendieck topologies (and the associated theory oftopoi) to work with étale sheaves. Recall that étale opens on a topologicalmanifold M are maps U → M which interpolate between the notions ofopen embedding U ⊂- M and covering space M̃ -- M . Grothendieckobserved that sections of a sheaf on different étale U (defined as globalsections of the pullback) satisfy a sheaf condition with respect to “étalecovers” (surjective étale maps

⊔Ui → X). He formalized the data of étale

opens and étale covers into the category-theoretic notion of “site” (herecalled Grothendieck topology). All functorial assignments

(étale open)→ vector space

satisfying the sheaf condition with respect to étale covers are called sheavesin the étale topology. For an algebraic variety X, one can similarly studysheaves, either in the Zariski topology or in the étale topology (where

6

the étale maps U → X are also required to be algebraic). Grothendieckobserved that for smooth manifolds, sheaves in the étale topology coincidewith sheaves in the ordinary topology (with respect to open subsets andopen covers). But in the context of algebraic geometry, sheaves in the étaletopology are in fact very different from sheaves in the Zariski topology,and (in the case where the base field is C) are a much better approximation(especially for doing homological algebra) to the category of topologicalsheaves on the underlying space.

In section 5, we introduce a new pair of Grothendieck topologies Θ, Θand a pair of equivalences of (Abelian) categories

QcohT (X) ∼= Shv(Θ)

Qcoh(X) ∼= Shv(Θ).

Since coherent sheaves are fundamentally (at least locally) an algebraic ob-ject, it is remarkable that they admit an interpretation in this topologicallanguage, and this algebro-topological relatinoship should be understoodas a mirror symmetric phenomenon at the Abelian cataegory level: themore so because the Grothendieck topology Θ is closely related to theétale topology on the topological manifold S involved in the coherent-constructible correspondence (and Θ, correspondingly, is related to theordinary topology on S̃).

We briefly sketch how the topologies Θ,Θ are defined, and how theyproduce a mirror symmetry functor. If X is an n-dimensional toric varietyand L is an ample line bundle on X, X inherits a Kähler structure fromthe projective embedding associated to the full pencil for L. Given theKähler structure together with the toric action gives a polytope ∆L ⊂ Rn:the image of the moment map on X (viewed as a symplectic variety)associated with the (real) Lie algebra action. These polytopes can beinterpreted in a more algebraic way, as the convex hulls in Rn of theweights of (homogeneous) sections of L. The key observation which relatesQcohT (X) with topos theory is the fact that, for fixed F , the assignmentF 7→ HomQcohT (L,F) behaves in a very sheaf-like way with respect to ∆L

as we vary L. Indeed, if a collection of polytopes ∆L1 , . . . ,∆Lk happen tocover another moment polytope ∆L , then so long as for each collectionof indices i1, . . . , ik

each intersection⋂j

∆(Lij ) is itself a moment polytope ∆Li1,...,id(P1)

(something that happens in a positive density of cases), then for anyequivariant sheaf F ∈ QcohT (X) we have

the diagram Homeq(L,F)→⊕i

Hom(Li,F)⇒ Hom(Lij ,F) is an equalizer.

(P2)Implicit in this statement is that the arrows above can be canonicallydefined: this follows from the fact that equivariant line bundles on X (upto isomorphism) are in bijection with equivariant Cartier divisors, whichform a partially ordered set.

7

The property (P1) is too restrictive to turn the poset of moment poly-topes into a Grothendieck topology. Instead what we do is weaken thecondition (P1) and consider all equivariant line bundles on X. In this cat-egory diagrams of the form (P2) have a natural generalization, and thisturns out to be a bona fide Grothendieck topology. For our purposes, it isconvenient to extend the poset of equivariant Cartier divisors to a largerposet, where we allow some components of multiplicity of +∞ (but not−∞). We denote these (Cartier) “quasidivisors,” QDiv. Such “divisors”are still associated with line-bundle-like sheaves (though now possibly qua-sicoherent rather than coherent), and when the resulting sheaf is ampleits moment polytope is a convex unbounded polytope in Rn (thought ofas having some faces at ∞). We write the resulting Grothendieck topol-ogy (QDiv, Θ) or Θ for short. A very useful property of the topology Θis “existence of (enough) points”, which says that this category possessesa collection of stalk functors which are conservative (i.e. which distin-guish isomorphisms of modules). The “mirror” to the stalk functors arefunctors from the category QcohT (X) to Vect, which turn out to be func-tors of (graded) sections on standard affine subsets. This means that the“stalkwise local” point of view on sheaves on Θ corresponds under ourequivalence of Abelian categories to the “affine local” point of view onQcoh(X)! The interplay between these two points of view (especially af-ter generalizing this picture to Novikov varieties) will be one of the mostimportant technical tools in this paper.

The Grothendieck topology (QDiv,Θ) on equivariant quasidivisors hasaction by the character lattice X∗(T ) ∼= Zn, which can be thought of ei-ther as shifts by equivariant characters or equivalently as shifts by homo-geneous principal divisors. We can “quotient out” (in a category-theoreticsense) by this equivariance to get a new Grothendieck topology (QDiv,Θ).In the same way that objects of QDiv behave as a generalization of poly-topes in Rn, objects of QDiv behave as the same polytopes, but nowmapping as étale opens to the quotient Rn/Zn = S. Using the dualityresult in section 2 (together with some formal nonsense from Appendix[topoi and equivariance]) we deduce that sheaves on the Grothendiecktopology (QDiv,Θ) are equivalent as an Abelian category with the cate-gory of non-equivariant sheaves, Qcoh(X).

Note (in the equivariant case) that if it were the case that all objectsof QDiv correspond to convex polyhedra in a functorial way compatiblewith open covers, then we would have a map of Grothendieck topologiesB : Opens(Rn) → Θ with the “pullback” of a divisor being the interiorassociated polytope (an object of the category underlying the topology,Open(Rn)). In fact, this is true if and only if X is an affine toric variety,in which case the derived functors of the pullback and pushforward withrespect to B coincide with the [4] ccc functor B∗ = ccc : Db QcohT (X)→Db Constr(Rn) and its adjoint. When X is not affine, it is no longer truethat all line bundles are ample, and so the convex polytope ∆L cannot bedefined in general. Instead, what becomes associated to a general divisor isa certain derived sheaf of sets. In the language of∞-categories these give a(finite) derived correspondence Bder : Opens(Rn)→ Θ. It is still possibleto define pullback and pushforward along this correspondence, so longas one is dealing with derived categories. The pullback and pushforward

8

functors (Bder)∗, (Bder)∗ with respect to this correspondence will then be

dg functors, which provide us with the ccc functor and its adjoint in thegeneral case. In cases where there is no ambiguity, we still write B∗, B∗for the derived functors (Bder)

∗, (Bder)∗.

1.3 Novikov toric varieties and almost local sheavesRecall that a toric variety is a compactification of the torus T := Spec k(Zn).A Novikov toric variety is a compactification (in a non-Noetherian sense)of the Novikov torus, TNov := Spec k(Rn). Here k(Rn) is the group al-gebra. We denote elements p ∈ k(Rn) “Novikov polynomials” and writep =

∑λ∈Rn pλt

λ with almost all pλ = 0. The notation is in analogy with“Novikov series”, which are certain infinite sums

∑prt

r over r ∈ R, whichappear on the Fukaya side of most mirror symmetry statements in a re-lated, but not entirely analogous, context.

Given a full-dimensional cone Λ ⊂ Rn (closed, convex with nonemptyinterior, and invariant with respect to multiplication by R+,) we definek(Λ) to be the (unital, commutative) semigroup algebra of Λ. We observethat in this case the ring of functions k(Rn) on the Novikov torus is alocalization of k(Λ): indeed, in order to get k(Rn) from k(Λ) it is sufficientto invert any tλ for λ in the interior of Λ. If Σ ⊂ Rn is a polyhedral fanthen the k[σ∨] are all partial completions of TNov which glue to forma variety, XNov

Σ . If Σ is a rational polyhedral fan with (ordinary) toricvariety X associated to Σ then we write XNov = XNov

Σ the “Novikovvariety associated to X”. A general Novikov fan will not be rational andin particular will not be associated to an ordinary toric variety.

Now the short exact sequence of groups Zn → Rn → Rn/Zn gives ashort exact sequence of group schemes Spec k(Rn/Zn) → Spec k(Rn) →Spec k(Zn). We introduce the notation Spec k(Rn/Zn) =: S∼. Now TNovacts on XNov

Σ for any fan Σ in a similar way to how T acts on a toricvariety. This means that (Sn)∼ ⊂ TNov also acts on XNov

Σ . For Σ arational fan with associated toric variety X, the quotient XNov/S∼ is astack which compactifies (in a suitable sense) TNov/S∼ ∼= T . In fact, thisstack should be thought of as having underlying variety X and stabilizerisomorphic to Spec k(Rk/Zk) on each n − k-dimensional open orbit inX. Thus one possible point of view on toric varieties (and the point ofview we essentially take here) is that in fact the variety XNov is a “morefundamental” object than X and X is simply the space

“XNov//S∼”

underlying the stack XNov/S∼.We will be interested not in the entire category Qcoh(XNov) but rather

a slightly smaller subcategory, Qcoha(XNov) of “almost local modules”.The qualifier “almost” comes from Faltings’s theory of almost geometry(see [3]) transposed to a global context. In the terminology of [3], if Ris a valuation ring with non-discrete valuation group, with residue field kand fraction field K, we say that a module V over R is almost local if therestriction map V (= Hom(R, V ))→ Hom(a, V ) is an isomorphism. Whenthe valuation is not discrete, this property is strictly weaker than being

9

a module over K: for example, the maximal ideal a is itself almost local,but never a module over K. In fact, in the non-discretely valued case, thecategory Moda(R) of almost local modules is much closer to the categoryof R-modules than to K-modules, as there is a Serre exact sequence ofcategories Mod(k) → Mod(R) → Moda(R) (whereas in the discretelyvalued case, the Serre kernel of the functor Mod(R)→ Mod(K) would bethe much larger category of all modules with support some thickening ofthe special fiber). In this paper, a coherent sheaf over a Novikov varietywill be called almost local if its restriction (as well as the derived functorsof the restriction) to any proper closed equivariant Novikov subvarietyis trivial. Define the ring of Novikov taylor series to be the completionR̂ := k̂(R≥0) of the semigroup ring R := k(R≥0). One can then showthat almost locality of a sheaf F is equivalent to almost locality in thesense of [3] of certain completions of F over the non-discretely valuedring R̂. Similarly to the case of perfectoids, we have a Serre sequence ofcategories Qcoh(∂XNov) → Qcoh(X) → Qcoha(X), where Qcoh(∂XNov)is the category of sheaves over the (closed) boundary X \ TNov. Despitethe fact that in the definition of almost locality we “puncture” the closedfiber of the complement XNov

Σ \TNov, the category of almost local sheaveson XNov

Σ depends in a nontrivial way on the fan Σ and is in general quitefar from the category of sheaves on TNov. When Σ is a regular rational fanwith toric variety X = XΣ the (usual) category of quasicoherent sheaveson X will be naturally a full subcategory of Qcoha(XNov

Σ ).We work with three equivariant flavors of Qcoha(XNov

Σ ) : other thanQcoha(XNov

Σ ) itself, we also consider equivariance under the Novikov torusQcoha(XNov

Σ )TNov and under the subgroup S∼ (= Spec k(Rn/Zn)) ⊂TNov. With these three equivariant flavors, we prove the following results.Let Σ be a fan in Rn and let Λ := ∪σ∈Σσ ⊂ Rn be its support. ViewT ∗(Rn) as Rn × Rn with the second coordinate the normal coordinate,and similarly view T ∗S ∼= S × Rn, where S := Rn

Zn . Then the manifoldRn × Λ ⊂ T ∗Rn gives a valid singular support condition for sheaves onRn, and similarly S × Λ ⊂ T ∗S is a valid singular support condition forS. (Here we should think of the coordinate of the ambient space of thefan as being the second, “momentum” coordinate.)

Theorem 4. Then

Db Qcoha(X)TNov ∼= Db ShvRn×Λ(Rn), (1)

Db Qcoha(X)S∼ ∼= Db ShvS×Λ(S), and (2)

Db Qcoha(X) ∼= Db ShvRn×Λ(Rn)Rn

. (3)

The notation Shv here means sheaves in the topological sense (whichmight not be constructible with respect to any stratification). The cat-egory ShvRn×Λ(Rn)R

n

is the category of Rn-equivariant sheaves on Rn,which includes much more than just constant sheaves: for example, thedirect sum of skyscraper sheaves at all points of Rn is naturally Rn-equivariant. Lines (2) and (3) above follow via some formal nonsense(see section 2) from (1), which can be considered the main result of thispaper.

10

1.4 Statement of main resultsHere we collect all of our results together. Let Σ be a toric fan (withcones that polyhedral but are allowed to not be rational) in the spaceNR with dual space MR, with “support” Λσ := ∪σσ ⊂ NR. Let A ⊂ NR(for most of the paper, we will work with A = NR) be a dense subgroupwhich contains M ⊂ MR and such that each cone σ ∈ Σ is spanned byits intersection with A. The group scheme TNov−A is defined to be theaffine group scheme with ring of functions k[A]. Write SA = A/M (whenA = MR this is the mirror torus); it is a group, and (ignoring any topologyon S), we define S∼A to be the affine group scheme Spec(k[SA]). We havean inclusion S∼A ⊂ TNov−A (with quotient T = Spec(k[M ]), as shownin Lemma 12). Associated to the fan Σ we define a TNov-equivariantscheme XNov−A

Σ which compactifies TNov. The scheme XNov−AΣ supports

a quasicoherent category Qcoh(XNovΣ ) and inside an Abelian subcategory

Qcoha(XNovΣ ) of “almost local” quasicoherent sheaves, which is equivalent

to the almost category (in the sense of [3]) associated to the idempotentsheaf of defining ideals for the boundary ∂ ⊂ Xσ. There is a sense in whichan additive category over k can be equivariant with respect to a groupscheme G, and the category Qcoha(XNov

Σ ) is equivariant with respect toTNov−A, and hence also with respect to S∼A ⊂ TNov−A. In particular, wecan define categories of equivariant objects Qcoha(XNov−A

Σ )TNov−A andQcoha(XNov−A

Σ )S∼A with respect to these actions. On the other hand, let

Shv(MR) be the category of all topological sheaves on MR. This categoryhas equivariance with respect to A ⊂MR (viewed as a discrete group andacting by transations) and also with respect to M ⊂ A. We can alsodefine categories of equivariant objects with respect to these actions, andobserve that since the action ofM onMR is discrete and free, the categoryof equivariant objects Shv(MR)M ∼= Shv(S) for S := MR/M .

With this set-up we prove the following mirror symmetry results. (Ananalogue for the dense subgroup A of theorem 4):Theorem 5. Suppose the fan Σ is compete, i.e. ΛΣ = NR. Then we havethe following three equivalences of derived categories.

Db Qcoha(XNov−AΣ )TNov−A ∼= Db Shv(MR). (4)

Db Qcoha(XNov−AΣ )S

∼A ∼= Db Shv(MR)M

( ∼= Db Shv(S)). (5)

Db Qcoha(XNov−AΣ ) ∼= Db Shv(MR)A (6)

It is from part (5) above, applied to the dense subgroup A = Q⊗M ⊂M , that we deduce the log-perfectoid mirror symmetry interpretation(Theorem 3).

2 Equivariant Categories and Spectral Pon-trjagin DualityIn this section we collect some technical categorical results which willallow us to change equivariance conditions in mirror symmetry results.

Recall that if G is a topological commutative group, its Pontrjagingdual is G∨ := Hom(G,U(1)). This is equivalent to the unitary spectrum

11

of G, with product structure corresponding to tensor product of unitaryrepresentations. A closely related concept in algebraic geometry: if Gis a finite group scheme over a field k, then the group of its charactersHom(G,Gm) is also a finite affine commutative group scheme, called the“Cartier dual” G∨ to G. If G = SpecH for H a finite dimensional commu-tative, cocommutative Hopf algebra then G∨ ∼= Spec(H∗). In this section,we will study categorical actions associated with an intermediate kind ofduality we call spectral Pontrjagin duality, which is between a discretecommutative group G and a dual group scheme G∼ over k.

Definition 1. If G is a discrete commutative group, write G∼ := Spec k(G)for the affine group scheme with function ring the group algebra of G,viewed as a Hopf algebra.

Remark 1. When G is a finite group, the duality G → G∼ is a specialcase of Cartier duality. Indeed, the discrete group scheme G × ptk andSpec k(G) are Cartier dual finite group schemes. However, when G is aninfinite group, the dual Hopf algebra k(G)∗ is too large an algebra, andSpec(k(G)∗) does not behave, e.g. from the point of view of representationtheory, in the same way as G. It might be possible to express the dualityG� G∼ as a special case of a generalization of Cartier duality if we passto a larger category than affine group schemes: e.g., by thinking of G×ptkas a group object in Ind-schemes. However, in order to avoid introducingunnecessary formalism in this section we will restrict ourselves to studyingthe representation theoretic behavior of the duality functor G 7→ G∼ fromdiscrete groups to affine groups schemes.

2.1 Categorical equivarianceDefinition 2. Suppose C is a category and G is a group. A (strict) G-equivariant structure on C is a strict representation of G on endofunctorsof C. I.e., it is a collection of functors X 7→ gX (g ∈ G) with isomor-phisms h(gX) ∼= (hg)X and a cocycle condition that says that the twodifferent natural isomorphisms i(h(gX)) ∼=(ihg) X coincide.

Definition 3. Suppose C is a k-linear category and G is an affine groupscheme over k. Suppose further that C is Abelian and has all colimits(this is not strictly necessary, but simplifies exposition). Write CG forthe category of objects of C fibered over the scheme G, i.e. with O(G)action. A G-action on C is then an Abelian functor A : C → CG togetherwith an isomorphism A ◦ A ∼= µ∗A of functors C → CG×G. Here µ∗ :CG → CG×G is the pullback functor along the multiplication map of schemesµ : G×G→ G. It is well defined because pullback is a colimit (actually, atensor product) and C has all colimits.

Writing µ3 : G×G×G→ G for the (associative) multiplication map,we have two natural isomorphisms A ◦ A ◦ AX ∼= µ∗3AX for X ∈ C anobject. We require these to be equal.

A consequence of these axioms is that the composition C A- CGfib- C

is the identity, where CGfib- C is the fiber at 1 ∈ G functor (well-defined

because of the existence of colimits).

12

Definition 4. For G a discrete group, let EG be the category with objectsXg | g ∈ G and a unique invertible map Xg → Xh for any g, h. Thiscategory has an obvious (and unique) G-equivariant structure with hXg =hgX. If C is an arbitrary G-equivariant category, define the category CGof G-equivariant objects in C to be the category FunG(EG, C) of functorsstrictly compatible with G-equivariance. More concretely, an object of CGis an object X ∈ C with isomorphisms ρg : X ∼= gX satisfying ρ1 = 1 andsuch that the diagram

Xρg

∼- gX

ghX

ρgh ∼

?∼- g(hX)

gρh ∼

?

(7)

commutes. Here gρh is obtained by applying the g-action functor to thearrow X

∼ρh

- hX.

Now we make a similar definition for a group scheme G.Definition 5. Suppose G is a group scheme and C is an Abelian G-equivariant category with all colimits. We say that a G-equivariant objectof C is an object X of C together with a functorial isomorphism of G-fibered objects (objects in CG): A(X) ∼= XG. Here XG is the pullback of Xalong the map G→ ptk .

The isomorphism is required to satisfy an analogue of the commuta-tivity restriction of the diagram (7) written in a fibered way: namely, thediagram

Xρ - XG

XG

ρ

?∆- XG×G

A(ρ)

?

(8)

must be commutative.

It is easy to check that these definitions of G-invariants as well asG-invariants are functorial. Further, the following lemma is standard:

Lemma 6. If C, C′ are G-equivariant categories as above and C → C′ isa G-equivariant functor which is an equivalence of categories, then thefunctor CG → (C′)G is an equivalence of categories. Similarly, if C, C′are G-equivariant categories and C → C′ is a G-equivariant functor whichis nonequivariantly an equivalence, then CG → (C′)G is an equivalence ofcategories.

We will extensively use the following lemma, which essentially saysthat if we consider a category C with trivial G∼-action then equivariantobjects are the same as G-graded objects.

13

Definition 6. 1. Given a category C, write Cgr G for the category ofG-graded objects of C (i.e. collections of G-tuples objects of C withHom({Xg}, {Yg}) :=

∏g∈G Hom(Xg, Yg).)

2. If C has symmetric monoidal structure and all colimits, the categoryCgr G has graded (unsigned) symmetric monoidal structure, with

({Xg} ⊗ {Yg})h := ⊕g1+g2=hXg1 ⊗ Yg2 .

3. Similarly, if C is a module category over the symmetric monoidalcategory a then agr G acts on Cgr G in a similar way.

Definition 7. 1. Write CG for the category of “sheaves of C-objectsover the scheme G”, i.e. objects of C with OG-action.

2. If C is symmetric monoidal with colimits, define symmetric monoidalstructure on CG by F⊗G := µ∗(F�G). Here if F ,G are objects withOG action then F � G is F ⊗ G viewed as an object with OG ⊗OG-action and the functor µ∗ pulls back the OG⊗OG action to OG actionvia the comultiplication map µ∗ : OG → OG ×OG.

3. Simiarly, if C is a module category over the symmetric monoidalcategory a then aG acts on CG in a similar way.

Lemma 7. Suppose G is a commutative discrete group scheme. Write

G := G∼.

Suppose that C is an Abelian category with all colimits and further, supposeevery object of C can be filtered by compact subobjects. We view C as botha G-equivariant and a G-equivariant category with trivial action.

1. We have an equivalence of categories CG ∼= CG. This equivalenceof categories is compatible with monoidal structure if C is monoidaland with module category structure if C is a module category overthe monoidal category a.

2. We have an equivalence of categories CG ∼= Ggr G. This equivalenceof categories is compatible with monoidal structure if C is monoidaland with module category structure if C is a module category overthe monoidal category a.

Proof. Part (1) follows from the definition. Part (2) can be rephrased asthe following proposition:

Proposition 8. Every representation of G is diagonalizable.

To prove this, filterG by finitely-generated Abelian subgroupsGi (withi an index in some totally ordered set). Write Gi := G∼i , a projectivesystem converging to G. Suppose that V is a representation of G, and letVi ⊂ V be the maximal subrepresentation on which the G action factorsthrough Gi. Now we know from standard algebraic geometry (and theclassification of finitely generated abelian groups) that representations ofGi are diagonalizable. It suffices to show that the Vi filter V , equivalentlythat any compact U ⊂ V is stabilized by the kernel of some map G→ Gi.Indeed, recall that the structure of a G-representation is defined by a

14

map V → VG = V ⊗ k(G). By compactness, the image of U ⊂ V willland in some finite sum ⊕Ni=1t

gi , which lies inside some finitely generatedpiece V (Gi). This proves the proposition, and hence the lemma. Thecompatibility with tensor product is clear.

Corollary 9. An algebra in the category of G-representations is equivalentto a G-graded algebra and an algebra in the category of G-representationsis equivalent to a G-graded algebra (in the obvious sense: i.e. a sheaf a overG with algebra structure A� A→ A lying over the map µ : G×G→ G).

It follows from this result that the category VectG is symmetric monoidallyequivalent to the category of G-graded modules.

Definition 8. In particualr, we have a special class of objects χg ∈ VectG

for g ∈ G corresponding to one-dimensional vector spaces in degree g ∈ G.Definition 9. Similarly, the category VectG is equivalent to the categoryof sheaves over G, and in particular we have a special sheaf χG which weview as a family of one-dimensional representations fibered over G.

Note that the category Vect acts by tensor product on any abeliancategory over k with all colimits. It follows that VectG acts on the G-invariants CG and, respectively, VectG acts on the G-invariants CG for Ca G-equivariant, respectively, a G-equivariant Abelian category with allcolimits.

Lemma 10. If C is a G-equivariant category with all colimits then thecategory CG has G-action via the functors αg : X 7→ X ⊗ χg for g ∈ G.Simiarly, if C is a G-equivariant category with all colimits then the categoryCG has G-action via the functor AG : C → CG given by X 7→ X ⊗ χG.

Proof. Simple verification.

Now we are ready to prove the main theorem of this section.

Theorem 11 (Categorical Pontrjagin Duality). 1. Suppose C is a com-pactly generated G-equivariant category with all colimits. Then wehave an equivalence of categories (CG)G ∼= C

2. Suppose C is a compactly generated G-equivariant category with allcolimits. Then we have an equivalence of categories (CG)G ∼= C.

Proof. Suppose we are in case (1). Let O := k[G] be object of (VectG)G

with G-action by multiplication and evident G-equivariance. The functorX 7→ X⊗O gives us a functor C → (CG)G. In the other direction, note thatgiven an object E of (CG)G and forgetting the G action, we get an objectforgG(E) of CG with trivial action (as the action of G twists by charactersof G, which are trivial when we forget the G action). By lemma 7, thisgives the object of C underlying E a G-grading. It is a simple check thatthe functor E 7→ forgG(E)〈0〉 sending E to one of its graded componentsis an inverse functor to X 7→ X ⊗O.

Now suppose we are in case (2). Then (abusing notation) we can viewthe vector space O := k[G] as an object of (VectG)G in an evident way,giving a functor C → (CG)G with X 7→ X ⊗ O. In the other direction,observe once again that we have a forgetful functor forgG : (CG)G 7→ CG

15

where the G-action is trivial. We have a fiber functor fib : CG → C givenby taking the fiber at the point 0 ∈ G of C viewed as an object of CG. It isanother simple verification that fib◦forgG is an inverse to X 7→ X⊗O.

2.2 Short exact sequencesWe prove a couple more technical results about the behavior of categoricalequivariance with respect to short exact sequences of groups. Suppose0→ G′′ → G→ G′ → 0 is a short exact sequence of commutative groups.

Proposition 12. The sequence (G′)∼ → G∼ → (G′′)∼ is an exact se-quence of affine commutative group schemes.

Proof. Recall that a sequence G′ → G→ G′′ of affine commutative groupschemes is a short exact sequence if and only if the map G π- G′′ isflat and the map i : G′ → G maps G′ isomorphically to the fiber π−1(0).

Now writeG := G∼,G′ := (G′)∼,G′′ := (G′′)∼, with mapsG′ i- G π- G′′dual to the corresponding maps of groups. The map π∗ on functions isthe embedding k[G′′] → k[G] which evidently flat (in fact, free). Thefiber π−1(0) is Spec(k[G] ⊗k[G′′] k) where the map of rings k[G′′] → k isthe augmentation map. Evidently, k[G]⊗k[G′′] k ∼= k[G′], completing theproof.

Write G′ → G→ G′′ for the sequence of dual groups.

Lemma 13. 1. Suppose C is a small G-equivariant category. ThenCG′′is naturally equivalent to a G′-equivariant category and using

this model we have an equivalence of categories (CG′′

)G′ ∼= CG.

2. Suppose C is a small G-equivariant category. Then CG′is naturally

equivalent to a G′-equivariant category and using this model we havean equivalence of categories (CG

′)G′′ ∼= CG.

Proof. The statement is of model theoretic nature, and is proven mostnaturally by introducing a model category structure on G-equivariant(resp., G-equivariant) categories with weak equivalences given by equiv-ariant maps which are equivalences of categories. We will give a proofwithout explicitly introducing this formalism.

Suppose we are in the case (1). Then recall that CG′′is defined as the

category of G′′-equivariant functors E → G′′, equivalently the categoryof strict G′′-fixed objects in the category Fun(EG′′ , C). Write C̃ for thecategory Fun(EG, C) viewed combinatorially (a set of objects and mor-phisms), as a G-equivariant category. We can choose a G′′-equivariantretract of the functor EG′′ → EG, giving a pair of inverse G′′-equivariantequivalences EG′′ ∼= EG. This pair of functors gives us an equivalence ofcategories between CG

′′and C̃strict G

′′, where the notation strict in the

superscript means we are taking invariants in a combinatorial sense (interms of invariant objects and morphisms). Using this model, the (com-binatorial) G-equivariant structure on C̃ restricts to an equivariant struc-ture on C̃strict G

′′which factors through to a G′-equivariance. We have

an equality of combinatorial objects(C̃strict G

′′)strict G′

= C̃strict G, and

16

a natural functor C̃strict G → C̃G (pre-composing with the G-equivariantfunctor EG → pt). Now we have(

C̃strict G′′)G′ ∼= Fun

(EG′ ,Fun (EG, C)strict G

′′)strict G′(9)

∼= Fun(EG×G′ , C)strict G, (10)

where G acts on EG×G′ diagonally and the isomorphism in the secondrow is a categorical adjunction result. Now EG×G′ is has a G-equivariant,equivariantly invertible category equivalence with EG, so the last term isisomorphic to EG.

Now we could prove part (2) of the lemma by reconstructing the argu-ment above in an algebro-geometric context. (Note that in fact, the resultholds for any short exact sequence of affine group schemes G′ → G→ G′′.)However, for simplicity we instead use an extension of Theorem 11 toprove it only for algebraic groups of the form G∼. Namely, we prove thefollowing proposition.

Proposition 14. We have an equivalence of categories CG′ ∼= (CG)G

′′.

Here recall that G acts on CG by character twists, and G′′ acts via theembedding G′′ ⊂ G.

Proof. Suppose X is an object of (CG)G′′. As seen in the proof of the-

orem 11, there is a G′′, i.e. a OG′′ , action on the nonequivariant objectforgG(X) ∈ C. Write fibforgG(X) for the fiber of forgG(X) over 0 ∈ G′′.Observe that this functor has G′ action. This induces a pair of functorsCG′� (CG)G

′′, which are inverse by an argument similar to our proof of

Theorem 11.

3 Algebra of ConesDefine MR := Rn. If Λ ⊂ MR is a subset of MR closed under addition(but not subtraction), the space k(Λ) with basis Λ has the structure ofan algebra, and k(∆) has the structure of a k(Λ)-module if ∆ ⊂ MR isa subset closed with respect to addition of elements of Λ. Furthermore,viewing MR as a discrete group, we have M∼R an algebraic group schemewith category of representations equivalent (in a monoidal way) to the cat-egory of MR-graded vector spaces. In particular, both k(Λ) and k(∆) forΛ ⊂MR,∆ ⊂MR as above are MR-graded, equivalently M∼R -equivariant.

We will be interested in this section in two cases: either when Λ is a(dual) polyhedral cone in MR or when it is the intersection of a (dual)polyhedral cone with a lattice M ⊂MR.

3.1 Direct limitsRecall from basic category theory that the functor Free : Sets→ Vect withX 7→ k(X) commutes with direct limits. It follows easily that its gradedanalogue Freegr MR : SetsMR → VectMR commutes with direct limits aswell. We deduce the following two results.

17

Proposition 15. 1. Suppose that U1 ⊂ · · · ⊂ Ui ⊂ . . . is a collectionof open sets indexed by a totally ordered index set i ∈ I. WriteU := ∪iUi. Then colim k(Ui) ∼= k(U).

2. Suppose that {Ui} is a collection of (arbitrary) subsets Ui ⊂ MRindexed by an arbitrary set i ∈ I. For i, j ∈ I write Uij := Ui ∩ Uj .Then the diagram

⊕i,j k(Uij) ⇒

⊕i k(Ui) → k(U) is a coequalizer

diagram.

From part (1) of the above, we deduce the following result.

Proposition 16. Suppose that Λ ⊂ MR is a sub-semigroup of MR and∆ is a Λ-equivariant subset. Let λ ∈ Λ be a point. Write tλ for thegenerator corresponding to the point λ ∈ MR. Then we have localizationsk(Λ)[(tλ)−1] ∼= k(Λ−Nλ) and k(∆)[(tλ)−1] ∼= k(∆−Nλ). Here Nλ is theset 0, λ, 2λ, . . . .

Inspired by this fact, we make the following definition. Suppose λ ∈MR is a vector and ∆ ∈ MR is a subset with ∆ + λ ⊂ ∆. Then we definethe localization ∆λ of ∆ by λ to be

∆λ := ∆− Nλ.

3.2 Polyhedral conesWrite NR := M∨R .

Definition 10. A subset Λ ⊂ NR is called a cone if Λ is closed, 0 ∈ Λ itis invariant with respect to positive dilations R+ ·Λ ⊂ Λ and Λ is convex.(Note that some authors use “cone” for something which is not necessarilyclosed and convex: a more consistent terminology might be “closed, convexcone”.)

Convexity implies that for any x, y ∈ Λ, the midpoint x+y2

is also inΛ. Dilation-invariance implies that x+y ∈ Λ as well, so that Λ is a unitalsemigroup.

We write down some standard definitions for cones.

Definition 11. 1. 〈Λ〉 is the vector space spanned by Λ, equivalently(by the semigroup property and dilation invariance,) Λ + (−Λ).

2. Λ± is the maximal vector space contained in Λ, equivalently Λ∩−Λ.

3. The “dual cone” to Λ ⊂ MR is the cone in the dual vector space NRof vectors that pair nonnegatively with Λ, i.e.

Λ∨ := {y ∈MR | 〈λ, y〉 ≥ 0∀λ ∈ Λ} ⊂MR.

4. The “orthogonal vector space” to a cone Λ is the orthogonal space toits span, Λ⊥ := 〈Λ〉⊥.

5. We say that Λ is sharp (also called “salient”) if Λ± = {0}.6. We say that Λ is full if it contains a non-empty open, equivalently if〈Λ〉 = MR.

Note that we allow cones which are not full, i.e. which are contained ina proper sub-vector space. For such cones we will slightly abuse notationas follows.

18

Definition 12. Given a cone Λ ⊂ MR, write Λ̊, the “interior of Λ” forthe maximal open set of 〈Λ〉 contained in Λ.

We collect here some simple combinatorial results about cones.Lemma 17. 1. The duality functor is contravariant with respect to in-

clusion.2. For any cone Λ, the double dual (Λ∨)∨ = Λ.

3. λ⊥ = 〈Λ〉⊥ = (Λ∨)±

4. Λ is full if and only if Λ∨ is sharp.5. Λ = MR if and only if it contains an open neighborhood of 0.

Definition 13. We say that a cone Λ is polyhedral if it is the set of so-lutions to a finite set of inequalities, equivalently an intersection of closedhalf-spaces.Definition 14. Given a lattice M ⊂ MR, we say that a polyhedral coneΛ is rational if it is the set of solutions to a finite set of inequalities withrational coefficients relative to a basis of M .Definition 15. We say that a polyhedral cone is regular if it is the positivequadrant

∑mi∈Z+ mixi for xi ∈ M ′ ⊂ M is the basis of some (possibly

lower-dimensional) sublattice M ′ = M ∩M ′R.Given a polyhedral cone Λ and a vector λ ∈ Λ, we can consider the

localization (defined above) Λλ := Λ− Nλ.Proposition 18. 1. Λλ = Λ + Rλ, and in particular is also a cone.

2. The dual cone Λ∨λ is the closed sub-cone

Λ∨ ∩ (R · λ)⊥.

Proof. The first part is obvious. For the second, note that since Λ∪±λ ∈Λλ, contravariance implies that Λ∨λ ⊂ Λ∨∩(R ·λ)∨. In the other direction,if we have some x ∈ Λ∨∩(R·λ)⊥ then 〈x, y〉 ≥ 0 for y ∈ Λ and 〈x,−λ〉 = 0so 〈x, y − rλ ≥ 0 for any r ∈ R.

If Λ is a cone, we say that Λ′ ⊂ Λ is a face of Λ either if Λ′ = Λ or ifΛ′ = H ∪ Λ, for H the boundary of a half-space which contains Λ. (Forus, a face is allowed to have arbitrary codimension.)Definition 16. Write Λ′ 4 Λ if Λ′ is a face of Λ.

From the above proposition, we get the following useful result.Lemma 19. Λ′ is a face of Λ if and only if (Λ′)∨ is a localization ofΛ∨. Further, if Λ is a rational polyhedral cone with respect to some latticeM ⊂Mλ then λ can be taken to be an integral vector λ ∈M ∩ Λ.

Proof. Suppose 0 6= λ ∈ Λ. Consider the hyperplane (R+λ)∨. Then onthe one hand, Λ ⊂ (R+λ)∨, on the other hand, by Proposition 18, we seethat the dual to the localization Λ∨λ = Λ∨ ∩ (Rλ)⊥ is the intersection ofthe localization with the boundary of (R+λ)∨. Thus the intersection willbe a face of Λ∨. In the other direction, by convexity, for any face σ ⊂ Λ∨

there is a nonnegative linear function on Λ∨ which is equal to zero exactlyon σ. Writing λ ∈ N∨R for a linear extension of this function to all of NR,we have λ ∈ Λ by nonnegativity of λ | Λ∨ and Λλ = σ∨. Obviously, if Λis rational (with respect to a lattice), λ can be taken to be rational.

19

Corollary 20. Suppose that Λ is a cone, and λ1, λ2 ∈ Λ are vectors. LetΛi := Λλi . Then (Λλ1+λ2)∨ = Λ∨1 ∩ Λ∨2

Since the {Λ′ | Λ′ 4 Λ} give a full set of faces of the polytope Λ in acombinatorial sense, we have the following result:

Proposition 21. For any polyhedral cone Λ, we have

Λ =⊔

Λ′4Λ

Λ̊′.

Now we can define a fan.

Definition 17. A fan Σ in NR is a finite nonempty set σ ∈ Σ of sharpcones in NR such that

1.⊔σ∈Σ σ̊ ⊂ NR (i.e., the cones are disjoint) and

2. if σ ∈ Σ and σ′ 4 σ is a face then σ′ ∈ Σ.

Any fan Σ comes naturally with the structure of a partially orderedset by boundary containment, with σ 4 σ′ if σ ⊂ σ′. Note since our conesare sharp and Σ is nonempty, the zero cone {0} ⊂ NR will be the initialelement of Σ. Given a pair σ, σ′ of cones in Σ, their intersection σ ∩ σ′is also in Σ. Thus, Σ is a finite poset with arbitrary intersections (i.e.,a meet-semilattice). We will use the notation Σk ⊂ Σ for the subset ofk-dimensional cones in Σ.

Most fans do not have a terminal element (when viewed as posets).Those that do are called affine. For any sharp cone σ, there is a uniquefan with terminal element σ. Namely, it follows from Proposition 21 thatthe set defined in the following definition is a fan:

Definition 18. Given a sharp polyhedral cone σ ⊂ NR, write Σσ, the“affine fan associated to Σ”, for the set of all faces σ′ 4 σ.

Definition 19. We say that a fan Σ is complete if⋂σ∈Σ σ̊ = MR.

Definition 20. We say that a fan is rational, resp., regular, if all itscones are rational, resp., regular.

A one-dimensional cone is called a ray. If ρ ⊂MR is a ray andM ⊂MRis a designated lattice ρ is rational with respect to M if and only if itcontains a nonzero integral point of M . In this case, the intersectionρ ∩M will be a semigroup isomorphic to N.Definition 21. For rational rays ρ, we use the notation gρ for the (unique)generator of ρ ∩M .

4 Toric Novikov Varieties and Almost Lo-cal SheavesSuppose that Σ is a fan. For σ ∈ Σ write

ONovσ := k(σ∨).

If σ is rational with respect to a lattice N ⊂ NR, we also define

Oσ := k(σ∨ ∩M),

20

with M := N∨ the dual lattice. If σ 4 σ′ we have a containment of ringsONovσ′ ⊂ ONovσ and Oσ′ ⊂ Oσ of both of these are defined.

Lemma 22. 1. The ONovσ , as well as (when defined) the Oσ, togetherwith the maps above give a representation of the poset Σop4 in rings.

2. For each pair τ 4 σ, the map of rings ONovσ → ONovτ , as well as(when defined) the map Oσ → Oτ , are localization maps.

3. If we have a quadruple of cones

σ

τ

>>

κ

σ′

>>

such that τ = σ ∩ σ′ then ONovσ ⊗ONovκONovσ′

∼= ONovτ , i.e. on thelevel of open subsets in SpecONovκ , we have

SpecONovσ ∩ SpecONovσ′ = SpecONovτ .

If the fan Σ is rational, we have an analogous result with ONov∗ re-placed by O∗.

The lemma above is precisely what is needed to show that the SpecONovσ ,respectively, when defined, the SpecOσ, glue together to produce a vari-ety, which we call XNov

Σ (resp., for rational Σ, the variety XΣ). In thespecial case where the fan is affine, Σ = Σσ, the glueing diagram has aterminal element, σ, and so the variety is simply the corresponding affineSpecOσ or SpecONovσ . We introduce notation:

Definition 22.Xσ := SpecOσ,

XNovσ := SpecONovσ .

Since all our rings as well as the maps between them, are MR graded(resp., in the rational case, M -graded), we get a TNov (respectively, inthe rational case, a T )-graded structure on XNov

Σ (resp., in the rationalcase, XΣ). The variety XΣ is the standard toric variety associated tothe rational fan Σ, with standard T action. We call the TNov-equivariantvariety XNov

Σ the (toric) Novikov variety associated to the fan Σ.With the glueing data above in hand, the following definition is stan-

dard:

Definition 23. A coherent sheaf F on XNovΣ is uniquely determined by

a collection of modules Fσ over ONovσ with coherent maps resστFσ → Fτfor σ ⊂ τ indexed by the poset Σop4 , such that the adjoint map

Fσ ⊗ONovσONovτ → Fτ

is an isomorphism. Similarly, for rational Σ, a coherent sheaf on XNovis determined by a diagram Fσ of modules over Oσ indexed by Σ with ananalogous localization requirement.

21

Now from Corollary 20, we deduce the following result.

Lemma 23. A TNov-equivariant coherent sheaf on XNovΣ is uniquely de-

termined by a colletion Fσ as in definition 23 together with MR-gradingsof each Fσ which are compatible with the maps resστ .

4.1 The poset Mσ

Suppose σ is a cone.

Definition 24. Let Mσ := M/σ⊥ be the dual vector space to 〈σ〉.On Mσ the cone σ∨/σ⊥ is sharp, and thus defines a partially ordered

set structure as follows.

Definition 25. For m,m′ ∈ Mσ, we say that m 4σ m′ if m′ − m ∈σ∨/σ⊥.

Note that if m,m′ ∈M we can also define a category structure on Mitself with a single morphism σ → σ′ if σ′−σ ∈ σ⊥. This will be a preorder(poset with isomorphisms), and quotienting out by the isomorphisms willgive us back the poset Mσ. Sometimes we will abuse notation slightly anduse Mσ for the above preorder on M .

4.2 Almost locality?? In the Novikov setting, we will consider a certain full subcategoryQcoha(XNov

Σ ) ⊂ Qcoh(XNovΣ ) of “almost local” objects. We begin by

defining this category for the affine case, XNovσ = SpecOσ for a cone σ.

Write aσ ⊂ Oσ for the homogeneous ideal

aσ := k(̊σ∨) ⊂ k(σ).

The ideal aσ has the wonderful property

a2σ = aσ

(the corresponding property fails for ideals in the classical toric case), andso we can apply a small piece of the theory of “almost Mathematics” of[3].1 First, a definition.

Definition 26. 1. Define the category of almost local representations,Repa(Oσ) to be the full subcategory of Rep(Oσ) consisting of repre-sentations V such that the natural map V → HomOσ (aσ, V ) is anisomorphism.

2. Write Rep∂(Oσ) for the full subcategory of representations of V onwhich aσ acts by 0.

Lemma 24. 1. The category Rep∂(ONovσ ) is a Serre subcategory. (Notethat the corresponding statement does not hold in the classical toriccase!)

1Faltings considers a slightly different context: that of non-discretely normed valuationfields. In the case where the cone σ is a one-dimensional ray in R1, imposing the almostlocality condition on sheaves is equivalent to imposing the condition in [3] on their adiccompletion, the ring of Novikov series of positive valuation.

22

2. The functor V 7→ Hom(aσ, V ) takes an object of RepONovσ to anobject of RepaONovσ and is right adjoint to the inclusion of categoriesRepaONovσ → RepONovσ .

3. The triple of functors Rep∂(ONovσ )→ Rep(ONovσ )→ Repa(ONovσ ) isa Serre quotient diagram.

All of these are proven in [?]. From this we deduce that we have ashort exact sequence of categories (in the sense of Serre)

Rep∂(Oσ)→ Rep(Oσ)→ Repa(Oσ).

Given a representation V ∈ Rep(Oσ), write V a := HomOσ (aσ, V ). We sayV a is the “almost local representation associated to V ”.

Now for a fan Σ which is not necessarily affine, write Qcoha(X) forthe category of sheaves of modules F such that F | Xσ is almost local.Alternatively, define X∂ ⊂ X to be the variety glued out of SpecOσ/aσ,a union of toric Novikov varieties of lower dimension. Define a∂ to bethe sheaf of ideals defining this variety. The condition of F being almostlocal is equivalent to the condition that F → Hom(a∂ ,F) is locally anisomorphism. Note that once again contrary to our expectations fromclassical algebraic geometry, this category is different from requiring F tobe a pushforward of a sheaf on the open orbit TNov: the latter conditioncan be thought of as the vanishing of F on a “large” formal neighborhood,X∂̂ of the boundary, whereas this is vanishing on the boundary itself, withno thickening.

If V is a representation of Oσ and τ 4 σ is a subcone, then

(V ⊗Oσ Oτ )a ∼= V a ⊗Oσ Oτ .

This means that the functor V 7→ V a, as well as the functor V 7→ V∂ glueto functors on sheaves, F 7→ Fa and F 7→ F∂ .

4.3 Quasidivisors and R-quasidivisorsSuppose that X = XNov

σ is a toric variety.Definition 27. We define

DivRσ := 〈σ〉∗

and, if σ is rational, we define the lattice

DivZσ := (〈σ〉 ∩N)∗ ⊂ 〈σ〉∗.

We view elements of DivRσ as linear functions on σ (which extend

uniquely to 〈σ〉). In particular, we say that a divisor (or R-divisor) αis effective if it is nonnegative as a function on σ. Note that for a rationalfan, divisors α ∈ Divσ are in bijection with equivariant Cartier divisorson the affine toric variety Xσ.

If Σ is a general fan, we make the following definition. Write VΣ :=⋃σ∈Σ σ ⊂ N for the “support” of a fan. Write

Definition 28. DivRΣ := {α : VΣ → R | α|σ ∈ DivR

σ}. Similarly, when Σis rational,

DivZΣ := {α ∈ DivR

σ | α|σ ∈ DivZσ}.

23

We say that a (R-)divisor is effective if α is nonnegative and principalif α = λ | Vσ for some λ ∈MR = N∗R (viewed as functions on NR).

Note that ifX = XΣ is a toric variety, elements of DivZX are in bijection

with T -equivariant Cartier divisors on X. Indeed, recall that every irre-ducible equivariant variety in X is the closure of a torus orbit. In particu-lar, irreducible codimension-one equivariant subvarieties are Yσ := Xσ \ Tclassified by rays η ∈ Σ1. The divisor α ∈ DivZ

Σ in the sense above thencorresponds to the Weyl divisor

∑α(gη)Zη, where recall that gη ∈ η is

the unique generator of the lattice subgroup. It is a standard result inthe geometry of toric varieties, which we will not use, that a Weyl divisor∑dηZη is a Cartier divisor (and hence corresponds to an equivariant line

bundle) if and only if the dη = α(gη) for some function α ∈ DivZΣ . Fur-

ther, equivariant line bundles are in bijection with equivariant divisors,and non-equivariant divisors are in bijection with the quotient DivZ

ΣM

ofequivariant Cartier divisors by principal equivariant divisors.

Recall that to every Weyl divisor d on X we can associate a sheaf,consisting of rational functions on X with degree of pole (or zero) alongany irreducible codimension-one subvariety Z ⊂ X bounded locally bydη. When d is an equivariant divisor, this sheaf is naturally equivariantand when d is a Cartier divisor, this sheaf is a line bundle. Note that ifwe formally set some of the dη = +∞, there will still be a quasicoherent(but not coherent!) sheaf associated to d, where if dη = +∞, we considerfunctions with locally arbitrary poles allowed along Zη.

It will be useful for us to partially compactify the groups DivRΣ and

DivZΣ to a larger semigroup, called quasidivisors, which (in the rational

case) precisely corresponds to extending the set of (Cartier) divisors byallowing some of the divisor coefficients to be +∞. To this end, we makethe following definitions.

Definition 29. Define¬R := R ∪+∞ and

¬Z := Z ∪+∞.

These have the structure of ordered semigroups in an obvious way

(with ∞ + k = ∞ for any k ∈¬R), and we can rescale elements of

¬R by

R+, with 0 ·+∞ := 0. Now for Σ = Σσ an affine fan, define QDivRσ to be

the set of maps of semigroups σ →¬R which commute with scaling. This

is equivalent to the following definition.

Definition 30. QDivRσ is the set of functions α : σ →

¬R which take finite

values on a single face τ ⊂ σ (of arbitrary dimension ≥ 0) and are linearon that face.

We also define

Definition 31. If σ is rational QDivZσ ⊂ QDivR

σ are those functions which

take values in¬Z ⊂

¬R on σ ∩N .

If Σ is not affine, we define QDivRΣ and QDivZ

Σ to be functions VΣ →¬R

which satisfy the conditions above on each (closed) cone σ:

Definition 32. QDivRΣ, resp. (if Σ is rational), QDivZ

Σ, is the set of

functions α : VΣ →¬R such that α | σ ∈ QDivR

σ for each σ ∈ Σ, respectively,such that α | σ ∈ QDivZ

σ for each σ ∈ Σ.

24

We say that a (R-)quasidivisor is1. effective, if α takes values in R+ ∪∞;

2. finite, if α ∈ QDivR ⊂ QDiv;

3. principal, if α = 〈λ, ·〉 for some λ ∈ Λ (in particular, such quasidivi-sors are finite),

4. affine, if α = +∞ outside a single cone σ ∈ Σ;

5. semiample, if α is convex up, i.e. if α(x + y) ≤ α(x) + α(y) (in anordered semi-group sense), and

6. ample, if α is semiample and α(x + y) = α(x) + α(y) if and only ifeither α(x+ y) =∞ or x, y both belong to some cone σ.

4.4 Sheaves associated to quasidivisors and R-quasidivisorsWe make the following definition.

Definition 33. Given an R-quasidivisor α on an affine fan Σσ, definethe set

∆α(= ∆σα) := {λ ∈MR | 〈λ, y〉 ≤ −α(y)∀y ∈ σ.}

Note the negative sign −α in the definition. This is the same signthat comes in the definition of a line bundle associated to a divisor: thelarger the multiplicity of D in a divisor, the more negative the (ID-adic)valuation of functions allowed in the ideal.

All ∆α for α finite are shifts of σ∨. Indeed, there is some λα such thatα = 〈λα, ·〉 | σ, and then ∆α = σ∨ − λα. Note that the shift vector λα isnot in general unique, as σ∨ has additive symmetry by σ⊥: rather, it isunique up to an additive factor of σ⊥. Now define aσα := k(∆̊α). This isa graded Oσ-module isomorphic in a non-graded way to aσ. Recall that ageneral quasidivisor α is finite precisely on some face τ ⊂ σ, and then ∆α

is a shift of τ∨.

Definition 34. Given an R-quasidivisor α ∈ QDivRΣ, define the TNov-

equivariant sheaf O(−α) ∈ Qcoha(XNovΣ )TNov to be the sheaf which on

the affine XNovσ ⊂ XNov

Σ has sections k(∆α), and with obvious transitionmaps. Similarly if Σ is a rational fan and α ∈ QDivZ

Σ is an integralquasidivisor, define the T -equivariant sheaf O(α) to be the sheaf which onthe affine Xσ has sections k(∆α ∩M).

It is immediate to check that these indeed glue to give equivariantand, in the Novikov case, almost local, sheaves. These sheaves will be theobjects corresponding to opens of the Dolan topology, defined in the nextsection.

5 The Dolan Topology and the AbelianCoherent-Constructible CorrespondenceAssociated to any fan Σ we will define a Grothendieck topology ΘR onthe poset QDivR which we call the “Dolan-Novikov topology”. When Σis rational, we also define an integral subtopology on QDivZ, the “Dolan

25

topology”. The main result of this section will be the Abelian coherent-constructible correspondence, i.e. the equivalence of the equivariant al-most coherent category Qcoha(XNov)TNov with sheaves on ΘR. Similarly,we will show that when Σ is regular, we have a an equivalence in thenon-Novikov (classical algebro-geometric) context Qcoh(X)T ∼= Shv ΘZ.

5.1 Grothendieck topology on a posetIn this section, some reminders about Grothendieck topologies. The cate-gories QDivZ,QDivR on which our topologies are defined are particularlynice in some of the same ways in which topoi coming from a topologicalspace are nice: the underlying categories are posets, they have all inter-sections (i.e., they are “meet semi-lattices”) and the covering families weintroduce can be easily checked to be “strongly epic”. In this situation, thetheory simplifies and we use full advantage of these special simplificationsin our definitions and results.

Suppose a category C is a poset. Then a Grothendieck topology on Cconsists of the following data satisfying the following conditions.

Definition 35. Suppose C is a poset. A Grothendieck topology Θ on Cis a collection, CovX , for every object X ∈ C, of sets of subobjects U 4 X.The poset C and the set of covering subcategories is required to satisfy thefollowing conditions.

1. (Identity) The set {X} is a covering of X.

2. (Existence of pullbacks) Given any map U 4 X with an element ofsome U ∈ CovX , and any map Y → X, the pullback U ×X Y exists(for a poset, a pullback is also called a “meet”).

3. (Basechange) Given a U ∈ CovX and any map Y → X, the set ofbasechanged maps U ×X Y

ι×XY- Y is a covering of Y as Uι- Y

goes over objects of U .4. (Subdivision) If UX ∈ CovX is a covering of X and UU ∈ CovU is a

collection, for each U ∈ UX , of coverings, the “subdivided covering”⋃U∈U

UU

is a covering of X.

All the Grothendieck topologies we consider will be “strong”, i.e. have finitelimits, and covering families will be strong epimorphisms.

Definition 36. A map of strong Grothendieck topologies α : Θ→ Θ′ is afunctor of underlying categories in the opposite direction,2 α−1 : C′ → C,such that

1. (Covering.) If UY is a covering of Y then f−1(UY ) is a covering off−1(Y ) and

2. (Continuity.) The functor f−1 respects pullbacks2This is to model the “pullback of opens” functor UY 7→ f−1(UY ) ⊂ X associated to a

map of topological spaces f : X → Y

26

Note that for Grothendieck topologies which are not strong, the cover-ing and continuity properties need to be replaced by slightly more subtleproperties.

5.2 The Dolan and Novikov-Dolan topologiesDefinition 37. Let QDivR,4 be the category of quasidivisors with mor-phisms given by the partial order d 4 d′ if d(x) ≤ d(x) pointwise. Wesay that the set di cover d if di 4 d and the supremum sup(di(x)) = d(x)for any x ∈ NR. Write Θ(= ΘΣ) for QDiv with this notion of open cover.Define ΘZ(= ΘZ

Σ) for the category QDivZ ⊂ QDivR with induced partialorder and collection of open covers.

Lemma 25. The assignments of open cover Θ,ΘR define Grothendiecktopologies.

Proof. Note that Θ and ΘZ have all limits. Checking this is equivalentto checking the existence of a terminal element and fibered product. Theterminal element is the infinite quasidivisor,

d∞ : x 7→

{0, x = 0

∞, x 6= 0

Existence of fibered products in a poset is equivalent to existence of meet(highest lower bound), which is given by d ∩ d′ := inf(d, d′). Existence ofpullback follows. Identity, basechange and subdivision are obvious.

Now we are finally able to state the main result of this section.

Theorem 26 (Abelian coherent-constructible correspondence). Supposethat Σ is a fan. Then we have equivalences of Abelian categories

1. Shv(Θσ) ∼= Shva(XNovΣ )TNov ,

2. if Σ is rational, we have an equivalence of Abelian categories Shv(ΘZσ) ∼=

Shv(XΣ)T .

5.3 Push-pull functors for fan inclusionsSuppose Σ′ ⊂ Σ is a sub-fan (we will mostly be using affine Σ′ = Σσ forσ ∈ Σ a cell). We define two maps ισσ′ : ΘΣ′ → ΘΣ and πσσ′ : ΘΣ → ΘΣ′

as follows. Suppose d : VΣ → R is an R-quasidivisor and d′ : Vσ′ → R isan R-quasidivisor.Definition 38. We define

ι−1(d) := ι | V ′Σ

and

π−1(d) : y 7→

{d(y), y ∈ VΣ′

∞, else.

We write ισ for ιΣσ,Σ, and similarly for πσ.

Proposition 27. The functors ιΣΣ′ , πΣΣ′ are morphisms of Grothendiecktopologies.

27

We see this by checking compatibility with covers and intersections.Fixing a map Σ ⊂ Σ′, write ι∗, ι∗, π∗, π∗ for the pullback and pushfor-ward functors between Shv(Σ), Shv(Σ′). These sheaves have an extremelyunusual relation (compared to what one would expect for topological orétale sites):

Lemma 28. There is a natural isomorphism between the functors π∗, ι∗ :Shv(Σ)→ Shv(Σ′).

Proof. Given a sheaf V on Σ, denote Vd for Γ(V, d) for d ∈ QDiv(Σ).Suppose V is a sheaf on Σ and d′ is a divisor in QDivΣ′ . Then ι

σ∗ (V)d′ :=

Vι−1(d′)∼= Vd′|Σ, where d′ | Σ is a restriction of d′ viewed as a function on

the support of Σ′ to the support of Σ′. On the other hand, π∗σ(V)d′ :=lim→V ′d | π−1(d) < d′. But π−1(d) < d′ if and only if d < ι−1(d′), and in

particular the poset d′ | π−1(d′) < d has an initial element, d = ι−1(d′).Thus ι∗(V)∗σ ∼= π∗(V)σ, and it is an immediate check that this isomorphismis compatible with restriction maps and functorial.

This fact will be extremely useful for us, and implies in particular thatπ∗ has an easily accessible left adjoint ι∗ (such a functor can with somelegitimacy be called π!). Further, since ι∗ = π∗, both ι∗ and ι∗ are exact(this corresponds to the fact that both pullback and pushforward from anopen subvariety is exact).

5.4 PointsRecall that a Grothendieck topology has a notion of point which general-izes the notion of a point of a variety, and that for general Grothendiecktopologies points form a category rather than just a set, and the functor of“stalks” produces a representation of this category starting from any sheafV. When there are “enough points” (as we shall see to be the case for theNovikov-Dolan topology ΘR and its relatives), we can check isomorphismof sheaves on stalks. As we shall see, stalks at points in the Dolan topol-ogy correspond to (graded components of) restrictions to affine opens ofXNov.

Definition 39. A point of a Grothendieck topology (C,Θ) is a functor C →Oppt, where Oppt is the Boolean category {∅ → pt}, which is compatiblewith intersections and open covers3.

Now suppose Σ = Σσ is an affine fan, and say m ∈ MR/σ⊥ is a point

of the dual poset associated to σ. Then we define the following functorfrom QDivσ to Sets:

Þm(d) :=

{pt, ∆d 3 m∅, otherwise.

Write StÞσm(V) for the stalk of V at the point Þσm Now any map ofGrothendieck topologies sends points to points (in a functorial way). IfΣ is an arbitrary flag and σ ∈ Σ is a cone with m ∈ M/σ⊥ as before,

3Here again we are using the strong epimorphism property

28

write Þσm := ισ(Þm), for Þm the point of Θσ as above. Note that thisconstruction gives new points in Θσ itself, corresponding to pushforwardof points from Θτ for τ 4 σ.

Lemma 29. Let Σ be a fan. The collection Þσm is a separating collectionof points for ΘΣ.

Proof.

Lemma 30. Let d ∈ QDiv be a divisor and V ∈ ShvT be a sheaf. ThenΓ(d,V) ∼= lim

→ σ∈Σop4

St(Þdσσ ), via the evident diagram of inclusions Þdττ ⊂

Þdσσ for τ ⊂ σ cones, compatible with the inclusions Þdσσ ⊂ d.It is obvious that the characteristic sheaves of open sets kd are a sepa-

rating collection of sheaves for d ∈ QDiv. Now the colimit of all skyscrapersheaves δÞmσ for Þmσ ∈ d is kd, and so the Þmσ separate.

5.5 The Abelian coherent-constructible correspon-denceWe are ready to define a pair of inverse functors between Qcoha(XNov

Σ )TNov

and Shv(Θ). Namely, consider the collection of equivariant sheaves Od ford ∈ QDiv. We have obvious maps Od → Od′ for d 4 d′, which give arepresentation of the coset QDiv .

Lemma 31. The Od form a cosheaf of objects of Qcoha(XNovΣ ) over Θ.

Proof. Working locally, it is sufficient to show this for an affine openXσ ⊂ XΣ. Using almost locality, Hom(Od,F) = Hom(Od ⊗ a∂ ,F), withOd ⊗ a∂ locally given by k(∆̊d) Thus, using Proposition 15, it is enoughto show that the shifted open cones ∆̊d satisfy the sheaf property ford ∈ QDiv(Σσ), which is obvious.

Now associated to the cosheaf OQDiv, we get a pair of functors: F :Qcoha(OQDiv)TNov � Shv(ΘΣ) : G, defined as follows.

Γ(F (F), d

):= Hom(Od,F)

Γ(G(V), Xσ)〈m〉 := StÞσm(V).

We refine the Abelian ccc to the following lemma.

Lemma 32. The functors F,G defined above are mutually inverse equiv-alences, as are their integral analogues FZ, GZ.

We will prove this lemma bit by bit in the remainder of the section.We work mostly with the Novikov case and functors F,G and only treatthe classical case FZ, GZ separately when the arguments from the Novikovcase do not translate in an obvious way. Recall that we have sub-posetsDivR ⊂ QDivR and DivZ ⊂ QDivZ of “finite” divisors. Since these posetsare cofiltering, any cover of an element of DivR consists of other elementsof DivR and intersections in DivR agree with intersections in QDivR .WriteΘf

Z ,ΘfR for the induced Grothendieck topology on DivZ,DivR .We have the

following result:

29

Proposition 33. For any fan Σ, we have an equivalence of Abelian cat-egories Shv(Θf

R) ∼= Shv(ΘR) and, if Σ is rational, Shv(ΘfZ) ∼= Shv(ΘZ).

Proof. This follows from the fact that divisors are cofiltering (i.e. closedwith respect to taking subobjects) and generating in quasidivisors (i.e.any quasidivisor has a covering by divisiors).

Now we proceed in stages.

5.5.1 Affine, rational fan Σσ

Assume Σ = Σσ is affine and rational, and we are considering the siteΘZ. Then note that the topology Θf

Z is trivial — i.e. any open coverof a finite divisor d contains d itself as a terminal element. This meansthat Shv(ΘZ) ∼= Shv(Θf

Z) ∼= Rep(DivZ), where Rep(DivZ) is the categoryof representations of the poset DivΣ . On the other hand, an object Fof Qcoh(XΣ)T is an M -graded representation of the graded semigroupΛσ ∩M . Let F〈m〉 be the m-weight component of F , for m ∈ M . Thenthe action tλ takes F〈m〉 7→ F〈m + λ〉. The relations between λ ∈ Λamount to the statement that all maps F〈m〉 → F〈m + λ〉 obtained ascompositions of ρλi for λ1 + · · · + λk = λ are equal, i.e. that F〈·〉 is arepresentation of the poset Mσ, equivalently a presheaf on Θf

Z .

5.5.2 Affine, Novikov fan Σσ

Now consider the Novikov case. An object of Qcoh(XNovΣ )TNov is an

MR-graded representation of Λσ, equivalently as above a presheaf onΘf

R. However, in this case while ΘfR is still cofiltering and generating

(and hence has the same category of representations as ΘR), its cate-gory of sheaves and presheaves are not the same, as we have nontrivialcovers. For example for M = R and σ = R+, where Div ∼= R, anyupper-bounded collection of numbers which never attains its supremums, such as −1,−1/2,−1/3, . . . , will give a nontrivial cover of s. For gen-eral affine Σ = Σσ, we once again identify DivΣ

R with the poset MσR .

Covers of the form m1 4 m2 4 . . . converging to some m ∈ MσR will

generate the topology. The sheaf condition on a presheaf F is then thatlimF〈mi〉 ∼= F〈m〉 for m1 4 m2 4 . . . 4 m as above. Now note thatF〈m〉 = Homgr(k[∆m],F), so by the exactness property of Hom, we havelimF〈mi〉 = Homgr(k[∪i∆mi ],F). Now there is a sequence of mi suchthat ∪∆mi = ∆̊mi , so that Homgr(k[∆̊m],F) = Homgr(a ⊗ Om,F), sothe sheaf condition implies the map F → Hom(a,F) =

⊕m∈MR

Hom(a⊗Om,F) is an isomorphism, i.e. F is almost local. Conversely, supposeF ∼= Hom(a,F). Then Hom(a′,F) ∼= Hom(a′,Hom(a,F)) ∼= Hom(a ⊗a′,F) ∼= Hom(a,F) ∼= F .

5.5.3 General Σ

Suppose Σ is a Novikov fan and F ∈ Qcoha(XNov)TNov is an equivariantquasicoherent sheaf. Suppose that Þmσ is a point. Then

StÞmσ F (F) := lim→ d→Þmσ

F (F)d ∼= lim→ d→Þmσ

Hom(Od,F) ∼= Γ(X,F⊗Oσ)〈m〉.

30

Further, these identifications are functorial and compatible with restric-tion maps, establishing that G ◦ F ∼= Id. In the other direction, supposethat V is a sheaf on Θ. We have a restriction map Vd → ⊕Þmσ ∈dStÞ

mσ V

which glues to a map Vd → colimÞmσ ∈d StÞmσ V. Now using the definition

of the functor G and the affine Čech resolution for maps of quasicoherentsheaves, this colimit is canonically colimÞmσ ∈d StÞ

mσ V ∼= Homgr(Om,Vd).

This gives a map Id → F ◦ G of endofunctors of Shv Θ. To check it is anatural isomorphism, it is sufficient to see this on a separating collectionof stalks, specifically on the StÞmσ , which is obvious. An analogous ar-gument shows that FZ, GZ are inverse functors. This concludes the proofof Lemma 32 and hence of the Abelian coherent-constructible correspon-dence (Theorem 26).

6 The Equivariant Novikov Coherent-ConstructibleCorrespondenceIn previous sections, we have used again and again the geometric natureof the category QDivR

Σ and the polyhedra ∆d as combinatorial bookkeep-ing tools. In this section this geometric nature comes to the forefront.We will use this geometry to define the coherent-constructible correspon-dence (dg) functors, B∗ and B∗. It then turns out to be almost immediateto check, using the formalism of points we have developed and a resultin [4], that these functors are mutually inverse. From now on, we workwith the formalism of DG categories, and all limits, pushforwards andpullback functors we will work with will be replaced by their derived ana-logues (using the usual derived convention of taking injective resolutionsfor computing pushforwards). We work with the injective derived cat-egories D+(Shv(ΘR)), etc. using that all the categories we are workingwith have enough injectives; we think of Db as objects of D+ with finitelymany nonzero homology groups.

6.1 Affine resolution of sheavesBefore defining the mirror symmetry functors, we write down a resultwhich will be very useful for us for deducing results for sheaves on Σ fromaffine analogues.Lemma 34. Suppose the fan Σ is complete. Then the functor Id :Shv(ΘR)→ Shv(ΘR) is the homotopy inverse limit in the derived categoryof the diagram of exact functors ισ∗ ι∗σ indexed by Σop4 , and with connectingmorphisms given by adjunction.

Proof. It is enough to show the natural map is an isomorphism on stalks.Now note the equivalence of exact functors

StÞσm ◦ ισ′∗ ◦ ι∗σ) ∼= StÞσm ◦ π∗σ′ ◦ ι∗σ ∼= StÞσ∩σ

′m .

It follows that the diagram lim← σ′

StÞσm(lim←ισ′∗ ι∗σ′)∼= lim← σ′∈Σ

op4

StÞσ∩σ′

m col-

lapses to the subdiagram {σ′ | σ′ 4 σ}op, which has an initial element,σ.

31

6.2 Defining the Bondal functors B∗, B∗.

Let Op(MR) be the Grothendieck topology of open sets in MR. For σ ∈ Σa cone, we make the following definition.

Definition 40. We define a map of Grothendieck topologies Bσ : Op(MR)→ΘRσ, with B−1(d) := ∆̊d.

From this we get a pair of adjoint functors B∗σ : Shv(Θσ)→ Shv(MR)and Bσ∗ : Shv(MR)→ Shv(Θσ), whose derived functors constitute the fullcoherent-constructible correspondence for affine toric varieties.

Now if Σ is not affine, we can still get functors from Shv(ΘΣ) to andfrom Shv(MR) by pushing and pulling along the composed map of topolo-gies

Bσ,Σ := ισ ◦Bσ : MR → Θ.

Now for general topological spaces, maps of topologies themselves forma category (which should be considered as a full subcategory of functorsof topoi). In order to get our derived coherent-constructible functor, wewill put the maps Bσ,Σ into a diagram and take its colimit; this willgive us an object of some category of derived correspondences. For thesake of brevity, we will not define any such category, but rather explicitlydefine its pushforward and pullback functors B∗, B∗ as homotopy limitsand colimits (which we will write as explicit chain complexes) of suitablediagrams with entries Bσ,Σ∗ and B∗σ,Σ, respectively.

Recall that π∗ = ι∗, so ι∗ is left adjoint to π∗. In particular, we haveadjunction maps Id→ ι∗π∗, π∗ι∗ → Id and, adjointly, Id→ ι∗π∗, π∗ι∗ →Id. Now write BτΣ = ιτΣBτ = ισΣιτσπτσBσ. Then replacing each mapby its pushforward, we can compose it with the map Idσ → ιτσ∗ π

τσ∗ to get

a map rστ : BσΣ∗ → BτΣ

∗ of functors Shv(MR) → Shv(ΘR). Conversely,on pullbacks we compose with the counit map π∗τσ ◦ ι∗τσ → Idσ to get anatural transformation, qστ : B∗τΣ → B∗σΣ. Note that the functors qστ forma representation in functors of the poset Σ4 and the functors rστ form arepresentation of its opposite. We want to define the Bondal functor B∗as the homotopy limit of the BσΣ

∗ along the r maps and, adjointly, B∗ asthe homotopy colimit of the B∗σΣ along the q maps. Later on, we will givean explicit model for this colimit as a complex of functors applied to aninjective resolution of V.

6.3 Full faithfulness of B∗

From now on, assume that the Novikov fan Σ is complete. We usea category-theoretic trick (explained to the author by Roman Bezrukavnikov)to prove the following Lemma.

Lemma 35. The functor B∗R : Shv(ΘR)→ Shv(MR) is fully faithful (as afunctor of DG categories).

Recall that an objectX of a dg category C is compact if for any diagramof objects D : I → C, taking RHom with X preserves (homotopy) colimitalong I. Specifically, for small index category I, we require:

∀D : I → C admitting a direct limit, we haveRHom (X, colimi∈I D(i)) = colimi∈I RHom(X,D(i)).

32

Recall also that a collection of objects Xi colimit generate a DG categoryC if every object of C can be expressed as a colimit of the Xi. (Note thatthere are different notions of DG generation and this is the strongest).Then we have the following categorical proposition.

Proposition 36. Suppose F : C → D is a functor of DG categoriesthat commutes with colimits and Xi is a collection of compact objects thatcolimit generate C, such that F (Xi) are also compact. Then F is fullyfaithful if and only if Fij : RHomC(Xi, Xj) → RHomD(F (Xi), F (Xj)) isa quasiisomorphism for all pairs i, j.

Proof. For any pair of objects X,Y ∈ C, we can express X = limI Xi andY = limJ Xj . Then by compactness, RHom(F (X), F (Y ))

∼= colimIop RHom(F (Xi), F (Y )) (by colimit compatibility) (11)∼= colimIop colimJ RHom(F (Xi), F (Yj)) (by compactness of F (Xi) )

(12)∼= colimIop×J colim RHom(Xi, Yj) (by faithfulness on generators)

(13)∼= RHom(X,Y ) (by reverse arguments in C).

(14)

We apply this proposition with F the functor B∗ (which commuteswith colimits because it is a left adjoint) and take for the collection ofgenerators the sheaves Od, using the following proposition.

Proposition 37. The objects Od for d finite quasidivisors are a collectionof compact generators for D− Shv(ThetaR).

Proof. To see the Od are compact, using lemma 34 (or the affine-localnature of computing RHom on XNov

Σ ), it suffices to check that Od forfinite d is compact in the affine case Σ = Σσ. But in this case the Od areisomorphic to one-dimensional free modules, hence compact in Rep(Oσ)itself (and therefore, a fortiori, in Repa(Oσ)). Now any Od for d notnecessarily finite is a direct limit of finite Od. Thus in order to showthe finite Od colimit-generate, it suffices to show all Od colimit-generate.Once again using Lemma 34, we have V ∼= lim ισ∗Vσ together with thefact that finite inverse limits are (shifted) direct limits in the derivedcategory, we reduce to the affine case, Σ = Σσ. Now since the forgetfulfunctor from sheaves to presheaves is left exact, it commutes with directlimits and hence it suffices to show the Oσ colimit-generate the categoryof presheaves, i.e. equivariant modules over Oσ. But this is standard: anymodule has a left resolution by free modules. (Note: we are using thatthe category PreSh(Θ) has both enough projectives and enough injectives,and so for objects of the bounded derived category, Ext can be computedusing either definition).

It follows that in order to check derived full faithfulness, it suffices tocheck it on the full subcategory of Od. In fact, we can do one better using

33

Lemma 34 to reduce to pushforwards ι∗Oσd from affines. Now

RHom(ισ∗Oσd , ισ′∗ Oσ

′d′ ) ∼= RHomΘσ′ (ι

∗σ′ ι

σ∗Oι∗d)

∼= RHomΘσ′ (πσ′∗ ι

σ∗Oι∗(d),Oι′∗(d′)) ∼= RHomΘσ′ (Oι∗σ∩σ′ (d)Oσ′∗d′)

∼=

{C, σ ⊂ σ′ and m 4σ m′

0, else.

(We know how to RHom out of Oσ in Rep(Oσ), and view Oσ′∩σ′ ∈Rep(Oσ) as a direct limit of shifts of Oσ, to get a colimit expression(with all connecting maps isomorphisms) to compute the derived Hom’sabove. It follows by Proposition 3.3 of [4] that the functor B∗ is indeedfaithful on the O

π−1σ (d)

, from which we deduce faithfulness on finite Odand thence by finite generation on all of Shv(ΘR), completing the proofof faithfulness of B∗.

6.4 Essential surjectivityIt remains to check B∗ is essentially surjective, or equivalently the follow-ing lemma.

Lemma 38. The counit map B∗B∗ : Shv(MR)→ Shv(MR) is the identity.

Proof. Let S be a sheaf onMR. It suffices to check the map B∗B∗S → S isan isomorphism on stalks Stm form ∈MR, or, equivalently on dual stalks,(taking δm fo the skyscraper sheaf on a point m) that Hom(S, δm) →Hom(B∗B∗S, δm) is an isomorphism. Now we write Hom(B∗B∗S, δm) ∼=Hom(B∗S,B∗δm). In the affine case whereB is a genuine map of Grothendiecktopologies, Bσ∗ δm ∼= δB(m)

∼= δÞσm. Since B∗ is glued out of the Bσ∗ , we getB∗δm ∼= lim

←δÞσm. Now the stalk at StÞmσ of Bσ∗S is equivalent to the global

sections of S pulled back to π−1σ (StÞmσ )

(= ∩StÞmσ ∈π−1(∆̊σ

d)π−1σ (∆̊σ

d )). We

write this as St∆mσ (S).Using a stalkwise argument, we see that limσ St∆mσ (S)is the shifted stalk functor Stm, and the Lemma follows.

This concludes the proof of the equivariant Novikov ccc.

7 Singular Support Conditions and Sub-topologiesNow suppose that Σ is a regular fan. Recall that we have a sub-categoryQDivZ ⊂ QDivR with induced topology, which corresponds to a projectionmap on topologies, πRZ : ΘR → ΘZ. We define adjoint “integral Bondalfunctors”

B∗Z := B∗ ◦ π∗RZand

BZ∗ := πRZ

∗ ◦B∗.Since πRZ

∗ π∗RZ∼= Id : Shv(ΘZ)→ Shv(ΘZ), we see that π∗R is fully faithful,

and hence so is B∗Z . It follows from [4] that the image of B∗Z has singular

34

support contained in Λ. Thus it remains to check that if S is a complex ofsheaves with singular support contained in Λ, then B∗ZBZ

∗S ∼= S (via thenatural adjunction counit). Using Lemma ??, it is enough to check thatin the affine case Σ = Σσ, we have the identity BZ

∗ (S) ∼= BR∗ (S) (under

singular support restrictions on S). Equivalently, this is asking that thederived sections functor RΓ(∆d,S) does not change for d ∈ QDivR solong as the “highest lower bound” bdc ∈ QDivZ does not change. This isprecisely guaranteed by the singular support condition.

8 Log-Perfectoid Mirror Symmetry Inter-pretation

8.1 From Novikov varieties to perfectoidsSuppose A ⊂ Rn is an arbitrary dense Abelian group containing a latticeM . Define TNov−A for the algebraic group Spec k(A) (if A ⊂ Q⊗M, thisis a profinite cover of the torus T ). Write Γ := (A/M)∼(:= Spec k[A/M ])this should be thought of as an algebraic version of the Galois group ofthe cover TNov−A → T . For a rational and regular4 fan Σ, write XNov−A

Σ

for the space glued out of the affine Novikov pieces Spec k[A ∩ σ∨] (justreplacing MR by A in the definitions of XNov

Σ ). The category of quasi-coherent sheaves on XNov−A

Σ has a well-defined almost-local subcategory,QcohXNov−A

Σ (defined locally on affines as an idempotence condition forthe ideal aσ, as for XNov

Σ ). By repeating the arguments of sections 1-8above, we get the following results.

The categoryDb QcohaXNovΣ∼= Db Shv(ΘA), where ΘA(= QDivA,ΘA)

is the sub-topos of Θ on quasidivisors d ∈ QDivRσ such that for each σ ∈ Σ,

the restriction d | σ is the restriction to σ of an element of A ⊂M∗R (viewedas a function on σ ⊂ MR). Consider the map of Grothendieck topologiesπAR : ΘR → ΘA (with π−1(d) = d for d a quasidivisor).

We use the following lemma.

Lemma 39. The pushforward and pullback functors along πAR are mu-tually inverse isomorphisms between Shv(ΘA) and Shv(ΘR).

Proof. Since A is dense, the subtopology ΘA generates ΘR. In order to seethat they define equivalent categories, it is sufficient to check that everycover of a divisor d ∈ QDivR can be refined by a cover with all objectsin QDivA and that any two such covers have a common refinement (alsoonly involving objects in A). This is once again clear from the density ofA.

It follows that (QcohaXNov−AΣ )TNov−A ∼= Shv(MR) and hence, by the

Pontrjagin duality theorem in section 11, we get the following theorem,which implies the main theorem 5.

Theorem 40. In the setting above, we have an equivalence of derivedcategories

(QcohaXNov−AΣ )(A/M)∼ ∼= Shv(S).

4one can weaken both the rationality and the regularity conditions

35

8.2 Log-perfectoid sheavesInspired by Theorem 40, we write down here a general procedure for ob-taining a new quasicoherent category associated to a pair of spaces (U,X)with normal complex complement, and make some conjectures about in-dependence on choice of X. Assume for simplicity that the base field khas characteristic zero. Suppose that U is a variety and X is a compacti-fication. Suppose that ∂ := X \ U is a normal crossings divisor. Assumethat the algebraic fundamental group of U maps surjectively to the localfundamental group (spanned by ramification) of the neighborhood of anypoint of ∂. (If this is not the case, we can remove some extra divisors fromthe interior of U to enlarge the fundamental group, then glue in an invari-ant procedure, but we will not worry about this level of generality here.)Let ΓU be the geometric fundamental group of U . Let ΓN be a cofilteringfamily of finite quotient groups with inverse limit ΓU . Let U (N) be thecorresponding étale covers of U . For each boundary component ∂i ⊂ ∂,let IiN ⊂ ΓN be a ramification group associated to ∂i (note that it dependson the choice of a branch of the cover near ∂i). Now there is a uniquefinite flat extension of the family U (N) over U to a flat ΓN -equivariantfamily over U ∪ ∂̊i ⊂ X. The preimage of ∂̊i will be |ΓN/IiN | copies of ∂iwith multiplicity |IiN |. Glueing such extensions, we extend to a cover ofthe codimension-two subvariety

U ∪⋃∂̊i ⊂ X.

There is a unique extension of this family to an affine family over X:étale locally over any component of the preimage of some (possibly high-codimension) boundary component ∂0, the map will look isomorphic to aproduct of power maps (x1, . . . , xn) 7→ (xN1

1 , xN22 , . . . , xNnn ) : An → An.

Now the pro-sequence of affine maps X(N) → X will have a well-defined projective limit scheme X(∞) which is affine over X, and hasaction by Γ, where (abusing notation), we view Γ also as an affine projec-tive limit of affine (group) schemes. Let ∂(∞) ⊂ X(∞) be the boundary.Our condition on ramification behavior of Γ implies that the category ofsheaves on X(∞) pushed forward from ∂(∞) is a Serre subcategory. Wedefine Qcoha(X(∞)) to be the quotient of Qcoh(X(∞)) by the Serre sub-category of sheaves pushed forward from ∂(∞). This category has naturalaction by the group scheme Γ. Define Qcohlog-perf(X) to be the categoryof Γ-equivariant quasicoherent sheaves in Qcoha(X(∞)). Note that whileour definition uses the (possibly very large and non-commutative) Galoisgroup Γ, the stabilizers of the stacky points of X(∞)/Γ∞ will be tame.Specifically, they will all be products of ramification groups of several∂i, and in particular commutative. Thus we could define the categoryQcohlog-perf(X) starting not with the full Galois group Γ but with anycontinuous subgroup Γ′ of Γ which maps surjectively to local ramificationgroups (in fact, it should be possibly to replace surjectivity by a densityrequirement).

If U = T and X is a toric variety, the Galois group of T (viewed asa group scheme) is Γ ∼=

((Q/Z)∼

)n. A canonical choice of tower X(N)

is provided by the N -frobenius maps, induced from the maps of latticesNΣ → N ·NΣ without changing the fan (with apologies for the awkward

36

notation: N is an integer and NΣ is the lattice). Theorem 40 then im-plies that Qcohlog-perf(U,X) is equivalent to the category of quasicoherentsheaves on S. In particular, it is independent of choice of X. This leadsus to make the following conjecture.

Conjecture 1. Assume the base field k = C. Then for any U , the derivedcategory Db Qcohlog-perf(U,X) is independent of choice of (n.c.) compact-ification X.

If this is true, we would have a well-defined categoryDb Qcohlog-perf(U)depending only on an open variety, and this would perhaps be the rightcategory to study for some wider class of mirror symmetry statements foropen varieties. Further, whether of not this conjecture is true, SYZ mirrorsymmetry provides us with an abundance of pairs of varieties (U,X),where X is an algebraic Calabi-Yau manifold admitting an SYZ torusfibration with singularities, and U ⊂ X is the complement to the space ofsingular fibers (which has codimension 2 and is in fact algebraic). SupposeU,X is such a pair and letM be the symplectic manifold obtained from thedual torus filtration on the nonsingular fibers. Then we ask the followingquestion.

Question. Is there a way to define a “microlocal” Fukaya category Fuk?(M)on M (which for T ∗S1 would be equivalent to the derived category of alltopological sheaves on S1) such that there is an equivalence of categoriesFuk?(M) ∼= Db Qcohlog-perf(U,X)?

References[1] Alexey Bondal. Derived categories of toric varieties. In Convex and

Algebraic geometry, Oberwolfach conference reports, EMS PublishingHouse, volume 3, pages 284–286, 2006.

[2] James Dolan. tannakian correspondence for toric varieties (sketchfor a doctoral thesis).

[3] Gerd Faltings. p-adic hodge theory. Journal of the American Math-ematical Society, 1(1):255–299, 1988.

[4] Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, and Eric Za-slow. A categorification of morelliâĂŹs theorem. Inventiones math-ematicae, 186(1):79–114, 2011.

[5] Ofer Gabber and Lorenzo Ramero. Almost ring theory. Springer,2003.

[6] Maxim Kontsevich. Homological algebra of mirror symmetry. arXivpreprint alg-geom/9411018, 1994.

[7] Tatsuki Kuwagaki. The nonequivariant coherent-constructible corre-spondence for toric stacks. arXiv preprint arXiv:1610.03214, 2016.

[8] David Nadler. Fukaya categories as categorical morse homology. Sym-metry, Integrability and Geometry. Methods and Applications, 10,2014.

37

[9] David Nadler and Eric Zaslow. Constructible sheaves and the fukayacategory. Journal of the American Mathematical Society, 22(1):233–286, 2009.

[10] David Treumann. Remarks on the nonequivariant coherent-constructible correspondence for toric varieties. arXiv preprintarXiv:1006.5756, 2010.

38