A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

DOMINICJOYCE ANDPAVELSAFRONOV

A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Tome XXVIII, n^o5 (2019), p. 831-908.

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A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Dominic Joyce⁽¹⁾ and Pavel Safronov⁽²⁾

ABSTRACT. — Pantev, Toën, Vaquié and Vezzosi [19] definedk-shifted symplectic derived schemes and stacksXfork∈Z, and Lagrangiansf:L→Xin them. They have important applications to Calabi–Yau geometry and quantization. Bussi, Brav and Joyce [7] and Bouaziz and Grojnowski [5] proved “Darboux Theorems” giving explicit Zariski or étale local models fork-shifted symplectic derived schemesX for k <0 presenting them as twisted shifted cotangent bundles.

We prove a “Lagrangian Neighbourhood Theorem” which gives explicit Zariski or étale local models for Lagrangiansf :L→ X ink-shifted symplectic derived schemesXfork <0, relative to the “Darboux form” local models of [7] forX. That is, locally such Lagrangians can be presented as twisted shifted conormal bundles.

We also give a partial result whenk= 0.

We expect our results will have future applications to shifted Poisson geom- etry [12], and to defining “Fukaya categories” of complex or algebraic symplectic manifolds, and to the categorification of Donaldson–Thomas theory of Calabi–Yau 3-folds and “Cohomological Hall Algebras”.

RÉSUMÉ. — Pantev, Toën, Vaquié et Vezzosi [19] ont défini des schémas et des champs dérivés symplectiquesk-décalésXpourk∈Z, et des Lagrangiensf:L→X en eux. Ils ont des applications importantes pour la géomètrie Calabi–Yau et la quantification. Bussi, Brav et Joyce [7] et Bouaziz et Grojnowski [5] ont prouvé des

« théorèmes de Darboux » donnant des modàles locaux précis Zariski ou étale pour les schémas dérivés symplectiquesk-décalésX pourk <0, les présentant comme des fibrés cotangent décalés tordus.

Nous prouvons un « théorème de voisinage Lagrangien » donnant des modèles locaux précis Zariski ou étale pour les Lagrangiensf :L →X dans les schémas dérivés symplectiquesk-décalésX pourk <0, par rapport à la « forme Darboux » de Bussi–Brav–Joyce pourX. C’est-à-dire, localement, ces Lagrangiens peuvent être présentés sous forme de fibrés conormaux décalés tordus. Nous donnons aussi un résultat partiel lorsquek= 0.

(*)Reçu le 4 août 2017, accepté le 30 août 2017.

(1) The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom — [email protected]

(2) Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland — [email protected]

This research was supported by EPSRC Programme Grant EP/I033343/1.

Article proposé par Bertrand Toën.

Nous espérons que nos résultats auront de futures applications à la géométrie de Poissonk-décalée de [12], à la définition de « catégories de Fukaya » de varié- tés symplectiques complexes ou algébriques, à la catégorification de la théorie de Donaldson–Thomas des variétés de Calabi–Yau de dimension 3, et au « Algèbres de Hall Cohomologiques ».

1. Introduction

Using Toën and Vezzosi’s theory of Derived Algebraic Geometry [23, 24, 25, 27, 26], Pantev, Toën, Vaquié and Vezzosi [19] definedk-shifted symplectic structuresωX on a derived scheme or stackX, fork∈Z. If X is a derived scheme andωX a 0-shifted symplectic structure, thenX =X is a smooth classical scheme andω_X ∈ H⁰(Λ²T^∗X) a classical symplectic structure on X. They proved that ifY is a Calabi–Yaum-fold then derived moduli stacks Mof (complexes of) coherent sheaves on Y have natural (2−m)-shifted symplectic structuresω_M.

Pantev et al. [19] also defined Lagrangians f : L → X in a k-shifted symplectic derived stack (X, ωX), and showed that fibre productsL×XM of Lagrangians f : L → X, g : M → X are (k−1)-shifted symplectic.

Calaque [11] proved that ifX is a Fano (m+ 1)-fold and Y ⊆X a smooth anticanonical divisor, so thatY is a Calabi–Yaum-fold, andL,Mare de- rived moduli stacks of (complexes of) coherent sheaves onX, Y with derived restriction morphism f : L → M, then L is Lagrangian in the (2−m)- shifted symplectic (M, ωM).

Recently, Calaque, Pantev, Toën, Vaquié and Vezzosi [12] have also de- veloped a related theory of k-shifted Poisson structures π_X on a derived scheme or stack X, for k ∈ Z, and coisotropics f : C → X in (X, π_X).

They prove [12, Th. 3.2.4] that the spaces ofk-shifted symplectic structures ωX and nondegeneratek-shifted Poisson structuresπX onX are equivalent.

Costello–Rozenblyum and Pridham [21] have also announced similar results.

For a symplectic manifold (X, ω), the classical Darboux Theorem chooses local coordinates (x1, . . . , xn, y1, . . . , yn) on X with ω=Pn

j=1ddRxjddRyj. Bussi, Brav and Joyce [7, Th. 5.18] proved a “k-shifted symplectic Dar- boux Theorem”, which for ak-shifted symplectic derivedK-scheme (X, ωX) with k < 0 chooses a cdga A^•, a Zariski open inclusion i : SpecA^• ,→ X, and coordinates xⁱ_j, y^k−i_j ∈ A^• with i^∗(ωX) ' (ω⁰,0,0, . . .) for ω⁰ = P

i,jd_dRxⁱ_jd_dRy_j^k−i. (Actually, all this only holds fork6≡2 mod 4, and for k≡2 mod 4 there is a more complicated expression also involving coordi- natesz^k/2_j .)

This was the foundation for a series of papers [2, 3, 4, 9, 8, 6, 7, 10, 13] concerning generalizations of Donaldson–Thomas theory for Calabi–Yau 3- and 4-folds, involving perverse sheaves, motives, and new enumerative invariants. It can also be used as part of a proof that k-shifted symplectic derived schemes carry nondegenerate k-shifted Poisson structures, though this was not used in [12, 21].

Given a LagrangianL ,→X in a symplectic manifold (X, ω), the classical Lagrangian Neighbourhood Theorem describesL, X, ω in local coordinates.

The purpose of this paper is to prove a “k-shifted symplectic Lagrangian Neighbourhood Theorem”, Theorem 3.7 below, which given a Lagrangianf : L→X in ak-shifted symplectic derivedK-scheme (X, ω_X) fork <0, and a “Darboux form” local descriptioni:SpecA^• ,→X,xⁱ_j, y^k−i_j ∈A^•,ω⁰= P

i,jd_dRxⁱ_jd_dRy_j^k−i for (X, ω_X) as in [7], chooses a cdga B^•, coordinates exⁱ_j, uⁱ_j, v^k−1−i_j ∈ B^•, a Zariski open inclusion j : SpecB^• ,→ X, and a cdga morphismα:A^• →B^• withxeⁱ_j =α(xⁱ_j) in a homotopy commutative diagram

SpecB^•

Specα

j //L

f SpecA^{• i} //X,

(1.1)

such that the pullbackj^∗(h_L) of the Lagrangian structureh_Lonf :L→X toSpecα:SpecB^•→SpecA^•using (1.1) hasj^∗(h_L)'(h⁰,0,0, . . .) with h⁰ =P

i,jd_dRuⁱ_jd_dRv_j^k−1−i. (Actually, all this only holds fork6≡3 mod 4, and fork≡3 mod 4 there is a more complicated expression also involving coordinatesw^(k−1)/2_j .) Theorem 3.11 also gives a partial result for k= 0.

Bouaziz and Grojnowski [5] proved their ownk-shifted symplectic Dar- boux Theorem independently of [7], showing that a k-shifted symplectic derivedK-scheme (X, ωX) fork <0 is (at least fork6≡2 mod 4) étale lo- cally equivalent to atwisted k-shifted cotangent bundle T_t^∗[k]Y, where Y is an affine derivedK-scheme, andt∈ O^k+1_Y with dt= 0 is used to “twist” the k-shifted cotangent bundleT^∗[k]Y. Remark 2.15 below relates their picture to that of [7].

In Remark 3.4 we interpret our “k-shifted Lagrangian Neighbourhood Theorem” in the style of Bouaziz and Grojnowski [5], by saying that if f : L→X is Lagrangian in ak-shifted symplectic (X, ω_X) fork <0, andX is locally modelled onT_t^∗[k]Y, thenf :L→X is (at least for k6≡3 mod 4) locally modelled on the inclusion morphismN_u/t^∗ [k](Z/Y)→T_t^∗[k]Y, where N_u/t^∗ [k](Z/Y) is the twisted k-shifted conormal bundle of a morphism of

affine derivedK-schemesg:Z→Y, andu∈ O^k_Z with du=−g^∗(t) is used to “twist”N^∗[k](Z/Y).

If the k-shifted symplectic derived K-scheme (X, ωX) is a point (SpecK,0) then Lagrangians f :L →X are just (k−1)-shifted symplec- tic derivedK-schemes (L, ωL). In this case, our Lagrangian Neighbourhood Theorem reduces to the Darboux Theorem of [7]. So the proof in Section 4 is a generalization of that in [7, §5.6], and runs parallel to [7] at several points.

Like the Darboux Theorem of [5, 7], our Lagrangian Neighbourhood The- orem should have important applications. For example, it gives local mod- els for moduli schemes of coherent sheaves on Fano (m+ 1)-folds X with restriction morphisms to moduli schemes of coherent sheaves on a Calabi–

Yau anticanonical hypersurfaceY ⊂X. We briefly discuss some conjectures which we hope our theorem will help to prove.

Conjecture 1.1. — Let (X, ω_X) be a (−1)-shifted symplectic derived C-scheme with an “orientation”. Then Bussi, Brav, Dupont, Joyce, and Szen- drői[6, Cor. 6.11] construct a natural perverse sheafP^•_X,ω

X onX =t₀(X), such that if(X, ωX)is locally modelled on a critical locusCrit(Φ :U →A¹), then P^•_X,ωX is locally modelled on the perverse sheaf of vanishing cycles PV^•_U,Φ.

Suppose f : L → X is a Lagrangian, with an “orientation” relative to that of X, and f is proper. Then we can define a natural element λL in the hypercohomologyH^vdimL(P^•_X,ωX). TheseλL satisfy certain composition laws for composition of Lagrangian correspondences.

The first author has an outline of a proof of Conjecture 1.1.

As suggested in [6, Rem. 6.15], we would like to define a “Fukaya cat- egory” F(S) of (derived) complex or algebraic Lagrangians L → S in a complex or algebraic symplectic manifold (S, ω) of dimension 2n, such that ifL, M are oriented Lagrangians inS then the morphismsL→M inF(S) are Hom^∗(L, M) = H^∗−n(P^•_L,M), where P^•_L,M is the perverse sheaf on the

−1-shifted symplecticX=L×SM described above.

As in Ben-Bassat [2], if L, M, N are (derived) Lagrangians in (S, ω), then Y = L×_S M ×_S N → (L×_S M)×(M ×_S N)×(N ×_S L) is La- grangian in −1-shifted symplectic. The hypercohomology class λY asso- ciated to this in Conjecture 1.1 is what we need to define composition Hom^∗(M, N)×Hom^∗(L, M) →Hom^∗(L, N) of morphisms in the “Fukaya category”F(S). Amorim and Ben-Bassat [1] discuss this proposal and Con- jecture 1.1 in detail.

A stacky version of Conjecture 1.1 is what we need to define multiplication in a “Cohomological Hall Algebra” associated to a Calabi–Yau 3-fold in the sense of Kontsevich–Soibelman [15], defined using the perverse sheaves on Calabi–Yau 3-fold moduli stacks constructed in Ben-Bassat, Bussi, Brav and Joyce [3].

Conjecture 1.2. — Let U be a smooth K-scheme and Φ : U → A¹ a regular function. Then the derived critical locus Crit(Φ) is a −1-shifted symplectic derivedK-scheme. We can also define the Z2-graded dg-category of matrix factorizationsMF(U,Φ), as in Preygel[20] for instance.

Supposef : L →Crit(Φ) is a Lagrangian, with vdimL−dimU even, equipped with an “orientation” and a “spin structure”, and thatf is proper.

Then we can define an object µ_L ∈MF(U,Φ) associated to L. In this way we interpretMF(U,Φ)as a kind of “Fukaya category” of the−1-shifted sym- plectic derivedK-schemeCrit(Φ).

This is connected to the programme of Kapustin and Rozansky [14] for associating a 2-category to a complex symplectic manifold, locally described using matrix factorization categories.

For each of the Conjectures 1.1–1.2, using our Lagrangian Neighbourhood Theorem we can write down local models onLfor the coisotropic structure, and for λL and µL. The problem is to glue these local models together globally.

We begin in Section 2 with background material on Derived Algebraic Geometry and Pantev–Toën–Vaquié–Vezzosi’s shifted symplectic geometry.

Section 3 gives our main results. Theorem 3.2 in Section 3.1 shows that a morphismf :X→Y of derivedK-schemes is locally modelled onSpecα: SpecA^•→SpecB^•, whereA^•, B^•are cdgas andα:B^•→A^•a morphism, all in a particularly nice form.

Theorem 3.7 in Section 3.3 is our “Lagrangian Neighbourhood Theo- rem”, showing that Lagrangiansf :L→X in k-shifted symplectic derived K-schemes (X, ωX) for k <0 are locally modelled on explicit “Lagrangian Darboux form” examples given in Examples 3.3 and 3.5 in Section 3.2. The- orem 3.11 in Section 3.4 also gives a partial result fork= 0. Section 4 proves Theorems 3.2 and 3.7.

Conventions. — ThroughoutKwill be an algebraically closed field with characteristic zero. All classicalK-schemes are assumed locally of finite type, and all derived K-schemes X are assumed to be locally finitely presented.

Our sign conventions for cdgas, exterior forms, etc., follow Bussi, Brav and Joyce [7].

Acknowledgements

We would like to thank Dennis Borisov, Chris Brav, and Ian Grojnowski for helpful conversations.

2. Background material

We begin with some background material and notation needed later.

Some references are Toën and Vezzosi [23, 24, 25, 26, 27] for Sections 2.1–2.2, and Pantev, Toën, Vezzosi and Vaquié [19] for Sections 2.3–2.4, and Brav, Bussi and Joyce [7] for Section 2.5. Throughout the paper, K will be an algebraically closed field of characteristic zero.

2.1. Commutative differential graded algebras

Definition 2.1. — Writecdga_Kfor the category of commutative differ- ential gradedK-algebras in nonpositive degrees, andcdga^op

K for its opposite category. Objects of cdga_K are of the form · · · →A⁻² −→^d A⁻¹ −→^d A⁰. Here A^k for k = 0,−1,−2, . . . is the K-vector space of degree k elements of A^•, and we have a K-bilinear, associative, supercommutative multiplica- tion· : A^k×A^l →A^k+l fork, l 60, an identity 1 ∈A⁰, and differentials d :A^k→A^k+1 fork <0 satisfying

d(a·b) = (da)·b+ (−1)^ka·(db)

for alla∈A^k,b∈A^l. We write such objects as A^• or (A^∗,d).

Here and throughout we will use the superscript “^∗” to denote graded objects (e.g. graded algebras or vector spaces), where∗stands for an index in Z, so thatA^∗ means (A^k,k∈Z). We will use the superscript “^•” to denote differential gradedobjects (e.g. differential graded algebras or complexes), so thatA^• means (A^∗,d), the graded object A^∗ together with the differentiald.

Morphisms α : A^• → B^• in cdga_K are K-linear maps α^k : A^k → B^k for all k 6 0 commuting with all the structures on A^•, B^•. A morphism α: A^• → B^• is a quasi-isomorphism if H^k(α) :H^k(A^•) →H^k(B^•) is an isomorphism on cohomology groups for allk60.

Remark 2.2. — A fundamental principle of derived algebraic geometry is thatcdga_Kis not really the right category to work in, but instead one wants to define a new category (or better, ∞-category) by inverting (localizing) quasi-isomorphisms incdga_K.

In factcdga_Khas the additional structure of a simplicial model category, with weak equivalences quasi-isomorphisms, in which all objects are fibrant, and in which cdgasA^•withA^∗ free as a commutative gradedK-algebra are cofibrant. Then-simplices of the mapping space between two cdgas A^• and B^• are given by morphismsA^•→B^•⊗Ω^•(∆ⁿ), where Ω^•(∆ⁿ) is the cdga generated by elementssi of degree 0 andti of degree 1 fori= 0, . . . , nwith the relationsPsi= 1 andPti= 0 and the differential dsi=ti. Note that Ω^•(∆ⁿ) are concentrated in positive degrees, and are not elements ofcdga_K. We will writecdga^∞_K for the associated ∞-category, so that the homo- topy category Ho(cdga^∞_K) is the localized categorycdga_K[Q⁻¹] with quasi- isomorphisms inverted, an ordinary category. We will not go into any detail about model categories and∞-categories below, but here is some basic ori- entation on one issue relevant to this paper, for readers unfamiliar with these ideas. The objects ofcdga_K,cdga^∞_K,Ho(cdga^∞_K) are the same. IfA^•, B^•are objects, a morphismφ:A^• →B^• in cdga_K is also a morphism in cdga^∞_K and Ho(cdga^∞_K). However, a morphismφ^∞:A^•→B^• incdga^∞_K (or equiv- alently, in Ho(cdga^∞_K)) need not correspond to any morphismφ:A^•→B^• in cdga_K, unless A^• is cofibrant. If A^• is cofibrant, the mapping space in cdga^∞_K is given by the mapping space incdga_K.

Standard model cdgasA^• are “nearly cofibrant”. They have the property that ifφ^∞:A^•→B^•is a morphism incdga^∞_K withA^•standard model, such that H⁰(φ^∞) :H⁰(A^•)→ H⁰(B^•) can be lifted to a K-algebra morphism φ⁰:A⁰→B⁰, thenφ^∞can be lifted toφ:A^•→B^• incdga_K.

All this will be important because ifX ' SpecA^• and Y 'SpecB^• are affine derived K-schemes and f : Y → X is a morphism, then f ' Specφ^∞ for some morphismφ^∞:A^•→B^• incdga^∞_K. For our Lagrangian Neighbourhood Theorem in Section 3.3, we want to liftφ^∞ toφ:A^•→B^• incdga_K.

Definition 2.3. — Let A^• ∈ cdga_K, and write D(modA) for the de- rived category of dg-modules overA^•. Define a derivation of degreek from A^• to anA^•-moduleM^• to be a K-linear mapδ:A^•→M^• that is homoge- neous of degreek with

δ(f g) =δ(f)g+ (−1)^kdegff δ(g).

Just as for ordinary commutative algebras, there is a universal derivation into anA^•-module of Kähler differentials Ω¹_A•, which can be constructed as I/I²forI= Ker(m:A^•⊗A^•→A^•). The universal derivationδ:A^•→Ω¹_A•

is δ(a) = a⊗1−1⊗a ∈ I/I². One checks that δ is a universal degree 0 derivation, so that◦δ: Hom^•_A•(Ω¹_A•, M^•)→Der^•(A, M^•)is an isomorphism of dg-modules.

Note thatΩ¹_A• = (Ω¹_A•)^∗,d

is canonical up to strict isomorphism, not just up to quasi-isomorphism of complexes, or up to equivalence inD(modA).

Also, the underlying graded vector space(Ω¹_A•)^∗, as a module over the graded algebraA^∗, depends only onA^∗ and not on the differentialdinA^•= (A^∗,d).

Similarly, given a morphism of cdgas Φ : A^• → B^•, we can define the relative Kähler differentials Ω¹_B•/A^•.

The cotangent complex LA^• of A^• is related to the Kähler differentials Ω¹_A•, but is not quite the same. If Φ : A^• → B^• is a quasi-isomorphism of cdgas over K, then Φ_∗ : (Ω¹_A•)⊗A^• B^• → Ω¹_B• may not be a quasi- isomorphism of B^•-modules. So Kähler differentials are not well-behaved under localizing quasi-isomorphisms of cdgas, which is bad for doing derived algebraic geometry.

The cotangent complexLA^• is a substitute forΩ¹_A• which is well-behaved under localizing quasi-isomorphisms. It is an object inD(modA), canonical up to equivalence. We can define it by replacing A^• by a quasi-isomorphic, cofibrant cdga B^•, and then setting LA^• = (Ω¹_B•)⊗B^•A^•. We will be inter- ested in thep^th exterior powerΛ^pLA^•, and the dual(LA^•)^∨, which is called the tangent complex, and writtenTA^•= (LA^•)^∨.

There is ade Rham differential d_dR: Λ^pLA^• →Λ^p+1LA^•, a morphism of complexes, withd²_dR= 0 : Λ^pL^A•→Λ^p+2L^A•. Note that eachΛ^pL^A• is also a complex with its own internal differential d : (Λ^pLA^•)^k → (Λ^pLA^•)^k+1, andd_dR being a morphism of complexes means thatd◦d_dR= d_dR◦d.

Similarly, given a morphism of cdgas Φ : A^• → B^•, we can define the relative cotangent complexLB^•/A^•.

Definition 2.4. — Following [7, Def. 2.9], we will call A^• ∈ cdga_K of standard form if A⁰ is a smooth finitely generated K-algebra, and the cotangent module Ω¹_A0 is a free A⁰-module of finite rank, and the graded K-algebraA^∗ is freely generated over A⁰ by finitely many generators, all in negative degrees.

More explicitly, asA⁰is a smoothK-algebra,U =SpecA⁰is a smoothK- scheme. Suppose that U admits étale coordinates (x⁰₁, . . . , x⁰_m0) :U →A^m0. ThenΩ¹_A0 ∼=A⁰⊗_KhddRx⁰₁, . . . ,ddRx⁰_m0i_K is a free A⁰-module of rank m0. Suppose we are given elementsxⁱ₁, . . . , xⁱ_mi inAⁱ fori=−1,−2, . . . , k, such thatA^∗=A⁰[xⁱ_j :i=−1, . . . , k,j= 1, . . . , mi]is the gradedK-algebra freely generated over A⁰ by the generators xⁱ_j in degreei <0. Then A^• = (A^∗,d) is a standard form cdga. The differentialdon A^∗ is determined uniquely by the elementsdxⁱ_j∈Aⁱ⁺¹ fori=−1,−2, . . . , k andj= 1, . . . , m_i.

The virtual dimensionof A^• isvdimA^•=Pd

i=0(−1)ⁱmi ∈Z.

Then the Kähler differentialsΩ¹_A• are given as an A^∗-module by

Ω¹_A•∼=A^∗⊗_Khd_dRxⁱ_j:i= 0,−1, . . . , k, j= 1, . . . , m_ii_K. (2.1) As in[7, §2.3], an important property of standard form cdgasA^• is that they are sufficiently cofibrant that the Kähler differentialsΩ¹_A•provide a model for the cotangent complex LA^•, so we can take Ω¹_A• =LA^•, without having to replaceA^•by an unknown cdgaB^•. Thus standard form cdgas are convenient for doing explicit computations with cotangent complexes.

We say that a standard form cdgaA^•isminimalatp∈SpecA^•if all the differentials in the complex ofK-vector spaces Ω¹_A•|p are zero. This means that m_i = dimHⁱ LA^•|_p

for i = 0,−1, . . . , d, and A^• is defined using the minimum number of variablesxⁱ_j in each degreei= 0,−1, . . . ,compared to all other cdgas locally equivalent toA^• nearp.

2.2. Derived algebraic geometry and derived schemes

Definition 2.5. — WritedSt_K for the∞-category of derivedK-stacks (orD⁻-stacks) defined by Toën and Vezzosi[27, Def. 2.2.2.14],[23, Def. 4.2].

ObjectsX indSt_K are∞-functors

X :{simplicial commutative K-algebras} −→ {simplicial sets}

satisfying sheaf-type conditions. There is aspectrum functor Spec: (cdga^∞_K)^op−→dSt_K.

A derivedK-stackX is called an affine derivedK-schemeifX is equivalent in dSt_K to SpecA^• for some cdga A^• over K. As in [23, §4.2], a derived K-stackX is called a derivedK-schemeif it may be covered by Zariski open Y ⊆X with Y an affine derived K-scheme. Write dSch_K for the full ∞- subcategory of derived K-schemes in dSt_K, and dSch^aff_K ⊂dSch_K for the full∞-subcategory of affine derivedK-schemes. ThenSpecis an equivalence (cdga^∞_K)^op −→^∼ dSch^aff_K .

We shall assume throughout this paper that all derivedK-schemes X are locally finitely presentedin the sense of Toën and Vezzosi[27, Def. 1.3.6.4].

With this assumption, derived schemes have avirtual dimension vdimX, which is a locally constant function vdimX : X → Z. If X = SpecA^• for A^• a standard form cdga then vdimX = vdimA^•, for vdimA^• as in Definition 2.4.

There is a classical truncation functor t0 : dSch_K → Sch_K taking a derived K-scheme X to the underlying classical K-scheme X =t0(X). On

affine derived schemes dSch^aff_K this maps t₀ : SpecA^• 7→SpecH⁰(A^•) = Spec(A⁰/d(A⁻¹)).

Toën and Vezzosi show that a derivedK-schemeX has acotangent com- plexLX[27, §1.4],[23, §4.2.4–4.2.5]in a stable∞-categoryLqcoh(X)defined in[23, §3.1.7, §4.2.4]. We will be interested in the p^th exterior power Λ^pLX, and the dual(LX)^∨, which is called the tangent complexTX.

By apointof a derivedK-schemeX, writtenx∈X, we will always mean thatx∈X(K)is aK-point of the underlying classicalK-schemeX =t0(X).

WhenX=X is a classical scheme, the homotopy category of Lqcoh(X) is the triangulated categoryDqcoh(X)of complexes of quasicoherent sheaves.

These have the usual properties of (co)tangent complexes. For instance, if f :X →Y is a morphism indSch_K there is a distinguished triangle

f^∗(LY) ^Lf //LX //LX/Y //f^∗(LY)[1], whereLX/Y is therelative cotangent complex off.

Now supposeA^• is a cdga overK, and X a derivedK-scheme withX' SpecA^• indSch_K. Then we have an equivalence of triangulated categories Ho(L_qcoh(X)) ' D(modA^•), which identifies cotangent complexes LX ' LA^•. If alsoA^• is of standard form then LA^•'Ω¹_A•, soLX'Ω¹_A•.

Bussi, Brav and Joyce [7, Th. 4.1] prove:

Theorem 2.6. — SupposeX is a derivedK-scheme (as always, assumed locally finitely presented), andx∈X. Then there exist a standard form cdga A^• overKwhich is minimal atp∈SpecA^•,in the sense of Definition 2.4, and a Zariski open inclusioni:SpecA^•,→X withi(p) =x.

They also explain [7, Th. 4.2] how to compare two such standard form chartsSpecA^• ,→X, SpecB^• ,→X on their overlap in X, using a third chart.

2.3. PTVV’s shifted symplectic geometry

Next we summarize parts of the theory of shifted symplectic geometry, as developed by Pantev, Toën, Vaquié, and Vezzosi in [19]. We explain them for derivedK-schemesX, although Pantev et al. work more generally with derived stacks.

Given a (locally finitely presented) derivedK-schemeXandp>0,k∈Z, Pantev et al. [19] define complexes of k-shifted p-forms A^p_K(X, k) and k- shifted closedp-forms A^p,cl_K (X, k). These are defined first for affine derived

K-schemes Y =SpecA^• forA^• a cdga overK, and shown to satisfy étale descent. Then for general X, k-shifted (closed) p-forms are defined as a mapping stack; basically, ak-shifted (closed)p-formωonXis the functorial choice for allY,f of ak-shifted (closed)p-formf^∗(ω) onY wheneverY = SpecA^• is affine andf :Y →X is a morphism.

Definition 2.7. — Let Y ' SpecA^• be an affine derived K-scheme, forA^•a cdga overK. Ak-shiftedp-formonY fork∈Zis an elementω⁰_A•∈ (Λ^pLA^•)^k withdω_A⁰• = 0in (Λ^pLA^•)^k+1, so that ω_A⁰• defines a cohomology class [ω_A⁰•] ∈ H^k(Λ^pLA^•). When p= 2, we call ω_A⁰• nondegenerate if the induced morphismω⁰_A•·:TA^•→LA^•[k]is a quasi-isomorphism.

Ak-shifted closedp-form onY is a sequenceωA^•= (ω_A⁰•, ω_A¹•, ω_A²•, . . .) such that ωⁱ_A• ∈ (Λ^p+iLA^•)^k−i for i > 0, with dω⁰_A• = 0 and dω¹⁺ⁱ_A• + ddRω_Aⁱ• = 0 in (Λ^p+i+1L^A•)^k−i for all i > 0. Note that if ωA^• = (ω⁰_A•, ω¹_A•, . . .)is ak-shifted closedp-form thenω⁰_A• is ak-shiftedp-form.

Whenp= 2, we call ak-shifted closed 2-formωA^•ak-shifted symplectic formif the associated 2-form ω⁰_A• is nondegenerate.

If X is a general derived K-scheme, then Pantev et al. [19, §1.2] define k-shifted 2-forms ω_X⁰ , which may be nondegenerate, andk-shifted closed 2- formsω_X, which have an associated k-shifted 2-formω⁰_X, and where ω_X is called a k-shifted symplectic form if ω⁰_X is nondegenerate. We will not go into the details of this definition for generalX.

The important thing for us is this: if Y ⊆ X is a Zariski open affine derivedK-subscheme withY 'SpecA^•then ak-shifted symplectic formωX

onXinduces ak-shifted symplectic formωA^•onY in the sense above, where ωA^•is unique up to cohomology in the complex(Q

i>0(Λ²⁺ⁱLA^•)^∗−i,d+ddR).

As in [19, §2.1], in the stacky case, an important source of examples of shifted symplectic derived stacks are Calabi–Yau moduli stacks:

Theorem 2.8. — Suppose Y is a Calabi–Yau m-fold over K, and M the derived moduli stack of complexes of coherent sheaves on Y. Then M has a natural(2−m)-shifted symplectic formω_M.

2.4. Lagrangians in shifted symplectic derived schemes

Following Pantev et al. [19, §2.2], we define:

Definition 2.9. — Let (X, ωX) be a k-shifted symplectic derived K- scheme, and f : L → X a morphism of derived K-schemes. An isotropic structureonf is a homotopyhLfrom 0 tof^∗(ωX)in the complexA^2,cl_K (L, k),

regarded as a simplicial set. Truncating to the first term A^2,cl_K (L, k) → A²_K(L, k) gives a homotopyh⁰_L from 0 tof^∗(ω_X⁰ )inA²_K(L, k).

This induces a 2-commutative diagram inLqcoh(L):

TL

h0 L·

//

Tf

0

f^∗(T^X) ^f

∗(ω⁰_X)·

' //f^∗(L^X[k]) ^Lf[k] //L^L[k].

(2.2)

We say that h⁰_L is nondegenerate if (2.2) is homotopy Cartesian (equiva- lently, homotopy co-Cartesian), and then we say thatL (with its morphism f :L→X and isotropic structureh_L) is Lagrangianin (X, ω_X).

An alternative way to explain the nondegeneracy ofh⁰_L is to note that it induces a natural morphismχ:TL/X→LL[k−1]via the diagram

TL

h0 L·

//

Tf

0

f^∗(TX) ^f

∗(ω⁰_X)·

' //

f^∗(LX[k]) ^Lf[k] //LL[k]

TL/X[1],

χ[1]

11 (2.3)

andh⁰_L is nondegenerate ifχ:TL/X →LL[k−1]is a quasi-isomorphism.

Now suppose that X ' SpecA^• and L ' SpecB^• are affine, and f is induced by a morphism α : A^• → B^• in cdga_K, and ωX lifts to ωA^• = (ω⁰_A•, ω¹_A•, ω²_A•, . . .) in (Q

i>0(Λ²⁺ⁱLA^•[k])^∗−i,d + ddR) as in Defi- nition 2.7. Then we can write h_L as a sequence (h⁰, h¹, h², . . .) with hⁱ ∈ (Λ²⁺ⁱLB^•)^1+k−i fori= 0,1, . . . ,where h_L an isotropic structure is equiva- lent to the equations

α_∗(ω⁰_A•) = dh⁰, α_∗(ωⁱ_A•) = dhⁱ+ d_dRhⁱ⁻¹, i= 1,2, . . . . (2.4) Remark 2.10. —Let us discuss virtual dimensions of shifted symplectic derivedK-schemes and their Lagrangians. If (X, ω_X) is ak-shifted symplec- tic derivedK-scheme, it is easy to show (e.g. using the “Darboux Theorem”

in Section 2.5) that

(i) If k≡0 mod 4 then vdimX is even inZ. (ii) If k≡1 mod 4 then vdimX= 0.

(iii) Ifk≡2 mod 4 then vdimX can take any value inZ. (iv) If k≡3 mod 4 then vdimX= 0.

Now suppose f : L →X, hL is Lagrangian in (X, ωX). Then we find that:

(i⁰) If k≡0 mod 4 then vdimL= ¹₂vdimX.

(ii⁰) If k≡1 mod 4 then vdimLcan take any even value inZ.

(iii⁰) Ifk≡2 mod 4 then vdimX must be even (at least near the image ofLinX), and vdimL=¹₂vdimX.

(iv⁰) If k≡3 mod 4 then vdimLcan take any value inZ.

So ifk≡2 mod 4 and vdimXis odd then no Lagrangians exist in (X, ω_X).

Example 2.11. — Take X = SpecK to be the point ∗, regarded as a k-shifted symplectic derivedK-scheme with symplectic formωX = 0. Then LagrangiansL in (∗,0) are equivalent to (k−1)-shifted symplectic derived K-schemes (L, ωL).

Pantev et al. [19, Th. 2.10] prove:

Theorem 2.12. — Suppose(X, ωX)is ak-shifted symplectic derivedK- scheme, andf₁ :L1 →X and f₂ :L2→X are Lagrangians in (X, ωX).

Then the fibre productL1×f₁,X,f₂L2indSch_Khas a natural(k−1)-shifted symplectic structure.

In the stacky case, Calaque [11, §3.2] extends Theorem 2.8:

Theorem 2.13. — SupposeX is a Fano(m+1)-fold overK,andY ⊆X is a smooth anticanonical divisor, so thatY is a Calabi–Yau m-fold. Write L,M for the derived moduli stacks of complexes of coherent sheaves on X, Y,andf :L→Mfor the morphism of derived restriction fromX toY. Theorem 2.8 gives a(2−m)-shifted symplectic structureω_M on M. Then there is a natural isotropic structure h_L on f : L→M making L into a Lagrangian in(M, ω_M).

2.5. A shifted symplectic “Darboux Theorem”

Bussi, Brav and Joyce [7] prove “Darboux Theorems” fork-shifted sym- plectic derivedK-schemes (X, ωX) fork <0, which give explicit Zariski or étale local models for (X, ω_X). We will explain their main result in Theo- rem 2.18 below. First, in Examples 2.14 and 2.16 we define families of explicit

“Darboux form”k-shifted symplectic cdgasA^•, ω fork <0.

Example 2.14. — Let k =−1,−2, . . . , and set d = [(k+ 1)/2], so that d=k/2 ifk is even (giving k= 2d), and d= (k+ 1)/2 ifk is odd (giving k = 2d−1). Following [7, Examples 5.8 & 5.9], we will define a simple class of standard form cdgasA^• = (A^∗,d) equipped with explicitk-shifted symplectic formsω= (ω⁰,0,0, . . .), which we will call ofDarboux form.

Fix nonnegative integers m₀, m₋₁, m₋₂, . . . , m_d. Choose a smooth K- algebra A⁰ of dimension m₀. Localizing A⁰ if necessary, we may assume that there exist x⁰₁, . . . , x⁰_m

0 ∈ A⁰ such that d_dRx⁰₁, . . . ,d_dRx⁰_m

0 form a ba- sis of Ω¹_A0 over A⁰. Geometrically, U = SpecA⁰ is a smooth K-scheme of dimension m₀, and (x⁰₁, . . . , x⁰_m

0) : U → A^m0 are global étale coordinates onU.

DefineA^∗ as a commutative gradedK-algebra to be the free graded al- gebra overA⁰generated by variables

xⁱ₁, . . . , xⁱ_m

i in degreeifori=−1,−2, . . . , d, and

y₁^k−i, . . . , y_m^k−i_i in degreek−ifori= 0,−1, . . . , d. (2.5) So the upper index i in xⁱ_j, y_jⁱ always indicates the degree. The variables come in pairsxⁱ_j, y_j^k−i, with total degreek. We will define the differential d in the cdgaA^•= (A^∗,d) later.

As in Section 2.1, the spaces (Λ^pΩ¹_A•)^k and the de Rham differential d_dR upon them depend only on the commutative graded algebra A^∗, not on the (not yet defined) differential d. Note that Ω¹_A• is the freeA^∗-module with basis d_dRxⁱ_j,d_dRy^k−i_j fori= 0,−1, . . . , d andj = 1, . . . , m_i. Define an element

ω⁰=

d

X

i=0 m_i

X

j=1

ddRxⁱ_jddRy^k−i_j in (Λ²Ω¹_A•)^k. (2.6) Clearly ddRω⁰= 0 in (Λ³Ω¹_A•)^k.

Now choose a superpotential Φ inA^k+1, called theHamiltonian, which we require to satisfy theclassical master equation

d

X

i=−1 m_i

X

j=1

∂Φ

∂xⁱ_j

∂Φ

∂y_j^k−i = 0 inA^k+2. (2.7) Define the differential d onA^∗ by d = 0 onA⁰, and

dxⁱ_j = (−1)^(i+1)(k+1) ∂Φ

∂y_j^k−i, dy^k−i_j = ∂Φ

∂xⁱ_j,

i= 0, . . . , d, j = 1, . . . , mi. (2.8) Equation (2.7) implies that d◦d = 0.

ThenA^•= (A^∗,d) is a standard form cdga, as in Definition 2.4, with vdimA^•=

(2Pd

i=0(−1)ⁱmi, keven,

0, kodd,