arXiv:2010.14221v2 [math.SG] 10 Dec 2020 Introduction 1 Contents Examples 4 Locus Critical Derived the of Geometry Symplectic 3 Geometry Symplectic Derived 2 [email protected] ∗ . eie eoLcso hfe -om ...... 1-Forms . . . Shifted . . . . of . . Locus . . . Zero . 4.4 . . Points . Derived . Critical . . . non-Degenerate . . . of . 4.3 Family . . Point . Critical . . . Non-Degenerate 4.2 . . One ...... 4.1 ...... Locus . . . . Critical Intersections . . Derived . . Derived . . . and . Fibrations . 3.3 . . . Lagrangian . . . . are . . . Intersections . 3.2 Lagrangian ...... 3.1 ...... Stacks . . Linear . of . . . Complexes . Cotangent . . Relative . . . . . Fibration 2.4 . . Lagrangian . Structures . 2.3 Lagrangian Structures Symplectic 2.2 Shifted 2.1 MG nv otele,CR,Mnplir France Montpellier, CNRS, Montpellier, Univ. IMAG, eie arninFbaino h Derived the on Fibration Lagrangian Derived A ou a aua arninfibration Lagrangian natural a has locus hss npriua,w hwta o functional a for that show we particular, In phisms. baino h eie rtcllcsi eemndb h H the by determined is locus s we critical derived smooth, the is on locus critical fibration strict the and non-degenerate G esuytesmlci emtyo eie nescin o intersections derived of geometry symplectic the study We EuvratTitdCtnetBnls...... Bundles Cotangent Twisted -Equivariant rtclLocus Critical eebr1,2020 11, December li Grataloup Albin p n Abstract ´ 1 1 q SitdSmlci ...... Symplectic -Shifted Crit p f Ñ q f ∗ : X X ntecs where case the In . Ñ o htteLagrangian the that how A sinqartcform. quadratic essian k 1 h eie critical derived the , arninmor- Lagrangian f ...... f is . . . . . 22 16 30 30 24 22 19 17 16 11 10 4 2 8 4 1 Introduction

In the context of derived algebraic geometry ([11], [12], [17], [18], [19]), the notion of shifted symplectic structures was developed in [13] (see also [6] and [7]). This has proven to be very useful in order to obtain symplectic structures out of natural constructions. For example we obtain: • shifted symplectic structures from transgression procedures (Theorem 2.5 in [13]), for example, the AKSZ construction.

• shifted symplectic structures from derived intersections of Lagrangians structures (Section 2.2 in [13]).

• symplectic structures on various moduli spaces (Section 3.1 in [13]).

• quasi-symplectic groupoids (see [21]) inducing shifted symplectic structures on the quotient stack as explained in [6].

• symmetric obstruction theory as defined in [1] from p´1q-shifted symplectic derived stacks (see [14] for the obstruction theory on derived stacks and [13] for the sym- metric and symplectic enhancement thereof).

• the d-critical loci as defined by Joyce in [10]. All p´1q-shifted symplectic derived scheme induces a classical d-critical locus on its truncation (see Theorem 6.6 in [2]). Another very useful construction in derived geometry is the derived intersection of derived schemes or derived stacks (see [13]). This includes many constructions such as:

• the derived critical locus of a functional (see [13] and [20]). For an action functional, this amounts to finding the space of solutions to the Euler-Lagrange equations, as well as remembering about the symmetries of the functional.

• G-equivariant intersections. This includes the example of symplectic reduction which can be expressed as the derived intersection of derived quotient stacks (see Section 2.1.2 in [5]).

In this paper, we make a more precise study of the shifted symplectic geometry of derived critical loci, and more generally of the derived intersections of Lagrangian mor- phisms. In particular, the main theorem (Theorem 3.4) of this paper says that whenever the Lagrangian morphisms fi : Xi Ñ Z, i “ 1..2 look like "sections" in the sens that there exists a map r : Z Ñ X such that the composition maps r ˝ fi : Xi Ñ X are weak equivalences, then the natural morphism X1 ˆZ X2 Ñ X is a Lagrangian fibration (see [7]). We then specialise this result to various examples and show in particular that, for the derived critical locus of a non-degenerate functional on a smooth algebraic variety, the non-degeneracy of the Lagrangian fibration is related to the non-degeneracy of the Hessian quadratic form of the functional.

This paper starts, in Subsection 2, by recalling the basic definitions and properties of shifted symplectic structures, Lagrangian structures and Lagrangian fibrations. We also recall, in Section 2.4, basic properties of the relative cotangent complexes of linear stacks

2 that proves useful when we try to understand in more details the structure of Lagrangian fibrations on derived critical loci. In Section 3 we start by recalling the fact that a derived intersection of Lagrangian structures in a n-shifted symplectic derived Artin stacks is pn ´ 1q-shifted symplectic. Then, in Subsection 3.2, we state and prove the main theorem (Theorem 3.4) that roughly says that if the Lagrangian morphisms look like sections (up to homotopy), then the natural projection from the derived intersection has a structure of a Lagrangian fibration. A1 We then recall basic elements on the derived critical loci of a functional f : X Ñ k, and then try to describe the Lagrangian fibration structure on the natural map CritpfqÑ X obtained from the main theorem. Section 4 gives examples of applications of our main theorem. In particular, in Subsec- tions 4.1 and 4.2, we give a better description of the Lagrangian fibration on the derived critical loci for non-degenerate functionals. We show that the non-degeneracy condition of the Lagrangian fibration of the derived critical locus of a non-degenerate functional on a smooth algebraic variety is given by the non-degeneracy of the Hessian quadratic form.

Acknowledgements: I would like to thank Damien Calaque for suggesting this project; for all his help with it and for his revisions of this paper. I am also very grateful for everything he explained to me on the subject of derived algebraic geometry. I would also like to thank Pavel Safronov for his comments on the first version of this paper. This research has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 768679).

Notation:

• Throughout this paper k denotes a field of characteristic 0.

• cdga (resp. cdgaď0) denotes the 8-category of commutative differential graded algebra over k (resp. commutative differential graded algebra in non positive de- grees).

• cdgagr denotes the 8-category of commutative monoids in the category of graded gr complexes dgk . • A ´ Mod denotes the 8-category of differential graded A-modules for A P cdga.

• cdgaǫ´gr denotes the 8-category of graded mixed differential graded algebra. We denote the differential δ and the mixed differential ǫ or d “ dDR in the case of the De Rham complex of a derived Artin stack X, denoted DRpXq. We refer to [8] for ǫ´gr gr gr all the definitions of cdga , cdga , dgk and the De Rham complex (see also [13] but with a different grading convention).

• All the 8-categories above are localisations of model categories (see [8] for details on these model structures and associated 8-categories). All along, unless explicitly stated otherwise, all diagrams will be homotopy commutative, all functor will be 8-functors and all (co)limits will be 8-(co)limits.

3 • For X a derived Artin stack, QCpXq denote the 8-category of quasi-coherent sheaves on X.

• In this paper derived Artin stack are defined as in [19]. In particular all our derived Artin stacks are locally of finite presentation over Specpkq.

• We denote by LX the cotangent complex of a derived Artin stack X. We denote by T L_ L X :“ X :“ Hom p X , OX q its dual.

2 Derived Symplectic Geometry

2.1 Shifted Symplectic Structures Before going to symplectic structures, we make a short recall of differential calculus and (closed) differential p-forms in the derived setting. Recall from [13] that there are classify- ing stacks App‚, nq and Ap,clp‚, nq of respectively the space of n-shifted differential p-forms and the space of n-shifted closed differential p-forms. We use the grading conventions used in [8]. On a derived affine scheme Spec(A), the space of p-forms of degree n and the space of closed p-forms of degree n are defined respectively by

p A pA, nq :“ Mapcdgagr pkr´n ´ psp´pq, DRpAqq

and

p,cl A pA, nq :“ Mapcdgaǫ´gr pkr´n ´ psp´pq, DRpAqq . From [8], the de Rham complex of A, denoted DRpAq, can be described, as a graded # L # complex, by DRpAq » SymA Ar´1sp´1q where p´q is the functor forgetting the mixed structure cdgaǫ´gr Ñ cdgagr (we refer to [8] for more on the de Rham complex).

All along, we denote the internal differential, i.e. the differential on LA, by δ and the mixed differential, i.e. the de Rham differential, by d.

By definition, the space of p-forms of degree n on a derived stack X is the map- p ping space MapdSt pX, A p‚, nqq and the space of closed p-forms of degree n on X is p,cl MapdSt X, A p‚, nq . Now the following proposition says that in the case where X is a derived` Artin stack,˘ the spaces of shifted differential forms are spaces of sections of quasi-coherent sheaves on X.

Proposition 2.1 (Proposition 1.14 in [13]). Let X be a derived Artin stack over k and LX be its cotangent complex over k. Then there is an equivalence

p pL A pX, nq» MapQCpXq pOX , Λ X rnsq . In particular, p n p π0pA pX, nqq “ H pX, Λ LX q.

4 Remark 2.2. More concretely, we have from [6] and [8] an explicit description of (closed) p-forms of degree n on a geometric derived stack X. A p-form of degree n is given p by a global section ω P DRpXqppqrn ` ps » RΓ pp LX qrnsq such that δω “ 0. A closed p-form of degree n is given by a semi-infiniteŹ sequence ω “ ω0 ` ω1 `¨¨¨ with p`i ωi P DRpp`iqrn ` ps“ RΓ LX rn ´ is such that δω0 “ 0 and dωi “ δωi`1. Equivalently, being closed´´Ź means that¯ ω is¯ closed for the total differential D “ δ ` d R p`i L in the bi-complex DRpXqěprns» Γ iě0 X rns whose total degree is given by n ` p ` i. Note that the conditions´ imposedś ´Ź on ω are¯ equivalent¯ to saying that ω is a cocycle of degree n ` p for the total differential.

In general, we can also describe the spaces of (closed) differential forms as AppX, nq» p,cl DRppqpXqrns and A pX, nq » iěp DRpp`iqpXqrns , where iěp DRpp`iqpXqrns is ˇ ˇ ˇendowed withˇ the total differential.ˇś ˇ ś ˇ ˇ ˇ ˇ Remark 2.3. Given a map of derived Artin stack f : Y Ñ X, we define Ap,pclqpY {X, nq, the n-shifted (closed) p-forms on Y relative to X, to be the homotopy cofiber of the natural map Ap,pclqpX, nqÑ Ap,pclqpY, nq. For instance n-shifted relative p-forms are equivalent to p the derived global sections of LY rns, with the relative cotangent complex LY äX äX ´ ¯ ˚ defined as the homotopy cofiberŹ of the natural map f LX Ñ LY . We refer to [8] for more details on the relative n-shifted (closed) p-forms and the relative version of the De Rham complex.

For every closed p-form of degree n, ω, we have the underlying p-form of degree n given by ω0 obtained from the natural projection DRěp Ñ DRppq. It induces a morphism Ap,clp‚, nqÑ App‚, nq that forgets the higher differential forms defining the closure of ω.

We say that a p-form, ω0, of degree n can be lifted to a closed p-form of degree n if there exists a family of pp ` iq-forms ωi of degree n ´ i for all i ą 0, such that ω “ ω0 ` ω1 `¨¨¨ is closed in DRpXqěprns (i.e. Dω “ 0). In this situation, we can see that dω0 is a priori not equal to 0 but is in fact homotopic to 0 (dω0 “ D p´ ią0 ωp`iq). The choice of such a homotopy is the same as a choice of a closure of the p-formř of degree n. Being closed is therefore no longer a property of the underlying p-form of degree n but a structure added to it given by the higher forms. In other words, a closure of ω0 is given by a homotopy between dω0 and zero. The collection of all closures of a p-form of degree n forms a space:

Definition 2.4. Let α P AppX, nq then the space of all closures of α is called the space of keys of α denoted keypαq. It is given by the homotopy pull-back:

keypαq Ap,clpX, nq (1)

‹ α AppX, nq

The mixed differential of the de Rham graded mixed complex induces a map d : AppX, nqÑ Ap`1pX, nq. Since dω is d-closed and δ ˝ dω “ d ˝ δω “ 0, we get Dpdωq“ 0.

5 Therefore the image of dω through the inclusion DRpp`1qrnsÑ DRěp`1rns is a pp`1`nq- cocycle, that is a closed (p+1)-form of degree n with all higher forms being equal to zero. We obtain a map of spaces d : AppX, nqÑ Ap`1,clpX, nq.

We are now turning toward symplectic geometry. We now know what are (shifted) closed 2-forms we only need to mimic the notion of non-degeneracy to define symplectic structures. Definition 2.5 (Non-Degenerate 2-Form of Degree n). For a derived Artin n-stack X, L T L_ the cotangent complex X is dualisable. Therefore there is a tangent complex X “ X . We say that a (closed) 2-form of degree n is non-degenerate if the (underlying) 2-form ω0 of degree n induces a quasi-isomorphism: 5 T L ω0 : X Ñ X rns Definition 2.6 (Shifted Symplectic Forms). A n-shifted symplectic structure is a non-degenerate n-shifted closed 2-form on X. The main example of a symplectic manifold is the cotangent bundle. In our setting, we can speak of n-shifted cotangent stacks. It is a derived stack defined as linear stack ˚ associated to LX rns, T rnsX :“ ApLX rnsq (see Definition 2.7). It comes with a natural ˚ morphism πX : T rnsX Ñ X. We refer to [7] for a general account of shifted symplectic geometry on the cotangent stack. Definition 2.7 (Linear Stacks). Given F P QCpXq a quasi-coherent sheaf over a derived Artin stack, we can construct a linear stack denoted ApFq and defined, as a derived stack over X, by A ˚ pFqpf : SpecpAqÑ Xq :“ MapA-Mod pA, f Fq . Remark 2.8. Whenever F is co-connective, ApFq is equivalent to the relative spec- F_ ˚ trum SpecX SymOX . A map SpecpAq Ñ T X is equivalently given by a map ˚L χ : SpecpAqÑ` X together˘ with a section s P MapA´ModpA, χ X q.

˚ ˚ _ MapA´ModpA, χ Fq» MapA´AlgpSymAχ F , Aq _ :“ SpecX pSymAF qpχ : SpecpAqÑ Xq A morphism Y Ñ T ˚rnsX is determined by the induced morphism f : Y Ñ X (by ˚ ˚ composition with πX ) and a section s : Y Ñ f T rnsX which corresponds to an element ˚L ˚ s P MapQCpY q pOY , f X rnsq. In the case of a section s : X Ñ T rnsX, we get the L 1 identity Id : X Ñ X and a section s P MapQCpXq pOX , X rnsq » A pX, nq. This shows, using Proposition 2.1, that the space of sections of T ˚rnsX is exactly the space of 1-forms of degree n as expected. Example 2.9. Any ordinary smooth symplectic variety can be seen as a derived Artin with a 0-shifted symplectic structure. Conversely, any 0-shifted symplectic structure on an Artin stack representable by a smooth variety is equivalent to a symplectic structure in the classical sense on that variety.

6 Example 2.10. As in the classical case, we can construct the canonical Liouville 1- form. Consider the identity Id : T ˚rnsX Ñ T ˚rnsX. It is determined by the projection ˚ ˚ ˚L π : T rnsX Ñ X and a section λX P MapQCpT ˚rnsXq OT rnsX , π X rns . Since we have a ˚ ˚ natural map π LX rnsÑ LT ˚rnsX rns, λX induces a 1-form` on T rnsX called˘ the tautologi- cal 1-form. This 1-form induces a closed 2-form dλX which happens to be non-degenerate (see [7] for a proof of that statement).

This symplectic structure on the cotangent is universal in the sense that it satisfies the usual universal property.

˚ ˚ Lemma 2.11. Given a 1-form α : X Ñ T rnsX, we have that α λX “ α.

Proof. In general, if we take f : X Ñ Y , the pull-back of a n-shifted 1-form, β, is described by:

pdfq˚ T ˚rnsX f ˚T ˚rnsY T ˚rnsY

β ˚ f β f XY ˚ ˚ Taking into account the fact that λ factors through πX T rnsX, we consider the fol- lowing diagram:

pdαq˚ T ˚rnsX α˚T ˚T ˚rnsX T ˚T ˚rnsX

pdπ q˚ pdπ q˚ Id X X ˚ ˚ ˚ ˚ ˚ ˚ T rnsX “ α πX T rnsX πX T rnsX

λ˜ λ Xα T ˚rnsX

˚ This proves that the pull-back along α of λX seen as a 1-form of degree n on T rnsX ˚ ˚ ˚ is the same as the pull-back along α of the section λX : T rnsX Ñ πX T rnsX.

L We denote by α1 the associated section in MapQCpXq pOX , X rnsq of degree n. There ˚ is a one-to-one correspondence between sections of πX : T rnsX Ñ X and points of L MapQCpXq pOX , X rnsq. Now we use the fact that Id ˝ α “ α: L • On the one hand, α is completely described by α1 P MapQCpXq pOX , X rnsq. • On the other hand, the map Id : T ˚rnsX Ñ T ˚rnsX is described by the projection ˚ ˚ ˚ L π : T rnsX Ñ X and the section λX P MapQCpT ˚rnsXq OT rnsX , πX X . There- ˚ fore the composition Id ˝ α is also a section of πX and` is described by˘ α λX P L MapQCpXq pOX , X rnsq.

˚ This proves that α λX “ α1. Since these maps characterise the sections of πX they ˚ represent, we have α λX “ α.

7 2.2 Lagrangian Structures We recall from [13] the definition and standard properties of Lagrangian structures. We reproduce some easy proves here for the convenience of the reader. Definition 2.12 (Isotropic Structures). Let f : L Ñ X be a map of derived Artin stacks and suppose X has a n-shifted symplectic structure ω. An isotropic structure on f is a homotopy, in A2,clpL, nq, between f ˚ω and 0. Isotropic structures on f form a space described by the homotopy pull-back:

Isopfq ‹

f ˚ω

‹ 0 A2,clpL, nq Remark 2.13. More explicitly, an isotropic structure is given by a family of forms of ˚ total degree pp ` n ´ 1q, pγiqiPN with γi P DRpLqpp`iqrp ` n ` i ´ 1s, such that δγ0 “ f ω0 ˚ ˚ and δγi `dγi´1 “ f ωi. This can be rephrased as Dγ “ f ω, thus γ is indeed a homotopy between f ˚ω and 0. Definition 2.14 (Lagrangian Structures). An isotropic structure γ on f : L Ñ X is called a Lagrangian structure on f if the leading term, γ0, viewed as an isotropic structure ˚ on the morphism TL Ñ f TX , is non-degenerate. We say that γ0 is non-degenerate if the following null-homotopic sequence (homotopic to 0 via γ0) is fibered:

˚ ˚ TL f TX » f LX rns LLrns (2)

˚ 5 pf ω0q Remark 2.15. To say that that sequence is fibered can be reinterpreted as a more classical condition involving the conormal. The relative cotangent complex Lf rns, also denoted LL rns when f is clear from context, is the homotopy cofiber of the natural äX ˚ map f LX rns Ñ LLrns. Since QCpXq is a stable 8-categories, the homotopy fiber of ˚ f LX rns Ñ LLrns is Lf rn ´ 1s and the non-degeneracy condition can be rephrased by saying that the natural map TL Ñ Lf rn ´ 1s is a quasi-isomorphism. Example 2.16. A 1-form of degree n on an Artin stack X is equivalent to a section α : X Ñ T ˚rnsX. This section is a Lagrangian morphism if and only if α admits a closure, i.e. Keypαq is non-empty. This is Theorem 2.15 in [7]. Proposition 2.17 proves a part of Example 2.16. Proposition 2.17. There is a weak homotopy equivalence IsopαqÑ Keypαq between the space of isotropic structures on the 1-form α and the space of keys of α. Proof. keypαq A1,clpX, nq ‹

0 (3)

d ‹ α A1pX, nq dR A2,clpX, nq

8 The leftmost square is Cartesian by definition of keypαq in Definiion 2.4. By definition, ˚ the pull-back of the outer square is Isopαq because ddRα “ α ω (by universal property of the Liouville 1-form, Lemma 2.11). It turns out that the rightmost square is also Cartesian. This is simply saying that the space of closed 1-forms of degree n is the same as the space of 1-forms of degree n whose de Rham differential is homotopic to 0. We obtain that keypαq and Iso pαq are both pull-backs of the outer square and therefore are canonically homotopy equivalent. Remark 2.18. It turns out that Theorem 2.15 in [7] says that all the isotropic structures on α (or equivalently the lifts of α to a closed form) are in fact non-degenerate, which implies the statement in Example 2.16.

Lemma 2.19. Consider the map X Ñ‹n where ‹n is the point endowed with the canon- ical n-shifted symplectic structure given by 0. Then a Lagrangian structure on this map is equivalent to an pn ´ 1q-shifted symplectic structure on X.

Proof. Pick an isotropic structure γ on p. We know that γ is a homotopy between 0 and 0 which means that Dγ “ 0. Therefore γ is a closed 2-form of degree n ´ 1. We want to show that γ is non-degenerate as an isotropic structure if and only if it is non-degenerate as a closed 2-form on X. The non-degeneracy of the Lagrangian structure, as described in Remark 2.15, corresponds to the requirement that the natural map TX Ñ LX rn ´ 1s is a quasi-isomorphism. This map depends on γ0 and we want to show that this map is 5 in fact γ0. This map is the natural map that fits in the following homotopy commutative diagram:

TX LX rn ´ 1s 0

05 0 0 LX rns We can show that by strictifying the homotopy commutative diagram:

TX p˚ω5“0

0 LX rns Note that this diagram is already commutative but we see it as homotopy commutative using the homotopy γ0. We use the homotopy γ0 to strictify the previous diagram and we obtain:

TX p˚ω5“0 γ5 `0 0 L L L X rn ´ 1s‘ X rns pr X rns

The homotopy fiber and also strict fiber of the projection pr : LX rn ´ 1s‘ LX rns Ñ L L 5 T L X rns is X rn ´ 1s, and therefore the natural map we obtain is γ0 : X Ñ X rn ´ 1s.

Since the non-degeneracy condition of the isotropic structure γ is the same as saying 5 that the map γ0 is a quasi-isomorphism, we have shown that an isotropic structure γ is an

9 pn´1q-shifted symplectic structure on X if and only if it is non-degenerate as an isotropic structure on X Ñ‹n. Definition 2.20 (Lagrangian Correspondence, [5]). Let X and Y be derived Artin stacks with n-shifted symplectic structures. A Lagrangian correspondence from X to Y is given by a derived Artin stack L with morphims

L

XY and a Lagrangian structure on the map L Ñ X ˆ Y where X ˆ Y is endowed with the n-shifted symplectic structure π˚ ω ´ π˚ ω . For example, a Lagrangian structure on X X Y Y s s L Ñ X is equivalent to a Lagrangian correspondence from X to ‹.

As explained in [5] Section 4.2.2, these Lagrangian correspondences can be composed. If we take X0, X1 and X2 derived Artin stacks with symplectic structures and L01 and L12 Lagrangian correspondences from respectively X0 to X1 and X1 to X2. We can produce a Lagrangian correspondence L02 from X0 to X2 by setting L02 :“ L01 ˆX1 L12.

L02

L01 L12

X0 X1 X2

2.3 Lagrangian Fibration We recall in this section the definition and standard properties of Lagrangian fibrations ([6] and [7]).

Definition 2.21. Let f : Y Ñ X be a map of derived Artin stacks and ω a symplectic structure on Y . A Lagrangian fibration on f is given by:

• A homotopy, denoted γ, between ω{X and 0, where ω{X is the image of ω under the natural map A2,clpY, nqÑ A2,clpY {X, nq (see Remark 2.3).

• A non-degeneracy condition which says that the following sequence (homotopic to 0 via γ0) is fibered:

TY {X Ñ TY » LY rnsÑ LY {X rns

In particular, the non-degeneracy condition can be rephrased by saying that there is a ˚ canonical quasi-isomorphism αf : TY {X Ñ f LX rns (similar to the criteria for Lagrangian morphism in Remark 2.15) that makes the following diagram commute:

10 α f ˚ TY f LX rns 0 äX (4)

ω5 TY LY rns LY rns äX

˚ Example 2.22. The natural projection πX : T rnsX Ñ X is a Lagrangian fibration. The ˚ L Liouville 1-form is a section of πX X rns which is part of the fiber sequence:

˚ L L ˚ L ˚ πX X rnsÑ T rnsX rnsÑ T rnsX rns äX

Thus the 1-form induced by λX in LT ˚rnsX rns is homotopic to 0. The non-degeneracy äX condition is more difficult and is proven in Section 2.2.2 of [7]. It turns out that the mor- phism expressing the non-degeneracy condition, απX , is given by a canonical construction (Proposition 2.24) which does not depend on the symplectic structure. This is the content of Proposition 2.28.

Lemma 2.23. Let x : ‹n Ñ X be a point of X. Then, given a Lagrangian fibration ˚ structure on x, the non-degeneracy condition is given by a quasi-isomorphism x TX Ñ ˚ x LX rn ` 1s.

Proof. The Lagrangian fibration structure on ‹n Ñ X is a homotopy between 0 and 2,cl ‹ itself in A p näX, nq. As in the proof of Lemma 2.19, this is given by an element 2,cl ‹ γ P A p näX, n ´ 1q. Similarly to what was done in the proof of Lemma 2.19, we can show that γ is non-degenerate as a Lagrangian fibration if and only if it is non-degenerate as a closed 2-form of degree n. Again it boils down to the fact that the natural morphism in 5 T‹ L‹ the non-degeneracy criteria for Lagrangian fibrations is in fact γ0 : näX Ñ näX rn ´ 1s. ˚ T‹ T L‹ Moreover, we have natural equivalences, näX » x X r´1s and näX rn ´ 1s » ˚ x LX rns because the sequence

˚ T‹ T T näX ‹n » 0 x X rns

is fibered. This concludes the proof.

2.4 Relative Cotangent Complexes of Linear Stacks This section is devoted to the study of relative cotangent complexes of linear stacks. Given F P QCpXq a dualisable quasi-coherent sheaf over a derived Artin stack X, we consider its associated linear stack, ApFq (see Definition 2.7) and the goal of this section is to describe LApFq and its functoriality in F and X. äX Proposition 2.24. Let X be a derived Artin stack and F P QCpXq a dualisable quasi- coherent sheaf on X. We denote πX : ApFqÑ X the natural projection. Then we have:

L L ˚ _ πX » ApFq » πX F äX

11 Proof. We will show the result for any B-point y : SpecpBq Ñ ApFq and we write x “ π ˝ y : SpecpBqÑ X. We will show that for all M P B ´ Mod connective, we have

˚ ˚ _ Hom LA Hom F B´Mod y pFqä , M » B´Mod px , Mq ´ X ¯ ˚ First we observe that HomB´Mod y LApFq , M is equivalent, using the universal äX property of the cotangent complex, to´ the following¯ homotopy fiber at y:

A F A F hofibery HomdStä pSpecpB ‘ Mq, p qq Ñ HomdStä pSpecpBq, p qq ´ X X ¯ with B ‘ M denoting the square zero extension and SpecpB ‘ Mq Ñ X being the composition:

p SpecpB ‘ Mq SpecpBq x X

˚ LA Thus a map in HomB´Mod y pFqä , M is completely determined by a map ´ X ¯ Φ: SpecpB ‘ MqÑ ApFq making the following diagram commute:

SpecpBq y i SpecpB ‘ Mq Φ ApFq

p πX SpecpBq x X

˚ Hom LA Thus, we obtain that B´Mod y pFqä , M is equivalent to ´ X ¯ ˚ ˚ ˚ hofibersy pMapB‘M´Mod pB ‘ M,p x FqÑ MapB´Mod pB, x Fqq

˚ A where sy P MapB´Mod pB, x Fq is the section associated to y : SpecpBq Ñ pFq. The map is then given by precomposition with i˚. We can now observe that p˚x˚F “ ˚ ˚ x F ‘ x F bB M and that

˚ ˚ ˚ ˚ MapB‘M´Mod pB ‘ M,p x Fq» MapB´Mod pB, x F ‘ x F bB Mq We obtain ˚ Hom LA B´Mod y pFqä , M ´ X ¯ ˚ ˚ ˚ » hofiber pMapB´Mod pB, x F ‘ x F bB MqÑ MapB´Mod pB, x Fqq ˚ ˚ _ » MapB´Mod pB, x F bB Mq» MapB´Mod px F , Mq Now the result follows from the fact that the functor

12 B ´ Mod Fun B ´ Modď0, sSet ` ˘ N MapB´Mod pN, ‚q

is fully faithful and the fact that everything we did is natural in B.

Lemma 2.25. Let f : X Ñ Y be a morphism of derived Artin stacks. We consider F P QCpY q dualisable. Then there is a commutative square:

˚ Φ LApFq LApf ˚Fq äX äY

» »

˚ ˚ _ » ˚ ˚ _ Φ πY F πX f F with Φ the natural morphism in the following homotopy pull-back:

Apf ˚Fq» f ˚ApFq Φ ApFq

πX πY f XY ˚ ˚ _ ˚ ˚ _ and the lower horizontal equivalence Φ πY F Ñ πX f F being the equivalence coming from the fact that πY ˝ Φ » f ˝ πX . Proof. The first things we observe is that Apf ˚Fq » f ˚ApFq. We consider as before B-points:

y˜

y SpecpBq f ˚ApFq Φ ApFq

x πX πY f XY x˜ We want to show that the following diagram is commutative:

˚ ˚ HomB´Mod y LApf ˚Fq , M HomB´Mod y˜ LApFq , M äX äY ´ ¯ ´ ¯ (5) » »

˚ ˚ ˚ _ ˚ ˚ _ HomB´Mod py πX f F , Mq HomB´Mod py˜ πY F , Mq Using the universal property of the cotangent complex, the top horizontal arrow is naturally equivalent to the map

13 A ˚F A ˚F hofibery HomdStä pSpecpB ‘ Mq, pf qq Ñ HomdStä pSpecpBq, pf qq ´ X X ¯

A F A F hofibery˜ HomdStä pSpecpB ‘ Mq, p qq Ñ HomdStä pSpecpBq, p qq ´ Y Y ¯

˚ induced by HomdSt p´, Φq. A map ψ : SpecpB ‘ Mq Ñ Apf Fq in this homotopy fiber fits in the following commutative diagram:

SpecpBq y˜ i y SpecpB ‘ Mq Apf ˚Fq ApFq ψ Φ

p πX πY f SpecpBq x XY and the map between the homotopy fiber sends ψ to Φ ˝ ψ. Since the underlying map of ψ is πX ˝ ψ : SpecpB ‘ Mq Ñ X is x ˝ p and the underlying map of Φ ˝ ψ is πY ˝ Φ ˝ ψ : SpecpB ‘ MqÑ Y is f ˝ x ˝ p “ x˜ ˝ p, this map between the homotopy fiber of derived stacks is therefore naturally equivalent to the map:

˚ ˚ ˚ ˚ ˚ ˚ hofibersy pMapB‘M´Mod pB ‘ M,p x f Fqq Ñ HomB´Mod pB,p x f Fqq

˚ ˚F ˚ ˚F hofibersy˜ pMapB‘M´Mod pB ‘ M,p x˜ qq Ñ HomB´Mod pB,p x˜ qq

where sy and sy˜ are the sections associated to y and y˜ respectively. This map is in fact induced by the natural identification p˚x˜˚F » p˚x˚f ˚F (since x˜ “ f ˝ x). But following the steps of the proof of Proposition 2.24, this map is naturally equivalent to the map

˚ ˚ ˚ _ ˚ ˚ _ HomB´Mod py πX f F , MqÑ HomB´Mod py˜ πY F , Mq The natural equivalence we used are all the natural equivalences used in Proposition 2.24 which proves that the Diagram (5) is commutative. Now the result follows once again from the fact that the functor

B ´ Mod Fun B ´ Modď0, sSet ` ˘ N MapB´Mod pN, ‚q

is fully faithful and the fact that everything we did is natural in B.

14 Lemma 2.26. Let X be a derived Artin stacks. We consider F, G P QCpXq dualisable and h : F Ñ G. Then there is a commutative square:

ˆ˚ h LApGq LApFq äX äX

» » π˚ h_ ˚ _ X ˚ _ πX G πX F

with hˆ : ApGqÑ ApFq the map induced by F.

Proof. Every step of the proof of Proposition 2.24 is functorial in F.

Proposition 2.27. Let f : X Ñ Y be a morphism of derived Artin stacks. We consider F P QCpXq and G P QCpY q dualisable and a morphism h : f ˚F Ñ G. Then there is a commutative square:

ˆ˚ LApFq f LApGq äX äY

» » π˚ h_ ˚ _ X ˚ ˚ _ ˆ˚ ˚ _ πX F πX f G “ f πY G

Proof. It follows from Lemma 2.25 and Lemma 2.26.

T ˚ ˚ L Proposition 2.28. The quasi-isomorphism απX : T rnsX Ñ πX X rns of Example äX 2.22 expressing the non-degeneracy of the canonical Lagrangian fibration on the shifted cotangent stacks is the canonical quasi-isomorphism from Proposition 2.24.

Proof. First, since the cotangent bundle has a section, we have a split exact sequence:

˚ L L ˚ L ˚ πX X rns T rnsX rns T rnsX rns äX

L ˚ ˚ T Proposition 2.24 gives us canonical equivalences T rnsX rns » πX X . With this äX data, we can rewrite Diagram (4), up to weak equivalences, as the strictly commutative diagram

» T ˚ ˚ L T rnsX πX X rns 0 äX

˚ L ˚ T ω ˚ T ˚ L ˚ T πX X rns‘ πX X πX X ‘ πX X rns πX X

T ˚ ˚ L Through the canonical equivalence T rnsX Ñ πX X rns of Proposition 2.24, the äX T ˚ ˚ L ˚ T morphism T rnsX Ñ πX X rns‘ πX X simply becomes the natural inclusion and ω äX T ˚ ˚ L becomes the identity. This implies that απX : T rnsX Ñ πX X rns is the canonical äX equivalence of Proposition 2.24.

15 3 Symplectic Geometry of the Derived Critical Locus

In 3.1 and 3.2 we present a few results on the symplectic geometry of homotopy pull- backs of derived Artin stacks. These results apply in particular to the case of derived intersections of derived schemes. In Section 3.3 we study in more details the special case of derived intersections given by derived critical loci.

3.1 Lagrangian Intersections are pn ´ 1q-Shifted Symplectic Proposition 3.1 ([13], Section 2.2). Let Z be a derived Artin stack with a n-shifted symplectic structure ω. Let f : X Ñ Z and g : Y Ñ Z be morphisms with γ and h δ Lagrangian structures on f and g respectively. Then the homotopy pull-back X ˆZ Y possesses a canonical pn´1q-shited symplectic structure called the residue of ω and denoted Rpω,γ,δq.

h h Proof. We consider the maps p : X ˆZ Y Ñ X Ñ Z and q : X ˆZ Y Ñ Y Ñ Z. There is a homotopy h : p ñ q. It induces a homotopy h˚ : p˚ ñ q˚ in the mapping space 2,cl 2,cl h Map A pZ, nq, A pX ˆZ Y, nq . Moreover` the pull-backs of the˘ isotropic structures on f and g define paths γ : 0 ù ˚ ˚ 2,cl h ˚ ˚ ˚ ´1 p ω and δ : 0 ù q ω in A pX ˆZ Y, nq. Concatenating γ, u ω : p ω ù q ω and δ 2,cl h 2,cl h we get a loop at zero in A pX ˆZ Y, nq which can be seen as an element in π1pA pX ˆZ 2,cl h Y, nq, 0q» π0pA pX ˆZ Y, n ´ 1qq. This gives us the closed 2-form of degree pn ´ 1q we denoted by Rpω,γ,δq. From the equivalent point of view of the chain complex DRě2pXqrns, this is just saying that a homotopy between 0 and 0 in degree p ` n is just a cocycle in degree pp ` n ´ 1q, that is, a closed p-form of degree n ´ 1. h We need to show that this closed 2-form is non-degenerate. Denote π : X ˆZ Y Ñ Z h h to be p (or equivalently q). Denote pX : X ˆZ Y Ñ X and pY : X ˆZ Y Ñ Y the natural h morphisms. Then we get a commutative diagram in QCpX ˆZ Y q with exact rows:

˚ ˚ ˚ T h pr TX ‘ pr TY π TZ XˆZ Y X Y 5 5 Θγ ‘Θ ω Rpω,γ,δq0 δ 0 (6) ˚ ˚ ˚ L h rn ´ 1s pr Lf rn ´ 1s‘ pr Lgrn ´ 1s π LZ rns XˆZ Y X Y The middle and right vertical arrows are quasi-isomorphisms because γ and δ are Lagrangian structures and ω is a symplectic form and therefore, non-degenerate. This implies that Rpω,γ,δq is also non-degenerate.

Remark 3.2. Theorem 3.1 is in fact a consequence of the procedure of composition of Lagrangian fibrations. Consider the following composition of Lagrangian correspondences:

X ˆZ Y

XY

‹ Z ‹

16 The maps X Ñ Z ˆ ‹¯ and Y Ñ Y ˆ ‹¯ are Lagrangian correspondences because X Ñ Z and Y Ñ Z are Lagrangian. Therefore, by composition, X ˆZ Y Ñ‹ˆ ‹¯ is also a Lagrangian correspondence, thus X ˆZ Y Ñ‹ is Lagrangian. From Lemma 2.19, since the point is n-shifted symplectic, then X ˆZ Y is pn ´ 1q-shifted symplectic.

3.2 Lagrangian Fibrations and Derived Intersections Proposition 3.3. Suppose we have a sequence L Ñ Y Ñ X of Artin stacks and ω a n-shifted symplectic form on Y. Assume that f : L Ñ Y is a Lagrangian morphism and g : Y Ñ X is a Lagrangian fibration. Then there is a canonical quasi-isomorphism TL Ñ LL rn ´ 1s. äX äX

Proof. Consider the following commutative diagram:

˚ ˚ TL f TY pg ˝ fq TX

˚ TL f TY 0 äX äX » » » » » ˚ ˚ LL rn ´ 1s f LY rns f LY rns » äY äX

˚ LL rn ´ 1s pg ˝ fq LX rns 0 äX In the upper face, all squares are bi-Cartesian because both the outer square and the right most square are bi-Cartesian. All non-dashed vertical arrows are quasi-isomorphisms by assumption (because of the various non-degeneracy conditions). Focusing on the right hand cube, it sends the upper homotopy bi-Cartesian square to the bottom square which ˚ ˚ is also homotopy bi-Cartesian. The homotopy cofiber of pg ˝ fq LX rns Ñ f LY rns is ˚ ˚ ˚ f LY rns and we obtain a quasi-isomorphism pg ˝ fq TX Ñ f LY rns depicted as a äX äX dashed arrow. By the same reasoning, since the upper outer square is homotopy bi-Cartesian, it maps to the lower outer square who is also homotopy bi-Cartesian. Moreover, the homotopy ˚ fiber of the map LL rn ´ 1sÑ f LY rns is exactly LL rn ´ 1s. This proves that there äY äX äX is a canonical quasi-isomorphism TL Ñ LL rn ´ 1s. äX äX

Theorem 3.4. Let Y be a n-shifted symplectic derived Artin stack. Let fi : Li Ñ Y be Lagrangian morphisms (for i “ 1 ¨ ¨ ¨ 2) and π : Y Ñ X a Lagrangian fibration. Suppose that the maps π ˝ fi : Li Ñ X are weak equivalences. Then P : Z “ L1 ˆY L2 Ñ X is a Lagrangian fibration.

Proof. We summarize the notation in the following diagram:

17 p1 Z L1 F p2 f1

f2 L2 Y π

X We also denote P :“ π ˝ F : Z Ñ X.

2,cl Z We first show that the pn ´ 1q-symplectic form induced in A äX, n ´ 1 is ho- ˚ ´ ¯ motopic to zero. We have that fi ω{X is homotopic to zero in two ways. Either by pulling back the homotopy between ω{X and 0 (homotopy coming form the Lagrangian fibration ˚ Y Ñ X) or by sending the homotopies between 0 and fi ω (coming from the Lagrangian ˚ ˚ structure on Li Ñ Y ) to homotopies between 0 and pfi ωq{X „ fi ω{X .

Therefore, the loop around 0 that defines the pn ´ 1q shifted symplectic form on Z is homotopic to the constant loop at 0 in the space of closed 2-forms relative to X:

˚ ˚ ˚ ˚ 0 p1 f1 ω{X p1 f1 ω{X 0

0 0

Thus the pn ´ 1q-symplectic form is homotopic to 0 relative to X:

Rpω,γ,δq{X „ 0

We now need to show the non degeneracy condition of the isotropic fibration.Consider the following commutative diagram:

T ˚T ˚T ˚T ˚T Z p1 L1 ‘ p2 L2 F Y P X

» » » »

T ˚T ˚T ˚T Z p1 L1 ‘ p2 L2 F Y 0 äX äX äX äX

» » » »

L ˚L ˚L ˚L ˚L Z rn ´ 1s p1 L1ä ‘ p2 L2ä rn ´ 1s F Y rns F Yä rns ´ Y Y ¯ X

˚L ˚L ˚L ˚L P X rn ´ 1s p1 L1ä ‘ p2 L2ä rn ´ 1s P X rns 0 ´ X X ¯ (7) We start considering the upper face:

18 T ˚T ˚T ˚T ˚T Z p1 L1 ‘ p2 L2 F Y P X

T ˚T ˚T ˚T Z p1 L1 ‘ p2 L2 F Y 0 äX äX äX äX

All squares are bi-Cartesians and the three left most horizontal arrows of the top row form a fiber sequence. Therefore the three left most horizontal arrows of the lower row also form a fiber sequence.

In the full diagram, the vertical arrows are all quasi-isomorphisms, either using the non-degeneracy conditions or Proposition 3.3 applied to the sequences Li Ñ Y Ñ X. We find the dashed arrows by completing the homotopy bi-Cartesian squares.

This implies that the bottom face satisfies the same properties as the upper face. Moreover, LL » 0 because we assumed that the maps Li Ñ X are weak equivalences. iäX ˚ ˚ Therefore the bottom face contains the fiber sequence P LX rn ´ 1sÑ 0 Ñ P LX rns. We ˚ obtain a weak equivalence α : TZ Ñ P LX rn ´ 1s. äX We still need to show that α is the morphism used in the criteria for the non-degeneracy of the Lagrangian fibration. Recall that this morphism is given by means of the Diagram (4):

αP ˚ TZ P LX rn ´ 1s 0 äX

„ TZ LZ rn ´ 1s LZ rn ´ 1s äX

We want to prove that αP and α are homotopic. The relevant data extracted from the Diagram (7) is:

α ˚ TZ P LX rn ´ 1s 0 äX

„ TZ LZ rn ´ 1s LL rn ´ 1s‘ LL rn ´ 1s 1äY 2äY But since all the squares in the lower face of Diagram (7) are bi-Cartesians, we have that LL rn´1s‘LL rn´1s is naturally quasi-isomorphic to LZ rn´1s. This proves, 1äY 2äY äX by universal property of the pull-back, that α is homotopy equivalent to αP .

3.3 Derived Critical Locus A1 Given a derived Artin stack X and a morphism f : X Ñ k, we define the derived critical locus of f, denoted Critpfq, as the derived intersection of df : X Ñ T ˚X with the zero section 0 : X Ñ T ˚X. It is given by the homotopy pull-back:

19 Critpfq X

df (8) X0 T ˚X . Example 3.5. We recall from [4] that if X is a smooth algebraic variety, its derived critical locus can be described, as a derived scheme, by the underlying scheme given by the ordinary critical locus of f, that we denote S, together with the sheaf of cdgaď0 given L by the derived tensor product OX b OX , restricted to S. This derived tensor product OT ˚X is described by the homotopy push-out:

Sym T 0 O OX p X q X

df

L OX OX b OX OT ˚X Taking the derived tensor product amounts to replacing the 0-section morphism 0 : T O T ã T T SymOX X Ñ X by the equivalent cofibration SymOX X Ñ SymOX p X r1s‘ X q, where Sym T 1 T Id : T 1 T OX p X r s‘ X q has the differential induced by X r sÑ X . Then we take the strict push-out of this replacement. The use of these resolutions are well explained in [4] or [20]. We obtain:

L T OCritpfq :“ OX bO ˚ OX » SymO X r1s, ιdf T X |S X |S ´ ¯ ` ˘ where ιdf is the the differential on OCritpfq given by the contraction along df. The restriction to S denotes the fact that this is a derived scheme whose underlying scheme T is the strict critical locus. Observe that outside of the critical locus, SymOX X r1s, ιdf is cohomologically equivalent to 0. ` ˘

Remark 3.6. If we do not assume that X is smooth in Example 3.5, then LX usu- ally has a non trivial internal differential. As a sheaf of graded algebra, we still obtain T SymOX p X r1sq since the replacement is the same as a graded algebra but the differential is a priori be different and involve a combination of the internal differential on TX and the contraction ιdf . Remark 3.7. From Example 2.10, we know that T ˚X carries a canonical symplectic form of degree 0 and from Example 2.16 we know that both the 0 section and df have a Lagrangian structure. From Proposition 3.1, the derived intersection of these Lagrangian structures, namely the derived critical locus Crit(f), has a p´1q-shifted symplectic struc- ture.

Remark 3.8. When X is a derived Artin stack and df “ 0, we have that Critpfq » ˚ ˚ T r´1sX and ωCritpfq is the canonical p´1q-shifted symplectic structure on T r´1sX. In this situation, the strict critical locus is X itself, and the restriction to X of T T SymOX X r1s is therefore SymOX X r1s itself (with the differential being zero since df “ 0). Crit Spec Sym T 1 ˚ 1 Thus pfq» X OX X r s “ T r´ sX. ` ˘ 20 We want to understand the p´1q-shifted symplectic form on Critpfq. We use the universal property of the tautological 1-form (Lemma 2.11) to see that pdfq˚ω “ 0. Us- ing the resolution of the zero section, as in Example 3.5, ω induces a closed 2-form on RSpec Sym T 1 T Sym T 1 T X OX p X r s‘ X q . Since the differential on the resolution, OX p X r s‘ X q, ˚ is induced` by Id : TX Ñ TX r1s,˘ the tautological 1-form ω´1 on T r´1sX induces a closed RSpec Sym T 1 T 2-form on X OX p X r s‘ X q which is a homotopy between ω and 0. We then have that the p´` 1q-shifted symplectic˘ form is described by the following loop around 0:

ω´ 0 1 p˚ω “ 0 0 0

The proof of Proposition 3.1 tells us that the ω´1 is the (-1)-shifted symplectic form on Critpfq.

Remark 3.9. From Theorem 3.4, we have that π : CritpfqÑ X is a Lagrangian fibration. In the situation where df “ 0 and X is smooth, this Lagrangian fibration coincides with ˚ the canonical Lagrangian fibration on πX : T r´1sX Ñ X. In general, the morphism απ controlling the non-degeneracy condition of the Lagrangian fibration (see Diagram (4)) is still natural in the sens given by the following proposition.

Proposition 3.10. απ is equivalent to the following composition of equivalences:

0ˆβ 0 ˚ ˚ TCrit TX ˆT ˚ TX » 0 ˆT ˚ 0 0 ˆ L 0 » π L r´1s pfqä ä T X ä T X πX X X X X äX X äX

L ˚ ˚ L where β is the dual of the canonical equivalence T X » πX X of Proposition 2.24. äX Proof. The strategy here is to express the Diagram (4) as a pull-back of the same type of diagrams. It reduces the problem to proving the same statement but for the projection ˚ πX : T X Ñ X. But this Proposition is known for the Lagrangian fibration on the shifted cotangent stacks (this is a direct consequence of Proposition 2.28).

First we express LCritpfqr´1s as a pull-back above LT ˚X . This can be done by observing that all squares in the following diagram are bi-Cartesians:

LCritpfqr´1s LT ˚X 0 äX

LT ˚X LT ˚X LX äX

0 LX LCritpfq We write Diagram (4) for π : CritpfqÑ X as:

21 απ »0ˆα 0 πX ˚ 0 ˆT ˚ 0 0 ˆ L 0 0 T X πX X äX

IdˆprId ˚ ˚ ˚ ˚ T ˆT ˚ T L ˆL ˚ L L ˆL ˚ L X T X X T Xä T X T Xä T Xä T X T Xä äX X X X äX X

˚ ˚ We need to describe the morphism ω : T ˆT ˚ T Ñ L ˆL ˚ L . Critpfq X T X X T Xä T X T Xä äX X X Recall from Remark 2.15 and the proof of the non-degeneracy in Proposition 3.1 that ωCritpfq is the natural map completing Diagram (6). This map is therefore Θdf ˆω Θ0. T L L L ˚ Here Θh : X Ñ hr´1s» Xä ˚ r´1s» T X . T X äX

Finally, Proposition 2.28 shows that β is the same as απX . This completes the proof.

4 Examples

4.1 One Non-Degenerate Critical Point A1 Let X be a smooth algebraic variety over k and f : X Ñ k a map which is smooth everywhere except at a point x P X where there is a non degenerate critical point. The goal is to understand the Lagrangian fibration on CritpfqÑ X and show that it is related to the Hessian quadratic form of f at x. This section is a particular case of Section 4.2, and we only sketch what is happening in this case. We will be making the statements more precise and give complete proofs in Section 4.2. O The strict critical locus is ‹ :“ ‹, X äI where I is the ideal generated by the partial ´ ¯ derivatives of f, I “xdf.v, v P TX y. There is a natural morphism x˜ :“‹Ñ Critpfq such that the following diagram commutes:

x˜ ‹p´1q Critpfq

πX x X The ideal generated by the partial derivatives is maximal and the partial derivatives form a regular sequence. This implies that x˜ is an equivalence. For more details, this ˚ ˚ is the analogue of Proposition ??, where we prove that T r´1sS “ T r´1s‹ “ ‹p´1q is weakly equivalent to Critpfq.

Using Lemma 2.23, the Lagrangian fibration induced on ‹p´1q Ñ X is weakly equiv- 2,cl ‹ T alent to a closed 2-form in A äX, ´2 which induces a metric on xX. The non- degeneracy of the symmetric bilinear` form is˘ equivalent to the non-degeneracy of the La- ˚ ˚ grangian fibration, which says that the natural map x TX Ñ x LX is a quasi-isomorphism. We will show that this metric is in fact characterised by the Hessian quadratic form of f

22 at the critical point.

We want to describe the Lagrangian fibration obtained on ‹Ñ X by pulling back ω 2,cl Critpfq along x˜ the homotopy between ´1äX and 0 in A äX, ´1 . We obtain a 2,cl ‹ ´ ¯ homotopy between 0 and itself in A äX, ´1 . We will relate the Hessian quadratic form with the map αx defined to describe` the non-degeneracy˘ condition of Lagrangian fibrations (see Definition 2.21 and Diagram (4)). For Critpfq and ‹ this diagram becomes respectively

απ ˚ TCritpfq π LX r´1s 0 äX

5 ω´1 TCritpfq LCritpfqr´1s LCritpfq r´1s äX and

αx ˚ T‹ L äX x X r´1s 0

L‹ 0 0 äX r´1s. These two diagrams are supposed to represent the same Lagrangian fibration. We will pull-back along x˜ the diagram for Critpfq to the category of differential graded k-vector space (i.e. QCp‹q). We can compare αx and απX via the following commutative diagram:

α ˚ πX ˚ x˜ TCritpfq x LX r´1s 0 äX „

αx ˚ T‹ L‹ L äX äX » x X r´1s 0

5 ω´1 TCritpfq LCritpfqr´1s LCritpfq r´1s äX „ ψ „

L‹ L‹ L‹ 0 äX ‘ äX r´1s äX r´1s We can now look at these morphisms in local étale coordinates around x. We denote i ˚ i ˚ by X coordinates in X, pi a basis of x TX and ξ its associated shifted basis in x TX r1s. i We also denote by dX the dual basis of pi. We write kxay :“ kxa1, ¨ ¨ ¨ , any for the k- vector space with basis a1, ¨ ¨ ¨ , an. We get:

23 απX kxBξy kxdXy 0 Id

αx kxBξy kxdθy 0

5 ω´1 kxBξ, BX y kxdX,dξy kxdξy

„ ψ „ 0 kxdθ,dξy kxdξy Here, dθ is the standard shifted variable variable added to make the following pull-back square a strict pull-back:

kxdθy 0

kxdθ,dξy kxdξy This imposes δdξ “ dθ. To make the all diagram strictly commutative, we must have ψpdξq“ dξ. And to make ψ a map of chain complexes, we must have ψpdδξiq“ δψpdξiq“ i i i ´1 i j j δdξ “ dθ and therefore it imposes ψpdX q“ Hessx pfqpdX qpdX qdθ . This implies that ´1 i j j αxpBξi q“ Hessx pfqpdX qpdX qdθ .

4.2 Family of non-Degenerate Critical Points We consider a generalisation of the previous example where f may have a family of critical points which are all non-degenerate in the directions normal to the critical locus.

Let us fix some notations. We denote by S the strict critical locus, which comes with OX a closed immersion i : S Ñ X and whose algebra of functions is OS “ äI with I “xdf.v, v P TX y.

We assume that both X and S are smooth algebraic varieties, which implies that OS is reduced. We denote by Critpfq the derived critical locus of f and we get a canonical morphism λ : S Ñ Critpfq.

In order to define the Hessian quadratic form and the non-degeneracy condition, we need to assume that the closed immersion S ãÑ X has a first order splitting. Concretely, we assume in this section that the following fiber sequence splits:

˚ TS i TX TS r1s (9) äX

This assumption is necessary to be able to restrict Q to the normal part TS r1s. äX Definition 4.1. The Hessian quadratic form is defined by the symmetric bilinear map:

Q : Sym2 i˚T O OS X Ñ S pw, vq ÞÑ dpdf.vq.w

24 We define non-degeneracy to be along the "normal" direction to S, by considering the following diagram:

˚ TS i TX TS r1s äX 0 Q (10) Q 0 ˚ r LS r´1s i LX LS äX

Both rows are split fiber sequences (by assumption in Diagram (9)). The map TSr1sÑ LSr´1s is the zero map because Q restricted to TS is zero and since Q is symmetric, Q projected to LS is also zero. We obtain a map Q which corresponds to the map induced by Q on the normal bundle. Then the non-degeneracy condition is the requirement r that Q is a quasi-isomorphism.

Sincer the differential on OCritpfq is δ “ ιdf , we have the commutative diagram:

δ TX r1s OX

Id´1 d (11) Q TX LX We will abusively write Q “ d ˝ δ.

In general, the natural map λ : S Ñ Critpfq is not an equivalence. This is due to the fact that the partial derivatives of f will not in general form a regular sequence and therefore Critpfq has higher homology. The default to be a regular sequence comes from vector fields that annihilate df. Such vector fields are in fact vector fields on S when f is non-degenerate. With that idea in mind, we show that an equivalent description of Critpfq is given by T ˚r´1sS when Q is non-degenerate.

Proposition 4.2. There exists a natural map Φ: T ˚r´1sS Ñ Critpfq making the fol- lowing diagram commute:

T ˚r´1sS Φ Critpfq

πS π Si X

Proof. Under our first order splitting assumption (Diagram (9)), the natural map TS Ñ ˚ ˚ ˚ ˚ i TX admits a retract, and therefore the natural map i T X Ñ T S admits a section: T ˚S 99K i˚T ˚X. We consider the following diagram:

T ˚X i˚T ˚X T ˚S

0 0 0 X S S i We want to pull-back these zero sections along the maps induced by df represented by the vertical morphisms in the following commutative diagram:

25 T ˚X i˚T ˚X T ˚S

df i˚df“0 0 X S S i This induces the following morphisms between the pull-backs:

˚ Critpfq S ˆi˚T ˚X S T r´1sS We obtain a map Φ: T ˚r´1sS Ñ Critpfq. The maps we obtain come from the ˚ universal properties of the pull-backs therefore if we denote s0 : X Ñ T X the zero ˚ section, we have s0 ˝ πX ˝ Φ “ s0 ˝ i ˝ πS . If we compose by the projection πX : T X Ñ X, we get πX ˝ Φ “ i ˝ πS . Φ gives a relationship between the Lagrangian fibration structures on T ˚r´1sS Ñ S and CritpfqÑ X which we now analyse. The idea is to show that the difference between these Lagrangian fibrations is in fact controlled by Q (see Proposition 4.6 and Remark 4.8). r ˚ Lemma 4.3. Φ induces a morphism TT ˚r´1sS Ñ Φ TCritpfq that fits in the commu- äS äX tative diagram

˚ TT ˚r´1sS Φ TCritpfq äS äX (12) απS απX ˚L ˚ ˚ L ˚ ˚L πS Sr´1s Φ πX X r´1s» πSi X r´1s where the bottom horizontal arrow is the pull-back along πS of the section LSr´1s Ñ ˚ i LX r´1s in the dual of the split fiber sequence (9).

h Proof. The homotopy pull-back, Critpfq“ X ˆT ˚X X lives over X. We get the equiva- lences:

T » T h T » h » ˚ L Critpfq Xä ˆT ˚ Xä ‹ˆT ˚ ‹ πX X r´1s äX X T XäX X T XäX

Proposition 2.28, gives us the following commutative square:

TT ˚S TT ˚X äS äX

βS βX π˚s ˚L S ˚ ˚L πS S πSi X where s is the section in the dual of the split fiber sequence (9). From Proposition 3.10, we know that both απS and απ are the morphism induced by the morphisms βS and βX

26 in the previous diagram when taking the pull-back. We obtain the commutative diagram:

0ˆh 0 » βS T ˚ h ˚L T r´1sS 0 ˆT ˚ 0 π Sr´1s ä T S S S äS

0ˆh 0 ˚T » ˚ h βX ˚ ˚ L Φ Critpfq Φ 0 ˆT ˚ 0 Φ πX X r´1s äX T X ˆ äX ˙

where the composition of the horizontal maps are exactly απS and απX thanks to Proposition 3.10.

˚ Lemma 4.4. We first remark that Φ LCritpfq can be described, as a sheaf of graded com- plex, by

˚L T ˚L ˚T Φ Critpfq » SymOS p Sr1sqbOS pi X ‘ i X r1sq

where LX is generated by terms of the form dg with g P OX and TX r1s is generated ˚ by terms of the form dξ with ξ P TX r1s. Then, the internal differential on Φ LCritpfq is characterised by Q “ d ˝ ιdf via δpdξq“ Qpξq and δpdgq“ 0.

˚ ˚ T L T O ˚ Proof. The differential on SymOS p Sr1sqbOS pi X ‘ i X r1sq is T r´1sS -linear because T 1 T 1 O Sym T 1 ιdf is zero on S r s. Moreover, for ξ P X r sĂ Critpfq “ OX X r s, we have δ ˝dpξq“ d ˝ δpξq“ d ˝ ιdf pξq“ Qpξq (see Diagram (11)), and for g P OX , δ ˝ dpgq“ d ˝ δg “ 0. Lemma 4.5. The composition

α ˚ ˚T ˚T πX ˚ ˚ L πSi X r´1s Φ Critpfq Φ πX X r´1s äX

˚ is given by πSQ. Similarly, the composition

απS ˚T T ˚ ˚L πS Sr´1s T r´1sS πS Sr´1s äS

˚ is 0 (the restriction of πSQ to S). Proof. It is enough to proves this locally around any point x of S. The left morphism is the morphism fitting in the fiber sequence:

˚ ˚T ˚T ˚T πSi X r´1s Φ Critpfq Φ Critpfq äX Which gives us locally:

˚ ˚T ˚T ˚T πSi X,ipxqr´1s Φ Critpfq Φ Critpfq,x äX,x

απX » ˚ ˚T ˚ ˚ L ˚ ˚ L ˚ ˚ T πSi X,ipxqr´1s Φ πX X,ipxqr´1s Φ πX X,ipxqr´1s‘ φ πX X,ipxq

˚ The second row can be seen as the extension (by πS) of the fiber sequence:

27 ˚ ˚ ˚ ˚ i TX,ipxqr´1s i LX,ipxqr´1s i LX,ipxqr´1s‘ i TX,ipxq

˚ ˚ Since X and S are smooth, i TX,ipxqr´1s and i LX,ipxqr´1s are both quasi-isomorphic to complexes concentrated in a single degree. This imposes that the dashed arrow is equivalent to the connecting morphism of the induced long exact sequence in cohomology. ˚ Therefore, it is equivalent to the map that sends an element s of i TX,ipxqr´1s to its ˚ ˚ ˚ differential, in i LX,ipxqr´1s‘i TX,ipxq, which can in turn be seen as an element in i LX,ipxq. ˚ ˚ More concretely, denote s˜ any lift of s to an element in i LX,ipxqr´2s‘i TX,ipxqr´1s. Using Lemma 4.4, its differential is given by

˚ ˚ ˚ Qpsq“ Qps˜q P i LX,ipxqr´1sĂ i LX,ipxqr´1s‘ i TX,ipxq. ˚ We then apply πS to get the sequence we want. The second part of the statement is proven the same way. ˚ Proposition 4.6. The map TT ˚r´1sS Ñ Φ TCritpfq induced by Φ is an equivalence if and only if Q is non-degenerate.

˚T ˚ ˚L ˚T ˚ Proof. First, using the equivalences απ : Φ Critpfq Ñ πSi X r´1s and απS : Φ T r´1sS Ñ äX äS ˚L T ˚ ˚T πS Sr´1s, we can show that the cofiber of T r´1sS Ñ Φ Critpfq is equivalent to äS äX ˚ π LS r´2s. Then Lemma 4.3 and 4.5 ensure that the upper half of the following dia- S äX gram is commutative:

˚ ˚ ˚ ˚ π TSr´1s π i TX r´1s π TS S S S äX

Q r T ˚ ˚T ˚L (13) T r´1sS Φ Critpfq πS Sä r´2s äS äX X

˚ TT ˚r´1sS Φ TCritpfq F This diagram is then commutative and all rows and columns are cofiber sequences and ˚ in particular F is both the homotopy cofiber of TT ˚r´1sS Ñ Φ TCritpfq and the homotopy cofiber of Q. In particular, the homotopy cofiber of Q is zero if and only the homotopy ˚ cofiber of T ˚ Ñ Φ T is also zero. rT r´1sS Critpfq r We now decompose απ into a part along S and a part normal to S. This decomposition is by means of split fibered sequences coming from the split fiber sequence (9).

Proposition 4.7. When Q is non-degenerate, the maps expressing the non-degeneracy of the Lagrangian fibrations fit in the commutative diagram:

TT ˚r´1sS TCritpfq TS äS äX äX

απS απX Q

˚ ˚ ˚ r π LSr´1s π i LX r´1s LS r´1s S S äX where the rows are fiber sequences.

28 Proof. First, when Q is non-degenerate, the top horizontal sequence is fibered and comes from the following diagram:

T ˚ ˚T ˚T T r´1sS Φ Critpfq πS Sä äS äX X

˚ TT ˚r´1sS Φ TCritpfq 0

˚ ˚ ˚ ˚ π TS Φ i TX π TS r1s S S äX

where all rows and columns are fibered and the cofiber of the second row is 0 thanks to Proposition 4.6 since we assumed that Q is non-degenerate. Using Lemma 4.3 and Lemma 4.5, we obtain the following commutative diagram:

˚ ˚ ˚ ˚ π TSr´1s Φ i TX r´1s π TS S S äX 0 Q Q r

T ˚ ˚T ˚T T r´1sS Φ Critpfq πS Sä (14) äS äX X

απS απX

˚ ˚ ˚ ˚ π LSr´1s Φ i LX r´1s π LS r´2s S S äX The only map the dashed arrow can be, in order to make the diagram commutative, is Q.

Remarkr 4.8. If we do not assume Q non-degenerate, the cofiber F of the map TT ˚r´1sS Ñ ˚ Φ TCritpfq will be non zero. We will denote by G the fiber of the natural map F Ñ TS . äX Then we can rewrite Diagram (14) as

˚ ˚ ˚ ˚ π TSr´1s Φ i TX r´1s π TS S S äX 0 Q Q r ˚ TT ˚r´1sS Φ TCritpfq G äS äX

αN απS απX

˚ ˚ ˚ ˚ π LSr´1s Φ i LX r´1s π LS r´2s S S äX

The map αN : G Ñ LS r´2s represent the "difference" between the maps απ and äX

απS from the Lagrangian fibrations. αN is still related to Q in the sens that the following diagram is commutative: r

29 TS äX Q r αN G LS r´2s äX

Therefore the restriction of αN to TS is again Q. äX A1r A1 X3 Remark 4.9. As a non-example if we take f : Ñ sending X to 3 , the basic assumptions that made this section work are failing. The strict critical locus S is not smooth since it is a fat point, and the sequence (9) does not split.

4.3 Derived Zero Locus of Shifted 1-Forms Let X be a derived Artin stack and α P A1 pX, nq be a 1-form. If Keypαq is non-empty, Proposition 2.17 and Remark 2.18 ensure that the map α : X Ñ T ˚rnsX is a Lagrangian morphism. Using Theorem 3.4 the derived intersection Zpαq of α with the zero section gives us a Lagrangian fibration ZpαqÑ X. This example is a generalisation of the derived critical locus we described in 3.3.

4.4 G-Equivariant Twisted Cotangent Bundles For X a smooth scheme, a twisted cotangent stack is a twist of the ordinary cotangent 1 1 stack by a closed 2-form of degree 1 on X, α P H pX, ΩX q. Such a closed form has an underlying 1-form of degree 1 that corresponds to a morphism α : X Ñ T ˚r1sX. The twisted cotangent bundle associated to α is defined to be the following pull-back:

˚ Tα X X α

X0 T ˚r1sX We refer to [9] for more informations on the relation between this definition and the usual definition of twisted cotangent bundles. This is a particular case of the situation ˚ ˚ in Section 4.3 and as such, Tα X is 0-shifted symplectic and the map Tα X Ñ X has a Lagrangian fibration structure.

Now take G an algebraic group acting on the algebraic variety X. Consider a character 1,cl χ : G Ñ Gm. We have the logarithmic form on Gm given by a map Gm Ñ A p0q which sends z to z´1dz. We get a closed 1-form on G described by the composition:

1,cl G Ñ Gm Ñ A p0q This is also a group morphism, we can therefore pass to classifying spaces and obtain a 1-shifted closed 1-form on BG:

1,cl 1,cl αχ : BG Ñ BA p0q“ A p1q

30 We can consider the pull-back of α along the G-equivariant moment map:

T ˚X ä ˆ g˚ BG BG G äG ” ı ” ı α

˚ ˚ T X µ g ˚ äG äG » T r1sBG ” ı ” ı It turns out that the moment map µ is Lagrangian (see [6]), which implies (with Proposition 3.1) that this fiber product is 0-shifted symplectic. It turns out that we have an equivalence of shifted symplectic derived Artin stacks:

˚ T X ˚ X ä ˆ g˚ BG » Tα ä G äG G ” ı ” ı ” ı p X Where α denotes the pull-back of α to a 1-form of degree 1 on äG . Therefore, according to Theorem 3.4, the natural projection ” ı p

˚ X X Tα äG äG ” ı ” ı p is a Lagrangian fibration.

To show the equivalence above, we use the following composition of Lagrangian cor- respondences (see 2.20):

˚ X Tα äG ” ı p

T ˚X X äG äG ” ı ” ı

X g˚ ˆ X äG äG BG ” ı ” ı α

˚ ˚ X g ‹ T r1s äG äG ‹ ” ı ” ı The only thing we need to show is that this is a diagram of Lagrangian correspondences and therefore we need to show that all squares in this diagrams are pull-backs. The right most square is clearly a pull-back and we can recognise the pull-back square defining ˚ X Tα äG . ” ı X g˚ ˆ X p We are left to prove that ä ˆ ˚ ä is naturally equivalent to G T r1srXä s G ” ı G ” ı T ˚X äG . This follows from the sequence of natural equivalences ” ı

31 X g˚ ˆ X ä ˚ ä G ˆT r1srXä s G ” ı G ” ı X g˚ X » ä ˆ ˚ ä ˆ ä G T r1srXä s G r‹ä s G ” ı G ” ı G ” ı ˚ ˚ X X g ˚ X g » ä ˆ ˚ ä ˆ ä » T ä ˆ ä G T r1srXä s G r‹ä s G G r‹ä s G ” ı G ” ı G ” ı ” ı G ” ı ˚ ˚ ˚ ˚ ‹ ˚ T X g T X ˚ ‹ g » ä ˆ g ä ˆ ‹ä ä » ä ˆ g ä ˆ ‹ä ä G äG G r Gs G G äG G r Gs G “ ‰ ” ı ” ı ” ı ” ı ” ı “ ‰ ” ı T ˚X » äG ” ı where we use the fact that the following square is a pull-back:

˚ X T äG BG ” ı 0

T ˚X µ g˚ äG äG ” ı ” ı References

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