A Derived Lagrangian Fibration on the Derived Critical Locus
aa r X i v : . [ m a t h . S G ] D ec A Derived Lagrangian Fibration on the DerivedCritical Locus
Albin Grataloup ∗ December 11, 2020
Abstract
We study the symplectic geometry of derived intersections of Lagrangian mor-phisms. In particular, we show that for a functional f : X Ñ A k , the derived criticallocus has a natural Lagrangian fibration Crit p f q Ñ X . In the case where f isnon-degenerate and the strict critical locus is smooth, we show that the Lagrangianfibration on the derived critical locus is determined by the Hessian quadratic form. Contents p n ´ q -Shifted Symplectic . . . . . . . . . . . 163.2 Lagrangian Fibrations and Derived Intersections . . . . . . . . . . . . . . . 173.3 Derived Critical Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 G -Equivariant Twisted Cotangent Bundles . . . . . . . . . . . . . . . . . . 30 ∗ IMAG, Univ. Montpellier, CNRS, Montpellier, [email protected] Introduction
In the context of derived algebraic geometry ([11], [12], [17], [18], [19]), the notion ofshifted symplectic structures was developed in [13] (see also [6] and [7]). This has provento 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 thequotient stack as explained in [6].• symmetric obstruction theory as defined in [1] from p´ q -shifted symplectic derivedstacks (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´ q -shifted symplectic derivedscheme 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 ofderived 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, aswell as remembering about the symmetries of the functional.• G -equivariant intersections. This includes the example of symplectic reductionwhich can be expressed as the derived intersection of derived quotient stacks (seeSection 2.1.2 in [5]).In this paper, we make a more precise study of the shifted symplectic geometry ofderived 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 wheneverthe Lagrangian morphisms f i : X i Ñ Z , i “ .. look like "sections" in the sens thatthere exists a map r : Z Ñ X such that the composition maps r ˝ f i : X i Ñ X are weakequivalences, then the natural morphism X ˆ Z X Ñ X is a Lagrangian fibration (see[7]). We then specialise this result to various examples and show in particular that, forthe 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 theHessian quadratic form of the functional.This paper starts, in Subsection 2, by recalling the basic definitions and properties ofshifted symplectic structures, Lagrangian structures and Lagrangian fibrations. We alsorecall, in Section 2.4, basic properties of the relative cotangent complexes of linear stacks2hat proves useful when we try to understand in more details the structure of Lagrangianfibrations on derived critical loci.In Section 3 we start by recalling the fact that a derived intersection of Lagrangianstructures in a n -shifted symplectic derived Artin stacks is p n ´ q -shifted symplectic.Then, in Subsection 3.2, we state and prove the main theorem (Theorem 3.4) that roughlysays that if the Lagrangian morphisms look like sections (up to homotopy), then thenatural projection from the derived intersection has a structure of a Lagrangian fibration.We then recall basic elements on the derived critical loci of a functional f : X Ñ A k , andthen try to describe the Lagrangian fibration structure on the natural map Crit p f q Ñ 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 derivedcritical loci for non-degenerate functionals. We show that the non-degeneracy conditionof the Lagrangian fibration of the derived critical locus of a non-degenerate functional ona 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 foreverything he explained to me on the subject of derived algebraic geometry. I wouldalso 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 theEuropean Union’s Horizon 2020 research and innovation programme (Grant AgreementNo. 768679).
Notation: • Throughout this paper k denotes a field of characteristic .• cdga (resp. cdga ď ) denotes the -category of commutative differential gradedalgebra over k (resp. commutative differential graded algebra in non positive de-grees).• cdga gr denotes the -category of commutative monoids in the category of gradedcomplexes dg grk .• A ´ Mod denotes the -category of differential graded A -modules for A P cdga .• cdga ǫ ´ gr denotes the -category of graded mixed differential graded algebra. Wedenote the differential δ and the mixed differential ǫ or d “ d DR in the case of theDe Rham complex of a derived Artin stack X , denoted DR p X q . We refer to [8] forall the definitions of cdga ǫ ´ gr , cdga gr , dg grk and the De Rham complex (see also[13] but with a different grading convention).• All the -categories above are localisations of model categories (see [8] for detailson these model structures and associated -categories). All along, unless explicitlystated otherwise, all diagrams will be homotopy commutative, all functor will be -functors and all (co)limits will be -(co)limits.3 For X a derived Artin stack, QC p X q denote the -category of quasi-coherentsheaves on X .• In this paper derived Artin stack are defined as in [19]. In particular all our derivedArtin stacks are locally of finite presentation over Spec p k q .• We denote by L X the cotangent complex of a derived Artin stack X . We denote by T X : “ L _ X : “ Hom p L X , O X q its dual. 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 A p p‚ , n q and A p,cl p‚ , n q of respectively the space of n -shifted differential p -formsand the space of n -shifted closed differential p -forms. We use the grading conventions usedin [8]. On a derived affine scheme Spec (A), the space of p -forms of degree n and the spaceof closed p -forms of degree n are defined respectively by A p p A, n q : “ Map cdga gr p k r´ n ´ p sp´ p q , DR p A qq and A p,cl p A, n q : “ Map cdga ǫ ´ gr p k r´ n ´ p sp´ p q , DR p A qq . From [8], the de Rham complex of A , denoted DR p A q , can be described, as a gradedcomplex, by DR p A q » Sym A L A r´ sp´ q where p´q is the functor forgetting the mixedstructure cdga ǫ ´ gr Ñ cdga gr (we refer to [8] for more on the de Rham complex).All along, we denote the internal differential, i.e. the differential on L A , by δ and themixed 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-ping space Map dSt p X, A p p‚ , n qq and the space of closed p -forms of degree n on X is Map dSt ` X, A p,cl p‚ , n q ˘ . Now the following proposition says that in the case where X isa derived Artin stack, the spaces of shifted differential forms are spaces of sections ofquasi-coherent sheaves on X . Proposition 2.1 (Proposition 1.14 in [13]) . Let X be a derived Artin stack over k and L X be its cotangent complex over k. Then there is an equivalence A p p X, n q »
Map QC p X q p O X , Λ p L X r n sq . In particular, π p A p p X, n qq “ H n p X, Λ p L X q . emark 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 givenby a global section ω P DR p X q p p q r n ` p s » R Γ pp Ź p L X q r n sq such that δω “ . Aclosed p -form of degree n is given by a semi-infinite sequence ω “ ω ` ω ` ¨ ¨ ¨ with ω i P DR p p ` i q r n ` p s “ R Γ ´´Ź p ` i L X ¯ r n ´ i s ¯ such that δω “ and dω i “ δω i ` .Equivalently, being closed means that ω is closed for the total differential D “ δ ` d in the bi-complex DR p X q ě p r n s » R Γ ´ś i ě ´Ź p ` i L X ¯ r n s ¯ whose total degree is givenby n ` p ` i . Note that the conditions imposed on ω are equivalent to saying that ω is acocycle of degree n ` p for the total differential.In general, we can also describe the spaces of (closed) differential forms as A p p X, n q » ˇˇ DR p p q p X qr n s ˇˇ and A p,cl p X, n q » ˇˇˇś i ě p DR p p ` i q p X qr n s ˇˇˇ , where ś i ě p DR p p ` i q p X qr n s isendowed with the total differential. Remark 2.3.
Given a map of derived Artin stack f : Y Ñ X , we define A p, p cl q p Y { X, n q ,the n -shifted (closed) p -forms on Y relative to X , to be the homotopy cofiber of the naturalmap A p, p cl q p X, n q Ñ A p, p cl q p Y, n q . For instance n -shifted relative p -forms are equivalent tothe derived global sections of ´Ź p L Y ä X ¯ r n s , with the relative cotangent complex L Y ä X defined as the homotopy cofiber of the natural map f ˚ L X Ñ L Y . We refer to [8] for moredetails on the relative n -shifted (closed) p -forms and the relative version of the De Rhamcomplex.For every closed p -form of degree n , ω , we have the underlying p -form of degree n given by ω obtained from the natural projection DR ě p Ñ DR p p q . It induces a morphism A p,cl p‚ , n q Ñ A p p‚ , n q that forgets the higher differential forms defining the closure of ω .We say that a p -form, ω , of degree n can be lifted to a closed p -form of degree n if thereexists a family of p p ` i q -forms ω i of degree n ´ i for all i ą , such that ω “ ω ` ω ` ¨ ¨ ¨ is closed in DR p X q ě p r n s (i.e. Dω “ ). In this situation, we can see that dω is a priorinot equal to 0 but is in fact homotopic to 0 ( dω “ D p´ ř i ą ω p ` i q ). The choice of sucha homotopy is the same as a choice of a closure of the p -form of degree n . Being closed istherefore no longer a property of the underlying p -form of degree n but a structure addedto it given by the higher forms. In other words, a closure of ω is given by a homotopybetween dω and zero. The collection of all closures of a p -form of degree n forms a space: Definition 2.4.
Let α P A p p X, n q then the space of all closures of α is called the spaceof keys of α denoted key p α q . It is given by the homotopy pull-back: key p α q A p,cl p X, n q‹ A p p X, n q α (1)The mixed differential of the de Rham graded mixed complex induces a map d : A p p X, n q Ñ A p ` p X, n q . Since dω is d -closed and δ ˝ dω “ d ˝ δω “ , we get D p dω q “ .5herefore the image of dω through the inclusion DR p p ` q r n s Ñ DR ě p ` r n s is a p p ` ` n q -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 : A p p X, n q Ñ A p ` ,cl p X, n q .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 symplecticstructures. Definition 2.5 (Non-Degenerate 2-Form of Degree n ) . For a derived Artin n -stack X ,the cotangent complex L X is dualisable. Therefore there is a tangent complex T X “ L _ X .We say that a (closed) 2-form of degree n is non-degenerate if the (underlying) 2-form ω of degree n induces a quasi-isomorphism: ω : T X Ñ L X r n s Definition 2.6 (Shifted Symplectic Forms) . A n -shifted symplectic structure is anon-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 stackassociated to L X r n s , T ˚ r n s X : “ A p L X r n sq (see Definition 2.7). It comes with a naturalmorphism π X : T ˚ r n s X Ñ X . We refer to [7] for a general account of shifted symplecticgeometry on the cotangent stack. Definition 2.7 (Linear Stacks) . Given F P QC p X q a quasi-coherent sheaf over a derivedArtin stack, we can construct a linear stack denoted A p F q and defined, as a derivedstack over X , by A p F q p f : Spec p A q Ñ X q : “ Map
A-Mod p A, f ˚ F q . Remark 2.8.
Whenever F is co-connective, A p F q is equivalent to the relative spec-trum Spec X ` Sym O X F _ ˘ . A map Spec p A q Ñ T ˚ X is equivalently given by a map χ : Spec p A q Ñ X together with a section s P Map A ´ Mod p A, χ ˚ L X q . Map A ´ Mod p A, χ ˚ F q » Map A ´ Alg p Sym A χ ˚ F _ , A q : “ Spec X p Sym A F _ q p χ : Spec p A q Ñ X q A morphism Y Ñ T ˚ r n s X is determined by the induced morphism f : Y Ñ X (bycomposition with π X ) and a section s : Y Ñ f ˚ T ˚ r n s X which corresponds to an element s P Map QC p Y q p O Y , f ˚ L X r n sq . In the case of a section s : X Ñ T ˚ r n s X , we get theidentity Id : X Ñ X and a section s P Map QC p X q p O X , L X r n sq » A p X, n q . This shows,using Proposition 2.1, that the space of sections of T ˚ r n s X is exactly the space of -formsof degree n as expected. Example 2.9.
Any ordinary smooth symplectic variety can be seen as a derived Artinwith a -shifted symplectic structure. Conversely, any 0-shifted symplectic structure onan Artin stack representable by a smooth variety is equivalent to a symplectic structurein the classical sense on that variety. 6 xample 2.10. As in the classical case, we can construct the canonical Liouville 1-form. Consider the identity
Id : T ˚ r n s X Ñ T ˚ r n s X . It is determined by the projection π : T ˚ r n s X Ñ X and a section λ X P Map QC p T ˚ r n s X q ` O T ˚ r n s X , π ˚ L X r n s ˘ . Since we have anatural map π ˚ L X r n s Ñ L T ˚ r n s X r n s , λ X induces a 1-form on T ˚ r n s X called the tautologi-cal -form. This -form induces a closed -form dλ X 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 satisfiesthe usual universal property. Lemma 2.11.
Given a 1-form α : X Ñ T ˚ r n s X , we have that α ˚ λ X “ α .Proof. In general, if we take f : X Ñ Y , the pull-back of a n -shifted -form, β , is describedby: T ˚ r n s X f ˚ T ˚ r n s Y T ˚ r n s YX Y p df q ˚ f ˚ β f β Taking into account the fact that λ factors through π ˚ X T ˚ r n s X , we consider the fol-lowing diagram: T ˚ r n s X α ˚ T ˚ T ˚ r n s X T ˚ T ˚ r n s XT ˚ r n s X “ α ˚ π ˚ X T ˚ r n s X π ˚ X T ˚ r n s XX T ˚ r n s X p dα q ˚ Id p dπ X q ˚ p dπ X q ˚ ˜ λ α λ This proves that the pull-back along α of λ X seen as a 1-form of degree n on T ˚ r n s X is the same as the pull-back along α of the section λ X : T ˚ r n s X Ñ π ˚ X T ˚ r n s X .We denote by α the associated section in Map QC p X q p O X , L X r n sq of degree n . Thereis a one-to-one correspondence between sections of π X : T ˚ r n s X Ñ X and points of Map QC p X q p O X , L X r n sq . Now we use the fact that Id ˝ α “ α :• On the one hand, α is completely described by α P Map QC p X q p O X , L X r n sq .• On the other hand, the map Id : T ˚ r n s X Ñ T ˚ r n s X is described by the projection π : T ˚ r n s X Ñ X and the section λ X P Map QC p T ˚ r n s X q ` O T ˚ r n s X , π ˚ X L X ˘ . There-fore the composition Id ˝ α is also a section of π X and is described by α ˚ λ X P Map QC p X q p O X , L X r n sq .This proves that α ˚ λ X “ α . Since these maps characterise the sections of π X theyrepresent, we have α ˚ λ X “ α . 7 .2 Lagrangian Structures We recall from [13] the definition and standard properties of Lagrangian structures. Wereproduce 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 stacksand suppose X has a n -shifted symplectic structure ω . An isotropic structure on f is a homotopy, in A ,cl p L, n q , between f ˚ ω and 0. Isotropic structures on f form a spacedescribed by the homotopy pull-back: Iso p f q ‹‹ A ,cl p L, n q f ˚ ω Remark 2.13.
More explicitly, an isotropic structure is given by a family of forms oftotal degree p p ` n ´ q , p γ i q i P N with γ i P DR p L q p p ` i q r p ` n ` i ´ s , such that δγ “ f ˚ ω and δγ i ` dγ i ´ “ f ˚ ω i . This can be rephrased as Dγ “ f ˚ ω , thus γ is indeed a homotopybetween f ˚ ω and 0. Definition 2.14 (Lagrangian Structures) . An isotropic structure γ on f : L Ñ X is calleda Lagrangian structure on f if the leading term, γ , viewed as an isotropic structureon the morphism T L Ñ f ˚ T X , is non-degenerate. We say that γ is non-degenerate ifthe following null-homotopic sequence (homotopic to 0 via γ ) is fibered: T L f ˚ T X » f ˚ L X r n s L L r n s p f ˚ ω q (2) Remark 2.15.
To say that that sequence is fibered can be reinterpreted as a moreclassical condition involving the conormal. The relative cotangent complex L f r n s , alsodenoted L L ä X r n s when f is clear from context, is the homotopy cofiber of the naturalmap f ˚ L X r n s Ñ L L r n s . Since QC p X q is a stable -categories, the homotopy fiber of f ˚ L X r n s Ñ L L r n s is L f r n ´ s and the non-degeneracy condition can be rephrased bysaying that the natural map T L Ñ L f r n ´ s is a quasi-isomorphism. Example 2.16. A -form of degree n on an Artin stack X is equivalent to a section α : X Ñ T ˚ r n s X . This section is a Lagrangian morphism if and only if α admits aclosure, i.e. Key p α q 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
Iso p α q Ñ Key p α q between thespace of isotropic structures on the 1-form α and the space of keys of α .Proof. key p α q A ,cl p X, n q ‹‹ A p X, n q A ,cl p X, n q α d dR (3)8he leftmost square is Cartesian by definition of key p α q in Definiion 2.4. By definition,the pull-back of the outer square is Iso p α q because d dR α “ α ˚ ω (by universal propertyof the Liouville 1-form, Lemma 2.11). It turns out that the rightmost square is alsoCartesian. This is simply saying that the space of closed 1-forms of degree n is the sameas the space of 1-forms of degree n whose de Rham differential is homotopic to 0. Weobtain that key p α q and Iso p α q are both pull-backs of the outer square and therefore arecanonically homotopy equivalent. Remark 2.18.
It turns out that Theorem 2.15 in [7] says that all the isotropic structureson α (or equivalently the lifts of α to a closed form) are in fact non-degenerate, whichimplies 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 mapis equivalent to an p n ´ q -shifted symplectic structure on X .Proof. Pick an isotropic structure γ on p . We know that γ is a homotopy between 0 and0 which means that Dγ “ . Therefore γ is a closed 2-form of degree n ´ . We want toshow that γ is non-degenerate as an isotropic structure if and only if it is non-degenerateas a closed 2-form on X . The non-degeneracy of the Lagrangian structure, as describedin Remark 2.15, corresponds to the requirement that the natural map T X Ñ L X r n ´ s is a quasi-isomorphism. This map depends on γ and we want to show that this map isin fact γ . This map is the natural map that fits in the following homotopy commutativediagram: T X L X r n ´ s
00 0 L X r n s We can show that by strictifying the homotopy commutative diagram: T X L X r n s p ˚ ω “ Note that this diagram is already commutative but we see it as homotopy commutativeusing the homotopy γ . We use the homotopy γ to strictify the previous diagram andwe obtain: T X L X r n ´ s ‘ L X r n s L X r n s γ ` p ˚ ω “ The homotopy fiber and also strict fiber of the projection pr : L X r n ´ s ‘ L X r n s Ñ L X r n s is L X r n ´ s , and therefore the natural map we obtain is γ : T X Ñ L X r n ´ s .Since the non-degeneracy condition of the isotropic structure γ is the same as sayingthat the map γ is a quasi-isomorphism, we have shown that an isotropic structure γ is an9 n ´ q -shifted symplectic structure on X if and only if it is non-degenerate as an isotropicstructure on X Ñ ‹ n . Definition 2.20 (Lagrangian Correspondence, [5]) . Let X and Y be derived Artin stackswith n -shifted symplectic structures. A Lagrangian correspondence from X to Y isgiven by a derived Artin stack L with morphims LX Y and a Lagrangian structure on the map L Ñ X ˆ s Y where X ˆ s Y is endowed with the n -shifted symplectic structure π ˚ X ω X ´ π ˚ Y ω Y . For example, a Lagrangian structure on 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 X , X and X derived Artin stacks with symplectic structures and L and L Lagrangian correspondences from respectively X to X and X to X . We can producea Lagrangian correspondence L from X to X by setting L : “ L ˆ X L . L L L X X X 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 symplecticstructure 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 thenatural map A ,cl p Y, n q Ñ A ,cl p Y { X, n q (see Remark 2.3).• A non-degeneracy condition which says that the following sequence (homotopic to0 via γ ) is fibered: T Y { X Ñ T Y » L Y r n s Ñ L Y { X r n s In particular, the non-degeneracy condition can be rephrased by saying that there is acanonical quasi-isomorphism α f : T Y { X Ñ f ˚ L X r n s (similar to the criteria for Lagrangianmorphism in Remark 2.15) that makes the following diagram commute:10 Y ä X f ˚ L X r n s T Y L Y r n s L Y ä X r n s α f ω (4) Example 2.22.
The natural projection π X : T ˚ r n s X Ñ X is a Lagrangian fibration. TheLiouville 1-form is a section of π ˚ X L X r n s which is part of the fiber sequence: π ˚ X L X r n s Ñ L T ˚ r n s X r n s Ñ L T ˚ r n s X ä X r n s Thus the 1-form induced by λ X in L T ˚ r n s X ä X r n s is homotopic to 0. The non-degeneracycondition 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, α π X , is given by a canonical construction(Proposition 2.24) which does not depend on the symplectic structure. This is the contentof Proposition 2.28. Lemma 2.23.
Let x : ‹ n Ñ X be a point of X . Then, given a Lagrangian fibrationstructure on x , the non-degeneracy condition is given by a quasi-isomorphism x ˚ T X Ñ x ˚ L X r n ` s .Proof. The Lagrangian fibration structure on ‹ n Ñ X is a homotopy between 0 anditself in A ,cl p‹ n ä X, n q . As in the proof of Lemma 2.19, this is given by an element γ P A ,cl p‹ n ä X, n ´ q . Similarly to what was done in the proof of Lemma 2.19, we canshow that γ is non-degenerate as a Lagrangian fibration if and only if it is non-degenerateas a closed 2-form of degree n . Again it boils down to the fact that the natural morphism inthe non-degeneracy criteria for Lagrangian fibrations is in fact γ : T ‹ n ä X Ñ L ‹ n ä X r n ´ s .Moreover, we have natural equivalences, T ‹ n ä X » x ˚ T X r´ s and L ‹ n ä X r n ´ s » x ˚ L X r n s because the sequence T ‹ n ä X T ‹ n » x ˚ T X r n s is fibered. This concludes the proof. This section is devoted to the study of relative cotangent complexes of linear stacks. Given F P QC p X q a dualisable quasi-coherent sheaf over a derived Artin stack X , we considerits associated linear stack, A p F q (see Definition 2.7) and the goal of this section is todescribe L A p F q ä X and its functoriality in F and X . Proposition 2.24.
Let X be a derived Artin stack and F P QC p X q a dualisable quasi-coherent sheaf on X . We denote π X : A p F q Ñ X the natural projection. Then we have: L π X » L A p F q ä X » π ˚ X F _ roof. We will show the result for any B -point y : Spec p B q Ñ A p F q and we write x “ π ˝ y : Spec p B q Ñ X . We will show that for all M P B ´ Mod connective, we have
Hom B ´ Mod ´ y ˚ L A p F q ä X , M ¯ » Hom B ´ Mod p x ˚ F _ , M q First we observe that
Hom B ´ Mod ´ y ˚ L A p F q ä X , M ¯ is equivalent, using the universalproperty of the cotangent complex, to the following homotopy fiber at y : hofiber y ´ Hom dSt ä X p Spec p B ‘ M q , A p F qq Ñ Hom dSt ä X p Spec p B q , A p F qq ¯ with B ‘ M denoting the square zero extension and Spec p B ‘ M q Ñ X being thecomposition: Spec p B ‘ M q Spec p B q X p x Thus a map in
Hom B ´ Mod ´ y ˚ L A p F q ä X , M ¯ is completely determined by a map Φ :
Spec p B ‘ M q Ñ A p F q making the following diagram commute: Spec p B q Spec p B ‘ M q A p F q Spec p B q X i y Φ p π X x Thus, we obtain that
Hom B ´ Mod ´ y ˚ L A p F q ä X , M ¯ is equivalent to hofiber s y p Map B ‘ M ´ Mod p B ‘ M, p ˚ x ˚ F q Ñ Map B ´ Mod p B, x ˚ F qq where s y P Map B ´ Mod p B, x ˚ F q is the section associated to y : Spec p B q Ñ A p F q .The map is then given by precomposition with i ˚ . We can now observe that p ˚ x ˚ F “ x ˚ F ‘ x ˚ F b B M and that Map B ‘ M ´ Mod p B ‘ M, p ˚ x ˚ F q » Map B ´ Mod p B, x ˚ F ‘ x ˚ F b B M q We obtain
Hom B ´ Mod ´ y ˚ L A p F q ä X , M ¯ » hofiber p Map B ´ Mod p B, x ˚ F ‘ x ˚ F b B M q Ñ Map B ´ Mod p B, x ˚ F qq» Map B ´ Mod p B, x ˚ F b B M q » Map B ´ Mod p x ˚ F _ , M q Now the result follows from the fact that the functor12 ´ Mod Fun ` B ´ Mod ď , sSet ˘ N Map B ´ Mod p N, ‚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 QC p Y q dualisable. Then there is a commutative square: Φ ˚ L A p F q ä X L A p f ˚ F q ä Y Φ ˚ π ˚ Y F _ π ˚ X f ˚ F _» »» with Φ the natural morphism in the following homotopy pull-back: A p f ˚ F q » f ˚ A p F q A p F q X Y Φ π X π Y f and the lower horizontal equivalence Φ ˚ π ˚ Y F _ Ñ π ˚ X f ˚ F _ being the equivalence comingfrom the fact that π Y ˝ Φ » f ˝ π X .Proof. The first things we observe is that A p f ˚ F q » f ˚ A p F q . We consider as before B -points: Spec p B q f ˚ A p F q A p F q X Y y ˜ yx ˜ x Φ π X π Y f We want to show that the following diagram is commutative:
Hom B ´ Mod ´ y ˚ L A p f ˚ F q ä X , M ¯ Hom B ´ Mod ´ ˜ y ˚ L A p F q ä Y , M ¯ Hom B ´ Mod p y ˚ π ˚ X f ˚ F _ , M q Hom B ´ Mod p ˜ y ˚ π ˚ Y F _ , M q » » (5)Using the universal property of the cotangent complex, the top horizontal arrow isnaturally equivalent to the map 13 ofiber y ´ Hom dSt ä X p Spec p B ‘ M q , A p f ˚ F qq Ñ Hom dSt ä X p Spec p B q , A p f ˚ F qq ¯ hofiber ˜ y ´ Hom dSt ä Y p Spec p B ‘ M q , A p F qq Ñ Hom dSt ä Y p Spec p B q , A p F qq ¯ induced by Hom dSt p´ , Φ q . A map ψ : Spec p B ‘ M q Ñ A p f ˚ F q in this homotopyfiber fits in the following commutative diagram: Spec p B q Spec p B ‘ M q A p f ˚ F q A p F q Spec p B q X Y y ˜ yip ψ π X Φ π Y x f and the map between the homotopy fiber sends ψ to Φ ˝ ψ . Since the underlyingmap of ψ is π X ˝ ψ : Spec p B ‘ M q Ñ X is x ˝ p and the underlying map of Φ ˝ ψ is π Y ˝ Φ ˝ ψ : Spec p B ‘ M q Ñ Y is f ˝ x ˝ p “ ˜ x ˝ p , this map between the homotopy fiberof derived stacks is therefore naturally equivalent to the map: hofiber s y p Map B ‘ M ´ Mod p B ‘ M, p ˚ x ˚ f ˚ F qq Ñ Hom B ´ Mod p B, p ˚ x ˚ f ˚ F qq hofiber s ˜ y p Map B ‘ M ´ Mod p B ‘ M, p ˚ ˜ x ˚ F qq Ñ Hom B ´ Mod p B, p ˚ ˜ x ˚ F qq where s y and s ˜ y are the sections associated to y and ˜ y respectively. This map is in factinduced by the natural identification p ˚ ˜ x ˚ F » p ˚ x ˚ f ˚ F (since ˜ x “ f ˝ x ). But followingthe steps of the proof of Proposition 2.24, this map is naturally equivalent to the map Hom B ´ Mod p y ˚ π ˚ X f ˚ F _ , M q Ñ Hom B ´ Mod p ˜ y ˚ π ˚ Y F _ , M q The natural equivalence we used are all the natural equivalences used in Proposition2.24 which proves that the Diagram (5) is commutative. Now the result follows once againfrom the fact that the functor B ´ Mod Fun ` B ´ Mod ď , sSet ˘ N Map B ´ Mod p N, ‚q is fully faithful and the fact that everything we did is natural in B .14 emma 2.26. Let X be a derived Artin stacks. We consider F , G P QC p X q dualisableand h : F Ñ G . Then there is a commutative square: ˆ h ˚ L A p G q ä X L A p F q ä X π ˚ X G _ π ˚ X F _» » π ˚ X h _ with ˆ h : A p G q Ñ A p F q 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 QC p X q and G P QC p Y q dualisable and a morphism h : f ˚ F Ñ G . Then there is acommutative square: L A p F q ä X ˆ f ˚ L A p G q ä Y π ˚ X F _ π ˚ X f ˚ G _ “ ˆ f ˚ π ˚ Y G _» » π ˚ X h _ Proof.
It follows from Lemma 2.25 and Lemma 2.26.
Proposition 2.28.
The quasi-isomorphism α π X : T T ˚ r n s X ä X Ñ π ˚ X L X r n s of Example2.22 expressing the non-degeneracy of the canonical Lagrangian fibration on the shiftedcotangent 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: π ˚ X L X r n s L T ˚ r n s X r n s L T ˚ r n s X ä X r n s Proposition 2.24 gives us canonical equivalences L T ˚ r n s X ä X r n s » π ˚ X T X . With thisdata, we can rewrite Diagram (4), up to weak equivalences, as the strictly commutativediagram T T ˚ r n s X ä X π ˚ X L X r n s π ˚ X L X r n s ‘ π ˚ X T X π ˚ X T X ‘ π ˚ X L X r n s π ˚ X T X » ω Through the canonical equivalence T T ˚ r n s X ä X Ñ π ˚ X L X r n s of Proposition 2.24, themorphism T T ˚ r n s X ä X Ñ π ˚ X L X r n s ‘ π ˚ X T X simply becomes the natural inclusion and ω becomes the identity. This implies that α π X : T T ˚ r n s X ä X Ñ π ˚ X L X r n s is the canonicalequivalence of Proposition 2.24. 15 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 derivedintersections of derived schemes. In Section 3.3 we study in more details the special caseof derived intersections given by derived critical loci. p n ´ q -Shifted Symplectic Proposition 3.1 ([13], Section 2.2) . Let Z be a derived Artin stack with a n -shiftedsymplectic structure ω . Let f : X Ñ Z and g : Y Ñ Z be morphisms with γ and δ Lagrangian structures on f and g respectively. Then the homotopy pull-back X ˆ hZ Y possesses a canonical p n ´ q -shited symplectic structure called the residue of ω and denoted R p ω, γ, δ q .Proof. We consider the maps p : X ˆ hZ Y Ñ X Ñ Z and q : X ˆ hZ Y Ñ Y Ñ Z . Thereis a homotopy h : p ñ q . It induces a homotopy h ˚ : p ˚ ñ q ˚ in the mapping space Map ` A ,cl p Z, n q , A ,cl p X ˆ hZ Y, n q ˘ .Moreover the pull-backs of the isotropic structures on f and g define paths γ : 0 ù p ˚ ω and δ : 0 ù q ˚ ω in A ,cl p X ˆ hZ Y, n q . Concatenating γ , u ˚ ω : p ˚ ω ù q ˚ ω and δ ´ we get a loop at zero in A ,cl p X ˆ hZ Y, n q which can be seen as an element in π p A ,cl p X ˆ hZ Y, n q , q » π p A ,cl p X ˆ hZ Y, n ´ qq .This gives us the closed 2-form of degree p n ´ q we denoted by R p ω, γ, δ q . From theequivalent point of view of the chain complex DR ě p X qr n s , this is just saying that ahomotopy between 0 and 0 in degree p ` n is just a cocycle in degree p p ` n ´ q , that is,a closed p -form of degree n ´ .We need to show that this closed 2-form is non-degenerate. Denote π : X ˆ hZ Y Ñ Z to be p (or equivalently q). Denote p X : X ˆ hZ Y Ñ X and p Y : X ˆ hZ Y Ñ Y the naturalmorphisms. Then we get a commutative diagram in QC p X ˆ hZ Y q with exact rows: T X ˆ hZ Y pr ˚ X T X ‘ pr ˚ Y T Y π ˚ T Z L X ˆ hZ Y r n ´ s pr ˚ X L f r n ´ s ‘ pr ˚ Y L g r n ´ s π ˚ L Z r n s R p ω,γ,δ q Θ γ ‘ Θ δ ω (6)The middle and right vertical arrows are quasi-isomorphisms because γ and δ areLagrangian structures and ω is a symplectic form and therefore, non-degenerate. Thisimplies that R p ω, γ, δ q is also non-degenerate. Remark 3.2.
Theorem 3.1 is in fact a consequence of the procedure of composition ofLagrangian fibrations. Consider the following composition of Lagrangian correspondences: X ˆ Z YX Y ‹ Z ‹ 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 ˆ Z Y Ñ ‹ ˆ ¯ ‹ is alsoa Lagrangian correspondence, thus X ˆ Z Y Ñ ‹ is Lagrangian. From Lemma 2.19, sincethe point is n -shifted symplectic, then X ˆ Z Y is p n ´ q -shifted symplectic. 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 morphismand g : Y Ñ X is a Lagrangian fibration. Then there is a canonical quasi-isomorphism T L ä X Ñ L L ä X r n ´ s .Proof. Consider the following commutative diagram: T L f ˚ T Y p g ˝ f q ˚ T X T L ä X f ˚ T Y ä X L L ä Y r n ´ s f ˚ L Y r n s f ˚ L Y ä X r n s L L ä X r n ´ s p g ˝ f q ˚ L X r n s » » »» » » In the upper face, all squares are bi-Cartesian because both the outer square and theright most square are bi-Cartesian. All non-dashed vertical arrows are quasi-isomorphismsby assumption (because of the various non-degeneracy conditions). Focusing on the righthand cube, it sends the upper homotopy bi-Cartesian square to the bottom square whichis also homotopy bi-Cartesian. The homotopy cofiber of p g ˝ f q ˚ L X r n s Ñ f ˚ L Y r n s is f ˚ L Y ä X r n s and we obtain a quasi-isomorphism p g ˝ f q ˚ T X Ñ f ˚ L Y ä X r n s depicted as adashed arrow.By the same reasoning, since the upper outer square is homotopy bi-Cartesian, it mapsto the lower outer square who is also homotopy bi-Cartesian. Moreover, the homotopyfiber of the map L L ä Y r n ´ s Ñ f ˚ L Y ä X r n s is exactly L L ä X r n ´ s . This proves that thereis a canonical quasi-isomorphism T L ä X Ñ L L ä X r n ´ s . Theorem 3.4.
Let Y be a n -shifted symplectic derived Artin stack. Let f i : L i Ñ Y beLagrangian morphisms (for i “ ¨ ¨ ¨ ) and π : Y Ñ X a Lagrangian fibration. Supposethat the maps π ˝ f i : L i Ñ X are weak equivalences. Then P : Z “ L ˆ Y L Ñ X is aLagrangian fibration.Proof. We summarize the notation in the following diagram:17 L L Y X p Fp f f π We also denote P : “ π ˝ F : Z Ñ X .We first show that the p n ´ q -symplectic form induced in A ,cl ´ Z ä X, n ´ ¯ is ho-motopic to zero. We have that f ˚ i ω { X is homotopic to zero in two ways. Either by pullingback the homotopy between ω { X and 0 (homotopy coming form the Lagrangian fibration Y Ñ X ) or by sending the homotopies between and f ˚ i ω (coming from the Lagrangianstructure on L i Ñ Y ) to homotopies between 0 and p f ˚ i ω q { X „ f ˚ i ω { X .Therefore, the loop around 0 that defines the p n ´ q shifted symplectic form on Z ishomotopic to the constant loop at 0 in the space of closed 2-forms relative to X : p ˚ f ˚ ω { X p ˚ f ˚ ω { X
00 0
Thus the p n ´ q -symplectic form is homotopic to 0 relative to X : R p ω, γ, δ q { X „ We now need to show the non degeneracy condition of the isotropic fibration.Considerthe following commutative diagram: T Z p ˚ T L ‘ p ˚ T L F ˚ T Y P ˚ T X T Z ä X p ˚ T L ä X ‘ p ˚ T L ä X F ˚ T Y ä X L Z r n ´ s ´ p ˚ L L ä Y ‘ p ˚ L L ä Y ¯ r n ´ s F ˚ L Y r n s F ˚ L Y ä X r n s P ˚ L X r n ´ s ´ p ˚ L L ä X ‘ p ˚ L L ä X ¯ r n ´ s P ˚ L X r n s » » » »» » » » (7)We start considering the upper face: 18 Z p ˚ T L ‘ p ˚ T L F ˚ T Y P ˚ T X T Z ä X p ˚ T L ä X ‘ p ˚ T L ä X F ˚ T Y ä X All squares are bi-Cartesians and the three left most horizontal arrows of the top rowform a fiber sequence. Therefore the three left most horizontal arrows of the lower rowalso form a fiber sequence.In the full diagram, the vertical arrows are all quasi-isomorphisms, either using thenon-degeneracy conditions or Proposition 3.3 applied to the sequences L i Ñ Y Ñ X . Wefind 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, L L i ä X » because we assumed that the maps L i Ñ X are weak equivalences.Therefore the bottom face contains the fiber sequence P ˚ L X r n ´ s Ñ Ñ P ˚ L X r n s . Weobtain a weak equivalence α : T Z ä X Ñ P ˚ L X r n ´ s .We still need to show that α is the morphism used in the criteria for the non-degeneracyof the Lagrangian fibration. Recall that this morphism is given by means of the Diagram(4): T Z ä X P ˚ L X r n ´ s T Z L Z r n ´ s L Z ä X r n ´ s α P „ We want to prove that α P and α are homotopic. The relevant data extracted fromthe Diagram (7) is: T Z ä X P ˚ L X r n ´ s T Z L Z r n ´ s L L ä Y r n ´ s ‘ L L ä Y r n ´ s α „ But since all the squares in the lower face of Diagram (7) are bi-Cartesians, we havethat L L ä Y r n ´ s ‘ L L ä Y r n ´ s is naturally quasi-isomorphic to L Z ä X r n ´ s . This proves,by universal property of the pull-back, that α is homotopy equivalent to α P . Given a derived Artin stack X and a morphism f : X Ñ A k , we define the derived criticallocus of f , denoted Crit p f q , as the derived intersection of df : X Ñ T ˚ X with the zerosection X Ñ T ˚ X . It is given by the homotopy pull-back:19 rit p f q XX T ˚ X df (8). Example 3.5.
We recall from [4] that if X is a smooth algebraic variety, its derivedcritical locus can be described, as a derived scheme, by the underlying scheme given bythe ordinary critical locus of f , that we denote S , together with the sheaf of cdga ď givenby the derived tensor product O X b L O T ˚ X O X , restricted to S . This derived tensor productis described by the homotopy push-out: Sym O X p T X q O X O X O X b L O T ˚ X O X df Taking the derived tensor product amounts to replacing the 0-section morphism O X T X Ñ O X by the equivalent cofibration Sym O X T X ã Ñ Sym O X p T X r s ‘ T X q , where Sym O X p T X r s ‘ T X q has the differential induced by Id : T X r s Ñ T X . Then we take thestrict push-out of this replacement. The use of these resolutions are well explained in [4]or [20]. We obtain: O Crit p f q : “ ´ O X b L O T ˚ X O X ¯ | S » ` Sym O X T X r s , ι df ˘ | S where ι df is the the differential on O Crit p f q given by the contraction along df . Therestriction to S denotes the fact that this is a derived scheme whose underlying schemeis the strict critical locus. Observe that outside of the critical locus, ` Sym O X T X r s , ι df ˘ iscohomologically equivalent to 0. Remark 3.6.
If we do not assume that X is smooth in Example 3.5, then L X usu-ally has a non trivial internal differential. As a sheaf of graded algebra, we still obtain Sym O X p T X r sq since the replacement is the same as a graded algebra but the differentialis a priori be different and involve a combination of the internal differential on T X andthe contraction ι df . Remark 3.7.
From Example 2.10, we know that T ˚ X carries a canonical symplecticform of degree 0 and from Example 2.16 we know that both the 0 section and df have aLagrangian structure. From Proposition 3.1, the derived intersection of these Lagrangianstructures, namely the derived critical locus Crit (f), has a p´ q -shifted symplectic struc-ture. Remark 3.8.
When X is a derived Artin stack and df “ , we have that Crit p f q » T ˚ r´ s X and ω Crit p f q is the canonical p´ q -shifted symplectic structure on T ˚ r´ s X .In this situation, the strict critical locus is X itself, and the restriction to X of Sym O X T X r s is therefore Sym O X T X r s itself (with the differential being zero since df “ ).Thus Crit p f q » Spec X ` Sym O X T X r s ˘ “ T ˚ r´ s X .20e want to understand the p´ q -shifted symplectic form on Crit p f q . We use theuniversal property of the tautological -form (Lemma 2.11) to see that p df q ˚ ω “ . Us-ing the resolution of the zero section, as in Example 3.5, ω induces a closed 2-form on R Spec X ` Sym O X p T X r s ‘ T X q ˘ . Since the differential on the resolution, Sym O X p T X r s ‘ T X q ,is induced by Id : T X Ñ T X r s , the tautological 1-form ω ´ on T ˚ r´ s X induces a closed2-form on R Spec X ` Sym O X p T X r s ‘ T X q ˘ which is a homotopy between ω and 0. Wethen have that the p´ q -shifted symplectic form is described by the following loop around0: p ˚ ω “ ω ´ The proof of Proposition 3.1 tells us that the ω ´ is the (-1)-shifted symplectic formon Crit p f q . Remark 3.9.
From Theorem 3.4, we have that π : Crit p f q Ñ X is a Lagrangian fibration.In the situation where df “ and X is smooth, this Lagrangian fibration coincides withthe canonical Lagrangian fibration on π X : T ˚ r´ s X Ñ X . In general, the morphism α π controlling the non-degeneracy condition of the Lagrangian fibration (see Diagram (4)) isstill natural in the sens given by the following proposition. Proposition 3.10. α π is equivalent to the following composition of equivalences: T Crit p f q ä X T X ä X ˆ T T ˚ X ä X T X ä X » ˆ T T ˚ X ä X ˆ π ˚ X L X » π ˚ L X r´ s ˆ β where β is the dual of the canonical equivalence L T ˚ X ä X » π ˚ X L X of Proposition 2.24.Proof. The strategy here is to express the Diagram (4) as a pull-back of the same type ofdiagrams. It reduces the problem to proving the same statement but for the projection π X : T ˚ X Ñ X . But this Proposition is known for the Lagrangian fibration on the shiftedcotangent stacks (this is a direct consequence of Proposition 2.28).First we express L Crit p f q r´ s as a pull-back above L T ˚ X . This can be done by observingthat all squares in the following diagram are bi-Cartesians: L Crit p f q r´ s L T ˚ X ä X L T ˚ X ä X L T ˚ X L X L X L Crit p f q We write Diagram (4) for π : Crit p f q Ñ X as:21 ˆ T T ˚ X ä X ˆ π ˚ X L X T X ˆ T T ˚ X ä X T X L T ˚ X ä X ˆ L T ˚ X L T ˚ X ä X L T ˚ X ä X ˆ L T ˚ X ä X L T ˚ X ä Xα π » ˆ απX ˆ pr Id We need to describe the morphism ω Crit p f q : T X ˆ T T ˚ X ä X T X Ñ L T ˚ X ä X ˆ L T ˚ X L T ˚ X ä X .Recall from Remark 2.15 and the proof of the non-degeneracy in Proposition 3.1 that ω Crit p f q is the natural map completing Diagram (6). This map is therefore Θ df ˆ ω Θ .Here Θ h : T X Ñ L h r´ s » L X ä T ˚ X r´ s » L T ˚ X ä X .Finally, Proposition 2.28 shows that β is the same as α π X . This completes the proof. Let X be a smooth algebraic variety over k and f : X Ñ A k a map which is smootheverywhere except at a point x P X where there is a non degenerate critical point. Thegoal is to understand the Lagrangian fibration on Crit p f q Ñ X and show that it is relatedto 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 statementsmore precise and give complete proofs in Section 4.2.The strict critical locus is ‹ : “ ´ ‹ , O X ä I ¯ where I is the ideal generated by the partialderivatives of f , I “ x df.v, v P T X y . There is a natural morphism ˜ x : “ ‹ Ñ Crit p f q suchthat the following diagram commutes: ‹ p´ q Crit p f q X x ˜ x π X The ideal generated by the partial derivatives is maximal and the partial derivativesform a regular sequence. This implies that ˜ x is an equivalence. For more details, thisis the analogue of Proposition ?? , where we prove that T ˚ r´ s S “ T ˚ r´ s‹ “ ‹ p´ q isweakly equivalent to Crit p f q .Using Lemma 2.23, the Lagrangian fibration induced on ‹ p´ q Ñ X is weakly equiv-alent to a closed 2-form in A ,cl ` ‹ ä X, ´ ˘ which induces a metric on T x X . 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 ˚ T X Ñ x ˚ L X is a quasi-isomorphism.We will show that this metric is in fact characterised by the Hessian quadratic form of f
22t the critical point.We want to describe the Lagrangian fibration obtained on ‹ Ñ X by pulling backalong ˜ x the homotopy between ω ´ ä X and 0 in A ,cl ´ Crit p f q ä X, ´ ¯ . We obtain ahomotopy between 0 and itself in A ,cl ` ‹ ä X, ´ ˘ . We will relate the Hessian quadraticform with the map α x defined to describe the non-degeneracy condition of Lagrangianfibrations (see Definition 2.21 and Diagram (4)). For Crit p f q and ‹ this diagram becomesrespectively T Crit p f q ä X π ˚ L X r´ s T Crit p f q L Crit p f q r´ s L Crit p f q ä X r´ s α π ω and T ‹ ä X x ˚ L X r´ s
00 0 L ‹ ä X r´ s . α x These two diagrams are supposed to represent the same Lagrangian fibration. We willpull-back along ˜ x the diagram for Crit p f q to the category of differential graded k -vectorspace (i.e. QC p‹q ). We can compare α x and α π X via the following commutative diagram: ˜ x ˚ T Crit p f q ä X x ˚ L X r´ s T ‹ ä X L ‹ ä X » x ˚ L X r´ s T Crit p f q L Crit p f q r´ s L Crit p f q ä X r´ s L ‹ ä X ‘ L ‹ ä X r´ s L ‹ ä X r´ s α πX „ α x ω ψ „„ We can now look at these morphisms in local étale coordinates around x . We denoteby X i coordinates in X , p i a basis of x ˚ T X and ξ i its associated shifted basis in x ˚ T X r s .We also denote by dX i the dual basis of p i . We write k x a y : “ k x a , ¨ ¨ ¨ , a n y for the k-vector space with basis a , ¨ ¨ ¨ , a n . We get:23 xB ξ y k x dX y k xB ξ y k x dθ y k xB ξ , B X y k x dX, dξ y k x dξ y k x dθ, dξ y k x dξ y α πX Id α x ω ψ „„ Here, dθ is the standard shifted variable variable added to make the following pull-backsquare a strict pull-back: k x dθ y k x dθ, dξ y k x dξ y This imposes δdξ “ dθ . To make the all diagram strictly commutative, we must have ψ p dξ q “ dξ . And to make ψ a map of chain complexes, we must have ψ p dδξ i q “ δψ p dξ i q “ δdξ i “ dθ i and therefore it imposes ψ p dX i q “ Hess ´ x p f qp dX i qp dX j q dθ j . This implies that α x pB ξ i q “ Hess ´ x p f qp dX i qp dX j q dθ j . We consider a generalisation of the previous example where f may have a family of criticalpoints 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 witha closed immersion i : S Ñ X and whose algebra of functions is O S “ O X ä I with I “ x df.v, v P T X y .We assume that both X and S are smooth algebraic varieties, which implies that O S is reduced. We denote by Crit p f q the derived critical locus of f and we get a canonicalmorphism λ : S Ñ Crit p f q .In order to define the Hessian quadratic form and the non-degeneracy condition, weneed to assume that the closed immersion S ã Ñ X has a first order splitting. Concretely,we assume in this section that the following fiber sequence splits: T S i ˚ T X T S ä X r s (9)This assumption is necessary to be able to restrict Q to the normal part T S ä X r s . Definition 4.1.
The
Hessian quadratic form is defined by the symmetric bilinearmap: Q : Sym O S i ˚ T X Ñ O S p w, v q ÞÑ d p df.v q .w
24e define non-degeneracy to be along the "normal" direction to S , by considering thefollowing diagram: T S i ˚ T X T S ä X r s L S ä X r´ s i ˚ L X L S Q r Q (10)Both rows are split fiber sequences (by assumption in Diagram (9)). The map T S r s Ñ L S r´ s is the zero map because Q restricted to T S is zero and since Q is symmetric, Q projected to L S is also zero. We obtain a map r Q which corresponds to the map inducedby Q on the normal bundle. Then the non-degeneracy condition is the requirementthat r Q is a quasi-isomorphism.Since the differential on O Crit p f q is δ “ ι df , we have the commutative diagram: T X r s O X T X L X Id ´ δ dQ (11)We will abusively write Q “ d ˝ δ .In general, the natural map λ : S Ñ Crit p f q is not an equivalence. This is due tothe fact that the partial derivatives of f will not in general form a regular sequence andtherefore Crit p f q has higher homology. The default to be a regular sequence comes fromvector 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 Crit p f q is given by T ˚ r´ s S when Q is non-degenerate. Proposition 4.2.
There exists a natural map
Φ : T ˚ r´ s S Ñ Crit p f q making the fol-lowing diagram commute: T ˚ r´ s S Crit p f q S X π S Φ πi Proof.
Under our first order splitting assumption (Diagram (9)), the natural map T S Ñ i ˚ T X admits a retract, and therefore the natural map i ˚ T ˚ X Ñ T ˚ S admits a section: T ˚ S i ˚ T ˚ X . We consider the following diagram: T ˚ X i ˚ T ˚ X T ˚ SX S S i We want to pull-back these zero sections along the maps induced by df representedby the vertical morphisms in the following commutative diagram:25 ˚ X i ˚ T ˚ X T ˚ SX S S df i i ˚ df “ This induces the following morphisms between the pull-backs:
Crit p f q S ˆ i ˚ T ˚ X S T ˚ r´ s S We obtain a map
Φ : T ˚ r´ s S Ñ Crit p f q . The maps we obtain come from theuniversal properties of the pull-backs therefore if we denote s : X Ñ T ˚ X the zerosection, we have s ˝ π X ˝ Φ “ s ˝ i ˝ π S . If we compose by the projection π X : T ˚ X Ñ X ,we get π X ˝ Φ “ i ˝ π S . Φ gives a relationship between the Lagrangian fibration structures on T ˚ r´ s S Ñ S and Crit p f q Ñ X which we now analyse. The idea is to show that the difference betweenthese Lagrangian fibrations is in fact controlled by r Q (see Proposition 4.6 and Remark4.8). Lemma 4.3. Φ induces a morphism T T ˚ r´ s S ä S Ñ Φ ˚ T Crit p f q ä X that fits in the commu-tative diagram T T ˚ r´ s S ä S Φ ˚ T Crit p f q ä X π ˚ S L S r´ s Φ ˚ π ˚ X L X r´ s » π ˚ S i ˚ L X r´ s α πS α πX (12) where the bottom horizontal arrow is the pull-back along π S of the section L S r´ s Ñ i ˚ L X r´ s in the dual of the split fiber sequence (9) .Proof. The homotopy pull-back,
Crit p f q “ X ˆ hT ˚ X X lives over X . We get the equiva-lences: T Crit p f q ä X T X ä X ˆ h T T ˚ X ä X T X ä X ‹ ˆ h T T ˚ X ä X ‹ π ˚ X L X r´ s » » » Proposition 2.28, gives us the following commutative square: T T ˚ S ä S T T ˚ X ä X π ˚ S L S π ˚ S i ˚ L Xβ S β X π ˚ S s where s is the section in the dual of the split fiber sequence (9). From Proposition 3.10,we know that both α π S and α π are the morphism induced by the morphisms β S and β X
26n the previous diagram when taking the pull-back. We obtain the commutative diagram: T T ˚ r´ s S ä S ˆ h T T ˚ S ä S π ˚ S L S r´ s Φ ˚ T Crit p f q ä X Φ ˚ ˆ ˆ h T T ˚ X ä X ˙ Φ ˚ π ˚ X L X r´ s » ˆ hβS » ˆ hβX where the composition of the horizontal maps are exactly α π S and α π X thanks toProposition 3.10. Lemma 4.4.
We first remark that Φ ˚ L Crit p f q can be described, as a sheaf of graded com-plex, by Φ ˚ L Crit p f q » Sym O S p T S r sq b O S p i ˚ L X ‘ i ˚ T X r sq where L X is generated by terms of the form dg with g P O X and T X r s is generatedby terms of the form dξ with ξ P T X r s . Then, the internal differential on Φ ˚ L Crit p f q ischaracterised by Q “ d ˝ ι df via δ p dξ q “ Q p ξ q and δ p dg q “ .Proof. The differential on
Sym O S p T S r sq b O S p i ˚ L X ‘ i ˚ T X r sq is O T ˚ r´ s S -linear because ι df is zero on T S r s . Moreover, for ξ P T X r s Ă O Crit p f q “ Sym O X T X r s , we have δ ˝ d p ξ q “ d ˝ δ p ξ q “ d ˝ ι df p ξ q “ Q p ξ q (see Diagram (11)), and for g P O X , δ ˝ d p g q “ d ˝ δg “ . Lemma 4.5.
The composition π ˚ S i ˚ T X r´ s Φ ˚ T Crit p f q ä X Φ ˚ π ˚ X L X r´ s α πX is given by π ˚ S Q . Similarly, the composition π ˚ S T S r´ s T T ˚ r´ s S ä S π ˚ S L S r´ s α πS is (the restriction of π ˚ S Q to S ).Proof. It is enough to proves this locally around any point x of S . The left morphism isthe morphism fitting in the fiber sequence: π ˚ S i ˚ T X r´ s Φ ˚ T Crit p f q ä X Φ ˚ T Crit p f q Which gives us locally: π ˚ S i ˚ T X,i p x q r´ s Φ ˚ T Crit p f q ä X,x Φ ˚ T Crit p f q ,x π ˚ S i ˚ T X,i p x q r´ s Φ ˚ π ˚ X L X,i p x q r´ s Φ ˚ π ˚ X L X,i p x q r´ s ‘ φ ˚ π ˚ X T X,i p x q α πX » The second row can be seen as the extension (by π ˚ S ) of the fiber sequence:27 ˚ T X,i p x q r´ s i ˚ L X,i p x q r´ s i ˚ L X,i p x q r´ s ‘ i ˚ T X,i p x q Since X and S are smooth, i ˚ T X,i p x q r´ s and i ˚ L X,i p x q r´ s are both quasi-isomorphicto complexes concentrated in a single degree. This imposes that the dashed arrow isequivalent 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 ˚ T X,i p x q r´ s to itsdifferential, in i ˚ L X,i p x q r´ s‘ i ˚ T X,i p x q , which can in turn be seen as an element in i ˚ L X,i p x q .More concretely, denote ˜ s any lift of s to an element in i ˚ L X,i p x q r´ s ‘ i ˚ T X,i p x q r´ s . UsingLemma 4.4, its differential is given by Q p s q “ Q p ˜ s q P i ˚ L X,i p x q r´ s Ă i ˚ L X,i p x q r´ s ‘ i ˚ T X,i p x q . We then apply π ˚ S to get the sequence we want. The second part of the statement isproven the same way. Proposition 4.6.
The map T T ˚ r´ s S Ñ Φ ˚ T Crit p f q induced by Φ is an equivalence if andonly if Q is non-degenerate.Proof. First, using the equivalences α π : Φ ˚ T Crit p f q ä X Ñ π ˚ S i ˚ L X r´ s and α π S : Φ ˚ T T ˚ r´ s S ä S Ñ π ˚ S L S r´ s , we can show that the cofiber of T T ˚ r´ s S ä S Ñ Φ ˚ T Crit p f q ä X is equivalent to π ˚ S L S ä X r´ s . Then Lemma 4.3 and 4.5 ensure that the upper half of the following dia-gram is commutative: π ˚ S T S r´ s π ˚ S i ˚ T X r´ s π ˚ S T S ä X T T ˚ r´ s S ä S Φ ˚ T Crit p f q ä X π ˚ S L S ä X r´ s T T ˚ r´ s S Φ ˚ T Crit p f q F r Q (13)This diagram is then commutative and all rows and columns are cofiber sequences andin particular F is both the homotopy cofiber of T T ˚ r´ s S Ñ Φ ˚ T Crit p f q and the homotopycofiber of r Q . In particular, the homotopy cofiber of r Q is zero if and only the homotopycofiber of T T ˚ r´ s S Ñ Φ ˚ T Crit p f q is also zero.We now decompose α π into a part along S and a part normal to S . This decompositionis 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-degeneracyof the Lagrangian fibrations fit in the commutative diagram: T T ˚ r´ s S ä S T Crit p f q ä X T S ä X π ˚ S L S r´ s π ˚ S i ˚ L X r´ s L S ä X r´ s α πS α πX r Q where the rows are fiber sequences. roof. First, when Q is non-degenerate, the top horizontal sequence is fibered and comesfrom the following diagram: T T ˚ r´ s S ä S Φ ˚ T Crit p f q ä X π ˚ S T S ä X T T ˚ r´ s S Φ ˚ T Crit p f q π ˚ S T S Φ ˚ i ˚ T X π ˚ S T S ä X r s where all rows and columns are fibered and the cofiber of the second row is thanksto Proposition 4.6 since we assumed that Q is non-degenerate. Using Lemma 4.3 andLemma 4.5, we obtain the following commutative diagram: π ˚ S T S r´ s Φ ˚ i ˚ T X r´ s π ˚ S T S ä X T T ˚ r´ s S ä S Φ ˚ T Crit p f q ä X π ˚ S T S ä X π ˚ S L S r´ s Φ ˚ i ˚ L X r´ s π ˚ S L S ä X r´ s Q r Qα πS α πX (14)The only map the dashed arrow can be, in order to make the diagram commutative,is r Q . Remark 4.8.
If we do not assume Q non-degenerate, the cofiber F of the map T T ˚ r´ s S Ñ Φ ˚ T Crit p f q will be non zero. We will denote by G the fiber of the natural map F Ñ T S ä X .Then we can rewrite Diagram (14) as π ˚ S T S r´ s Φ ˚ i ˚ T X r´ s π ˚ S T S ä X T T ˚ r´ s S ä S Φ ˚ T Crit p f q ä X G π ˚ S L S r´ s Φ ˚ i ˚ L X r´ s π ˚ S L S ä X r´ s Q r Qα πS α πX α N The map α N : G Ñ L S ä X r´ s represent the "difference" between the maps α π and α π S from the Lagrangian fibrations. α N is still related to r Q in the sens that the followingdiagram is commutative: 29 S ä X G L S ä X r´ s r Qα N Therefore the restriction of α N to T S ä X is again r Q . Remark 4.9.
As a non-example if we take f : A Ñ A sending X to X , the basicassumptions that made this section work are failing. The strict critical locus S is notsmooth since it is a fat point, and the sequence (9) does not split. Let X be a derived Artin stack and α P A p X, n q be a 1-form. If Key p α q is non-empty,Proposition 2.17 and Remark 2.18 ensure that the map α : X Ñ T ˚ r n s X is a Lagrangianmorphism. Using Theorem 3.4 the derived intersection Z p α q of α with the zero sectiongives us a Lagrangian fibration Z p α q Ñ X . This example is a generalisation of the derivedcritical locus we described in 3.3. G -Equivariant Twisted Cotangent Bundles For X a smooth scheme, a twisted cotangent stack is a twist of the ordinary cotangentstack by a closed 2-form of degree 1 on X , α P H p X, Ω X q . Such a closed form has anunderlying 1-form of degree 1 that corresponds to a morphism α : X Ñ T ˚ r s X . The twisted cotangent bundle associated to α is defined to be the following pull-back: T ˚ α X XX T ˚ r s X α We refer to [9] for more informations on the relation between this definition and theusual definition of twisted cotangent bundles. This is a particular case of the situationin Section 4.3 and as such, T ˚ α X is -shifted symplectic and the map T ˚ α X Ñ X has aLagrangian fibration structure.Now take G an algebraic group acting on the algebraic variety X . Consider a character χ : G Ñ G m . We have the logarithmic form on G m given by a map G m Ñ A ,cl p q whichsends z to z ´ dz . We get a closed 1-form on G described by the composition: G Ñ G m Ñ A ,cl p q This is also a group morphism, we can therefore pass to classifying spaces and obtaina -shifted closed -form on B G : α χ : B G Ñ B A ,cl p q “ A ,cl p q
30e can consider the pull-back of α along the G -equivariant moment map: ” T ˚ X ä G ı ˆ ” g ˚ ä G ı B G B G ” T ˚ X ä G ı ” g ˚ ä G ı » T ˚ r s B G αµ It turns out that the moment map µ is Lagrangian (see [6]), which implies (withProposition 3.1) that this fiber product is -shifted symplectic. It turns out that we havean equivalence of shifted symplectic derived Artin stacks: ” T ˚ X ä G ı ˆ ” g ˚ ä G ı B G » T ˚ p α ” X ä G ı Where p α denotes the pull-back of α to a -form of degree on ” X ä G ı . Therefore,according to Theorem 3.4, the natural projection T ˚ p α ” X ä G ı ” X ä G ı is a Lagrangian fibration.To show the equivalence above, we use the following composition of Lagrangian cor-respondences (see 2.20): T ˚ p α ” X ä G ı” T ˚ X ä G ı ” X ä G ı” X ä G ı ” g ˚ ˆ X ä G ı B G ‹ T ˚ r s ” X ä G ı ” g ˚ ä G ı ‹ α The only thing we need to show is that this is a diagram of Lagrangian correspondencesand therefore we need to show that all squares in this diagrams are pull-backs. The rightmost square is clearly a pull-back and we can recognise the pull-back square defining T ˚ p α ” X ä G ı .We are left to prove that ” X ä G ı ˆ T ˚ r s r X ä G s ” g ˚ ˆ X ä G ı is naturally equivalent to ” T ˚ X ä G ı . This follows from the sequence of natural equivalences31 X ä G ı ˆ T ˚ r s r X ä G s ” g ˚ ˆ X ä G ı » ” X ä G ı ˆ T ˚ r s r X ä G s ” g ˚ ä G ı ˆr ‹ ä G s ” X ä G ı » ” X ä G ı ˆ T ˚ r s r X ä G s ” X ä G ı ˆr ‹ ä G s ” g ˚ ä G ı » T ˚ ” X ä G ı ˆr ‹ ä G s ” g ˚ ä G ı » “ ‹ ä G ‰ ˆ ” g ˚ ä G ı ” T ˚ X ä G ı ˆr ‹ ä G s ” g ˚ ä G ı » ” T ˚ X ä G ı ˆ ” g ˚ ä G ı “ ‹ ä G ‰ ˆr ‹ ä G s ” g ˚ ä G ı » ” T ˚ X ä G ı where we use the fact that the following square is a pull-back: T ˚ ” X ä G ı B G ” T ˚ X ä G ı ” g ˚ ä G ı µ References [1] Kai Behrend and Barbara Fantechi. Symmetric obstruction theories and Hilbertschemes of points on threefolds.
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