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A Derived Lagrangian Fibration on the Derived Critical Locus

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A Derived Lagrangian Fibration on the Derived Critical Locus A Derived Lagrangian Fibration on the Derived Critical Locus Albin Grataloup ∗ December 11, 2020 Abstract We study the symplectic geometry of derived intersections of Lagrangian mor- : A1 phisms. In particular, we show that for a functional f X Ñ k, the derived critical locus has a natural Lagrangian fibration Critpfq Ñ X. In the case where f is non-degenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form. Contents 1 Introduction 2 2 Derived Symplectic Geometry 4 2.1 Shifted Symplectic Structures . .... 4 2.2 LagrangianStructures . .. .. .. 8 2.3 LagrangianFibration . .. .. .. 10 2.4 Relative Cotangent Complexes of Linear Stacks . ....... 11 3 Symplectic Geometry of the Derived Critical Locus 16 3.1 Lagrangian Intersections are pn ´ 1q-Shifted Symplectic . 16 3.2 Lagrangian Fibrations and Derived Intersections . .......... 17 3.3 DerivedCriticalLocus . .. .. .. 19 arXiv:2010.14221v2 [math.SG] 10 Dec 2020 4 Examples 22 4.1 OneNon-DegenerateCriticalPoint . .... 22 4.2 Family of non-Degenerate Critical Points . ....... 24 4.3 DerivedZeroLocusofShifted1-Forms . .... 30 4.4 G-Equivariant Twisted Cotangent Bundles . 30 ∗IMAG, Univ. Montpellier, CNRS, Montpellier, France [email protected] 1 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.
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