Introduction to derived algebraic geometry: Forms and closed forms

Transcription

1 Introduction to derived algebraic geometry: Forms and closed forms Tony Pantev Gabrielle Vezzosi University of Pennsylvania Universitá degli Studi di Firenze Etats de la Recherche: Derived Algebraic Geometry and Interactions Toulouse, June 2017

2 Outline tangent and cotangent complex shifted forms and closed forms shifted symplectic and isotropic structures examples and constructions

3 Tangent complex X P dst C, x : SpecpCq Ñ X a point normalized chain complex ˆStalk TX,x of the of the homotopy fiber of tangent complex X pcrεsq Ñ X pcq over x simplicial abelian group

4 Tangent complex X P dst C, x : SpecpCq Ñ X a point normalized chain complex ˆStalk TX,x of the of the homotopy fiber of tangent complex X pcrεsq Ñ X pcq over x When X is a moduli stack: H 1 pt X,x q infinitesimal automorphisms of x; H 0 pt X,x q infinitesimal deformations of x; H 1 pt X,x q Ě obstructions of x.

5 Examples (i): X BG rpt {Gs ñ T X,pt gr1s. X moduli of vector bundles E on a smooth projective Y ñ T X,E RΓpY,EndpEqqr1s. X moduli of maps f from C to Y ñ T X,f RΓpC,f T Y q. X moduli of local systems E on a compact manifold Y ñ T X,E RΓpY,EndpEqqr1s.

6 Examples (ii): X derived intersection hą Lâ L 1 L 2 L 1 XL 2, O L1 M O M O L2 of smooth subvarieties L 1,L 2 Ă M in a smooth M ñ T X,x r T L1,x T L2,x T M,x s, 0 1 H 0 pt X,x q T L1 XL 2,x; H 1 pt X,x q failure of transversality.

7 Examples (iii): Special case: X derived zero locus Rzeropsq of s P H 0 pl,e q.

8 Examples (iii): Special case: X derived zero locus Rzeropsq of s P H 0 pl,e q. an algebraic vector bundle on a smooth variety L

9 Examples (iii): Special case: X derived zero locus Rzeropsq of s P H 0 pl,e q. Thus where: X L hą L M Z, i 1 L psym pe _ r1sq,s 5 q, Z t 0 X zeropsq is the scheme theoretic zero locus of s, i L : Z Ñ L is the natural inclusion, and s 5 is the contraction with s.

10 Examples (iv): In particular T X where M totpeq, and r i L T L i L T L i M, o, and s are the natural maps i L do`i L ds i M T M s, 0 1 i L L o i M Z M. i L s L

11 Examples (iv): In particular T X where M totpeq, and r i L T L i L T L i M, o, and s are the natural maps i L do`i L ds i M T M s, 0 1 Remark: There is a natural quasi-isomorphism «ff «ff T X i L T p sq 5 L i L E p sq T 5 L E Note: : E Ñ E bω 1 L is an algebraic connection which exists only locally and is not unique. However p sq Z is well defined globally and independent of the choice of. Z.

12 Cotangent complex A P cdga C, X SpecpAq P dst C, QA Ñ A a cofibrant (quasi-free) replacement ˆ ˆ cotangent complex Kähler differentials L X L A Ω 1 QA of QA If X P dst C is a general derived Artin stack, then X hocolimtspeca Ñ X u (in the model category dst C ) and Note: L X holim SpecAÑX L A L X P L qcoh px q - the dg category of quasi-coherent O X modules. X is locally of finite presentation iff L X is perfect. In this case T X L _ X HompL X,O X q.

13 p-forms A P cdga C, X SpecpAq P dst C, QA Ñ A a cofibrant (quasi-free) replacement. Then: pě0 ^p A L A pě0 Ω p QA - a fourth quadrant bicomplex with vertical differential d : Ω p,i QA Ñ Ωp,i`1 QA induced by d QA, and horizontal differential d DR : Ω p,i QA Ñ Ωp`1,i QA given by the de Rham differential. Hodge filtration: F q paq : pąq Ω p QA : still a fourth quadrant bicomplex.

14 (shifted) closed p-forms Motivation: If X is a smooth scheme/c, then Ω ěp X rps,d DR Ω p,cl X p forms in general. Definition:. Use pω ěp X rps,d DRq as a model for closed complex of closed p-forms on X SpecA: A p,cl paq : tot ś pf p paqqrps complex of n-shifted closed p-forms on X SpecA: A p,cl pa;nq : tot ś pf p paqqrn `ps Hodge tower: Ñ A p,cl paqr ps Ñ A p 1,cl paqr1 ps Ñ Ñ A 0,cl paq

15 (shifted) closed p-forms (ii) Explicitly an n-shifted closed p-form ω on X SpecA is an infinite collection ω tω i u iě0, ω i P Ω p`i,n i A satisfying d DR ω i dω i`1. Note: The collection tω i u iě1 is the key closing ω.

16 p-forms and closed p-forms Note: The complex A 0,cl paq of closed 0-forms on X SpecA is exactly Illusie s derived de Rham complex of A. There is an underlying p-form map A p,cl pa;nq Ñ ^pl A{k rns inducing H 0 pa p,cl paqrnsq Ñ H n px, ^pl A{k q. The homotopy fiber of the underlying p-form map can be very complicated (complex of keys): being closed is not a property but rather a list of coherent data.

17 Functoriality and gluing: the n-shifted p-forms 8-functor A p p ;nq : cdga C Ñ SSets : A ÞÑ Ω p QArns» p^p A L Aqrns the n-shifted closed p-forms 8-functor A p,cl p ;nq : cdga C Ñ SSets : A ÞÑ A p,cl paqrns A p p ;nq and A p,cl p ;nq are derived stacks for the étale topology. underlying p-form map (of derived stacks) A p,cl p ;nq Ñ A p p ;nq Notation: denotes Map C dgmod pc, q i.e. Dold-Kan applied to the ď 0-truncation - all our dg-modules have cohomological differential.

18 global forms and closed forms For a derived Artin stack X (locally of finite presentation {C) we have Definition: A p px q : Map dstc px,a p p qq is the space of p-forms on X; A p,cl px q : Map dstc px,a p,cl p qq is the space of closed p-forms on X; the corresponding n-shifted versions : A p px;nq : Map dstc px,a p p ;nqq A p,cl px;nq : Map dstc px,a p,cl p ;nqq an n-shifted (respectively closed) p-form on X is an element in π 0 A p px;nq (respectively in π 0 A p,cl px;nq)

19 global forms and closed forms (ii) Note: 1) If X is a smooth scheme there are no negatively shifted forms. 2) When X SpecA then there are no positively shifted forms. For general X they might exist for any n P Z.

20 global forms and closed forms (ii) underlying p-form map (of simplicial sets) A p,cl px;nq Ñ A p px;nq this map is not a monomorphism for general X, its homotopy fiber at a given p-form ω 0 is the space of keys of ω 0. If X is a smooth and proper scheme then this map is indeed a mono (homotopy fiber is either empty or contractible) ñ no new phenomena in this case. Theorem (PTVV): for X derived Artin, A p px;nq» Map Lqcoh px qpo X, p^pl X qrnsq (smooth descent) in particular an n-shifted p-form on X is an element in H n px, ^pl X q

21 global forms and closed forms (iii) Remark: If A P cdga is quasi-free, and X SpecA, then A p,cl ź px;nq Ω p`1 ˇ A rn is,d `d DR ˇˇˇˇˇ iě0 ˇ ˇtot Π pf p paqqrnsˇ NCpAqppqrn `ps

22 global forms and closed forms (iii) Remark: If A P cdga is quasi-free, and X SpecA, then A p,cl ź px;nq Ω p`1 ˇ A rn is,d `d DR ˇˇˇˇˇ iě0 ˇ ˇtot Π pf p paqqrnsˇ NCpAqppqrn `ps negative cyclic complex of weight p

23 global forms and closed forms (iii) Remark: If A P cdga is quasi-free, and X SpecA, then A p,cl ź px;nq Ω p`1 ˇ A rn is,d `d DR ˇˇˇˇˇ iě0 ˇ ˇtot Π pf p paqqrnsˇ NCpAqppqrn `ps Hence π 0 A p,cl px;nq HC n p paqppq.

24 global forms and closed forms (iv) Definition: Given a higher Artin derived stack the n-th algebraic de Rham cohomology of X is defined to be H n DR px q π 0A 0,cl px;nq. Remark: agrees with Illusie s definition in the affine case. if X is a higher Artin derived stack locally f.p., then H DR px q H DR pt 0X q algebraic de Rham cohomology of the underived higher stack t 0 X defined by Hartschorne s completion formalism.

25 global forms and closed forms (iv) Definition: Given a higher Artin derived stack the n-th algebraic de Rham cohomology of X is defined to be H n DR px q π 0A 0,cl px;nq. Remark: agrees with Illusie s definition in the affine case. if X is a higher Artin derived stack locally f.p., then H DR px q H DR pt 0X q algebraic de Rham cohomology of the underived higher stack t 0 X defined by Hartschorne s completion formalism. Corollary: Let X be a locally f.p. derived stack and let ω be an n-shifted closed p-form on X with n ă 0. Then ω is exact, i.e. rωs 0 P H n`p DR px q.

26 Examples (i): (1) If X SpecpAq is an usual (underived) smooth affine scheme, then A p,cl px;nq pτ ďn p Ω p d DR A Ω p`1 A 0 1 d DR qqrns, and hence $ & 0, n ă 0 π 0 A p,cl px;nq Ω p,cl A %, n 0 px q, n ą 0 H n`p DR Remark: The standard notation for forms is inadequate - if X Cˆ, then dz{z P π 0 A 1,cl px;0q and also dz{z P π 0 A 0,cl px;1q.

27 Examples (ii): (2) If X is a smooth and proper scheme, then π i A p,cl px;nq F p H n`p i DR px q. (3) If X is a (underived) higher Artin stack, and X Ñ X is a smooth affine simplicial groupoid presenting X, then π 0 A p px;nq H n pω p px q,δq with δ Čech differential. In particular if G is a complex reductive group, then # π 0 A p 0, n p pbg;nq psym g _ q G, n p.

28 Examples (iii): (4) Similarly A p,cl ź pbg;nq `Sym p`i g _ G rn `p 2isˇˇˇˇˇ, ˇ iě0 and so π 0 A p,cl pbg;nq # 0, if n is odd psym p g _ q G, if n is even.

29 Examples (iv): (5) If X Rzeropsq for s P H 0 pl,eq on a smooth L, then Ω 1 X E Z _ p sq 5 Ω 1 L Z, 1 0 and if we choose local flat algebraic connection on E we can rewrite Ω 1 X as a module over the Koszul complex: ^2E _ bω 1 L s 5 E _ bω 1 L ^2E _ be _ s 5 E _ be _ s 5 s 5 Ω 1 L E _ r,s 5 s Ω 1 L Z 0 E _ Z p sq 5 1

30 Examples (v): In the same way we can describe Ω 2 X as a module over the Koszul complex ^2E _ bω 2 L ^2E _ be _ bω 1 L E _ bω 2 L E _ be _ bω 1 L ^2E _ bs 2 E _ E _ bs 2 E _ Ω 2 L E _ bω 1 L S 2 E _ Ω 2 L Z 0 pe _ bω 1 L q Z S 2 E _ Z 1 2

31 Examples (v): In the same way we can describe Ω 2 X as a module over the Koszul complex ^2E _ bω 2 L ^2E _ be _ bω 1 L E _ bω 2 L E _ be _ bω 1 L ^2E _ bs 2 E _ E _ bs 2 E _ Ω 2 L E _ bω 1 L S 2 E _ Ω 2 L Z 0 pe _ bω 1 L q Z S 2 E _ Z forms of degree 1

32 Examples (v): In the same way we can describe Ω 2 X as a module over the Koszul complex ^2E _ bω 2 L ^2E _ be _ bω 1 L E _ bω 2 L E _ be _ bω 1 L ^2E _ bs 2 E _ E _ bs 2 E _ Ω 2 L E _ bω 1 L S 2 E _ Ω 2 L Z 0 pe _ bω 1 L q Z S 2 E _ Z 1 2 Note: The de Rham differnetial d DR : Ω 1 X Ñ Ω2 X d DR `κ, where κ is the Koszul contraction is the sum κ : ^ae _ bs b E _ Ñ ^a 1 E _ bs b`1 E _.

33 Examples (vi): Lemma [Behrend] If E Ω 1 L and so s is a 1-form, then a 2-form of degree 1 corresponds to a pair of elements α P pω 1 L q_ bω 2 L and φ P pω1 L q_ bω 1 L such that r,s5 spφq s 5 pαq. Take φ id P pω 1 L q_ bω 1 L. Suppose the local is chosen so that pidq 0 (i.e. is torsion free). Then r,s 5 spidq ds. Conclusion: The pair pα,idq gives a 2-form of degree 1 iff ds s 5 pαq, or equivalently ds Z 0, i.e. is an almost closed 1-form on L.

34 Symplectic structures Recall: For X a smooth scheme/c is a symplectic structure is an ω P H 0 px,ω 2,cl X q such that its adjoint ω5 : T X Ñ Ω 1 X is a sheaf isomorphism. Note: Does not work for X singular (or stacky or derived): T X and Ω 1 X are too crude as invariants and get promoted to complexes T X and L X. A form being closed is not just a condition but rather an extra structure.

35 Definition: X derived Artin stack locally of finite presentation (so that L X is perfect). A n-shifted 2-form ω : O X Ñ L X ^L X rns - i.e. ω P π 0 pa 2 px;nqq - is nondegenerate if its adjoint ω 5 : T X Ñ L X rns is an isomorphism (in D qcoh px q). The space of n-shifted symplectic forms SymplpX;nq on X {C is the subspace of A 2,cl px;nq of closed 2-forms whose underlying 2-forms are nondegenerate i.e. we have a homotopy cartesian diagram of spaces SymplpX,nq A 2,cl px,nq A 2 px,nq nd A 2 px,nq

36 Shifted symplectic structures: examples (i) Nondegeneracy: a duality between the stacky (positive degrees) and the derived (negative degrees) parts of L X. G GL n BG has a canonical 2-shifted symplectic form whose underlying 2-shifted 2-form is k Ñ pl BG ^L BG qr2s» pg _ r 1s ^g _ r 1sqr2s Sym 2 g _ given by the dual of the trace map pa,bq ÞÑ trpabq. Same as above (with a choice of G-invariant symm bil form on g) for G reductive over k. The n-shifted cotangent bundle T _ X rns : Spec X psympt X r nsqq has a canonical n-shifted symplectic form.

37 Shifted symplectic structures: examples (ii) Theorem: [PTVV] Let F be a derived Artin stack equipped with an n-shifted symplectic form ω P SymppF,nq. Let X be an O-compact derived stack equipped with an O-orientation rx s : HpX,O X q ÝÑ kr ds of degree d. If the derived mapping stack MAPpX,F q is a derived Artin stack locally of finite presentation over k, then, MAPpX, F q carries a canonical pn dq-shifted symplectic structure. Remark: 0) Analog to Alexandrov-Kontsevich-Schwarz-Zaboronsky result. 1) A d O-orientation on X is a kind of Calabi-Yau structure of dimension d; 2) A compact oriented topological d-manifold has an O-orientation of degree d (Poincaré duality).

38 Lagrangian structures Let py,ωq be a n-shifted symplectic derived stack. A lagrangian structure on a map f : X Ñ Y is path γ in A 2,cl px;nq from f ω to 0 that is non-degenerate (in a suitable sense), i.e. the induced map θ γ : T f Ñ L X rn 1s is an equivalence. Examples: usual smooth lagrangians L ãñ py,ωq where py,ωq is a smooth (0)-symplectic scheme. there is a bijection between lagrangian structures on the canonical map X Ñ pspeck,ω n`1 q and n-shifted symplectic structures on X (thus lagrangian structures generalize shifted symplectic structures)

39 Shifted symplectic structures: examples (iii) Theorem: [PTVV] Let pf, ωq be n-shifted symplectic derived Artin stack, and L i Ñ F a map of derived stacks equipped with a Lagrangian structure, i 1, 2. Then the homotopy fiber product L 1 ˆF L 2 is canonically a pn 1q-shifted derived Artin stack. In particular, if F Y is a smooth symplectic Deligne-Mumford stack (e.g. a smooth symplectic variety), and L i ãñ Y is a smooth closed lagrangian substack, i 1, 2, then the derived intersection L 1 ˆF L 2 is canonically p 1q-shifted symplectic.

40 Remark: An interesting case is the derived critical locus RCritpf q for f a global function on a smooth symplectic Deligne-Mumford stack Y. Here RCritpf q Y Y 0 df T _ Y

41 Local models (i) Recall: In classical symplectic geometry the local structure of a symplectic manifold is described by the Darboux theorem:

42 Local models (i) Recall: In classical symplectic geometry the local structure of a symplectic manifold is described by the Darboux theorem: a symplectic structure is locally (in the C 8 or analytic setting) or formally (in the algebraic setting) isomorphic to the standard symplectic structure on a cotangent bundle.

43 Local models (i) In the derived and stacky setting there are two natural incarnations of an n-shifted symplectic cotangent bundle: (a) The shifted cotangent bundle T M _ rns Spec {M `Sym OM pt M r nsq, equipped with n-th shift of the standard symplectic form; (b) The derived critical locus Rcritpwq of an n `1 shifted function w : M Ñ A 1 rn `1s, equipped with the inherited n-shifted symplectic form ω Rcritpwq. defined for general Lagrangian intersections in [PTVV 2012].

44 Local models (i) In the derived and stacky setting there are two natural incarnations of an n-shifted symplectic cotangent bundle: (a) The shifted cotangent bundle T M _ rns Spec {M `Sym OM pt M r nsq, equipped with n-th shift of the standard symplectic form; (b) The derived critical locus Rcritpwq of an n `1 shifted function w : M Ñ A 1 rn `1s, equipped with the inherited n-shifted symplectic form ω Rcritpwq. Note: (a) is a special case of (b) corresponding to the zero shifted function.

45 Local models (ii) Remark: Shifted cotangent bundles are too restrictive to serve as local models of shifted symplectic structures. Derived critical loci of shifted functions have enough flexibility to provide local models. This leads to a remarkable shifted version of the Darboux theorem:

46 Local models (ii) Theorem: [BBBJ 2013] Let X be a derived Deligne-Mumford stack, and let ω be an n-shifted symplectic structure on X, with n ă 0. Then, étale locally px,ωq is isomorphic to `Rcritpwq,ω Rcritpwq for some shifted function w : M Ñ A 1 rn `1s on a derived scheme M. O.Ben-Bassat, C.Brav, V.Bussi, D.Joyce

47 Local models (ii) Theorem: [BBBJ 2013] Let X be a derived Deligne-Mumford stack, and let ω be an n-shifted symplectic structure on X, with n ă 0. Then, étale locally px,ωq is isomorphic to `Rcritpwq,ω Rcritpwq for some shifted function w : M Ñ A 1 rn `1s on a derived scheme M. Question: Find additional geometric structures that will ensure a global existence of a potential?

48 Local models (iii) Answer: Potentials always exist in the presence of isotropic foliations.

49 Local models (iii) Theorem: [CPTVV] Let X be a derived stack, locally of f.p. and let ω be an n-shifted symplectic structure on X. Assume: ω is exact, i.e. rωs 0 P H DR px q; px,ωq is equipped with an isotropic foliation pl,hq. Then there exists a shifted function f : rx {L s Ñ A 1 rn `1s, and a symplectic map s : X Ñ Rcritpf q of n-shifted symplectic stacks, i.e. s ω Rcritpf q ω. Moreover, if pl,hq is Lagrangian, then s is étale.