Constructing shifted symplectic structures part 1Derived algebraic geometry Derived stacks (i) There is a category of derived schemes such that: It contains the category of schemes.

Constructing shifted symplectic structures

part 1

Tony Pantev

University of Pennsylvania

Geometry and Analysis of Moduli SpacesImperial College, January 6-10, 2020

Tony Pantev University of Pennsylvania

Universal symplectic structures

Outline

Outline

joint with Dima Arinkin and Bertrand Toen

shifted symplectic geometry

relative shifted symplectic structures



Derived algebraic geometry

Intrinsic geometry of moduli

Observation: Moduli spaces, when constructed as algebraicvarieties, or schemes, or algebraic spaces, or analytic spaces,tend to be pathological.

Better setting: Use derived algebraic stacks as parameterspaces in a moduli problem. Leads to nicer solutions of themoduli problem:

less singular,

easier to describe.

The price to pay: Heavy technology relying on highercategory theory, homotopical algebra, etc.




Derived stacks (i)

There is a category of derived schemes such that:

It contains the category of schemes.

Has fibered products and for affinesSpecpAq ˆSpecpCq SpecpBq “ SpecpA bL

C Bq,where A bL

C B is the derived tensor product: acommutative dg-algebra.

Note:

H0pA bLC Bq – A bC B .

H‚pA bLC Bq controlls the defect of transversality (excess

intersection).




Derived stacks (ii)

Definition: A derived scheme (over C) is a pair X “ pX ,O‚X

q,consisting of a topological space X and a sheaf O‚

Xof commu-

tative non-positively graded C-dg-algebras such that

the ringed space t0X “ pX ,H 0pO‚X

qq is a scheme;

for all i ď 0 the sheaves H ipO‚X

q are quasi-coherentH 0pO‚

Xq-modules.




Derived stacks (ii)



Xof commu-



qq is a scheme;



Xq-modules.

Intuition: X is an infinitesimal thickening of the ordinaryscheme t0X with extra nilpotent functions sitting in higherhomological degrees.




Derived stacks (ii)



Xof commu-



qq is a scheme;



Xq-modules.

Definition: An algebraic derived stack is a quotient of a derivedscheme by the action of a smooth groupoid.

Note: Derived stacks form a complicated 8-category dStC.




Tangent complex

X P dStC, x : SpecpCq Ñ X a point

˜

Stalk TX ,x ofthe tangentcomplex

¸

“

¨

˝

normalized chain complexof the homotopy fiber ofX pCrεsq Ñ X pCq over x

˛

‚

simplicial abeliangroup




Tangent complex

X P dStC, x : SpecpCq Ñ X a point

˜

Stalk TX ,x ofthe tangentcomplex

¸

“

¨

˝

normalized chain complexof the homotopy fiber ofX pCrεsq Ñ X pCq over x

˛

‚

When X is a moduli stack:

H´1pTX ,xq “ infinitesimal automorphisms of x ;

H0pTX ,xq “ infinitesimal deformations of x ;

H1pTX ,xq Ě obstructions of x .




Examples (i):

X “ BG “ rpt G s ñ TX ,pt “ gr1s.

X “ moduli of vector bundles E on a smooth projectiveY ñ TX ,E “ RΓpY ,EndpE qqr1s.

X “ moduli of maps f from C to Y ñTX ,f “ RΓpC , f ˚TY q.

X “ moduli of local systems E on a compact manifold Y

ñ TX ,E “ RΓpY ,EndpEqqr1s.




Examples (ii):

X “ derived intersection

L1

hą

M

L2 “

˜

L1 X L2, OL1

Lâ

OM

OL2

¸

of smooth subvarieties L1, L2 Ă M in a smooth M ñ

TX ,x “ r TL1,x ‘ TL2,x// TM,x s,

0 1

H0pTX ,xq “ TL1XL2,x ;H1pTX ,xq “ failure of transversality.




Examples (iii):Special case: X “ derived zero locus Rzeropsq ofs P H0pL,E q.





an algebraic vector bundleon a smooth variety L





Thus

X “ L

hą

M

L “`

Z , i´1L pSym‚pE_r1sq, s5q

˘

,

where:

Z “ t0X “ zeropsq is the scheme theoretic zero locus ofs,

iL : Z Ñ L is the natural inclusion, and

s5 is the contraction with s.




Examples (iv):In particular TX “ r i˚LTL ‘ i˚LTL

i˚Ldoì˚

Lds

// i˚MTM s,

0 1where

M “ totpE q, and

iM , o, and s are the natural maps Lo

Z

iL @@

iL

iM // M .

Ls

>>⑤⑤⑤⑤




Examples (iv):In particular TX “ r i˚LTL ‘ i˚LTL

i˚Ldoì˚

Lds

// i˚MTM s,

0 1where

M “ totpE q, and

iM , o, and s are the natural maps

Remark: There is a natural quasi-isomorphism

TX “

„

i˚LTL

p∇sq5

// i˚LE

“

„

TL

p∇sq5

//E

|Z

.

Note: ∇ : E Ñ E b Ω1L is an algebraic connection which

exists only locally and is not unique. However p∇sq|Z is welldefined globally and independent of the choice of ∇.




Cotangent complexA P cdgaC, X “ SpecpAq P dStC,QA Ñ A a cofibrant (quasi-free) replacement

ˆ

cotangent complexLX “ LA

˙

“

ˆ

Kahler differentialsΩ1

QA of QA

˙

If X P dStC is a general derived Artin stack, thenX “ hocolimtSpecA Ñ Xu (in the model category dStC) and

LX “ holimSpecAÑX LA

Note:

LX P LqcohpX q - the dg category of quasi-coherent OX

modules.

X is locally of finite presentation iff LX is perfect. In thiscase TX “ L_

X “ HompLX ,OX q.



Forms and closed forms

p-forms

A P cdgaC, X “ SpecpAq P dStC,QA Ñ A a cofibrant (quasi-free) replacement. Then:

‘pě0

Źp

A LA “ ‘pě0ΩpQA - a fourth quadrant bicomplex with

vertical differential d : Ωp,iQA Ñ Ωp,i`1

QA induced by dQA, and

horizontal differential dDR : Ωp,iQA Ñ Ωp`1,i

QA given by the deRham differential.

Hodge filtration: F qpAq :“ ‘pąqΩpQA: still a fourth

quadrant bicomplex.




(shifted) closed p-forms

Motivation: If X is a smooth scheme/C, thenΩp,cl

X –`

ΩěpX rps, dDR

˘

. Use pΩěpX rps, dDRq as a model for

closed p forms in general.

Definition:

complex of closed p-forms on X “ SpecA:Ap,clpAq :“ tot

ś

pF ppAqqrps

complex of n-shifted closed p-forms onX “ SpecA: Ap,clpA; nq :“ tot

ś

pF ppAqqrn ` ps

Hodge tower:¨ ¨ ¨ Ñ Ap,clpAqr´ps Ñ Ap´1,clpAqr1 ´ ps Ñ ¨ ¨ ¨ Ñ A0,clpAq




(shifted) closed p-forms (ii)

Explicitly an n-shifted closed p-form ω on X “ SpecA is aninfinite collection

ω “ tωiuiě0 , ωi P Ωpì ,níA

satisfyingdDRωi “ ´dωi`1.

Note: The collection tωiuiě1 is the key closing ω.




p-forms and closed p-formsNote:

The complex A0,clpAq of closed 0-forms on X “ SpecAis exactly Illusie’s derived de Rham complex of A.

There is an underlying p-form map

Ap,clpA; nq Ñľp

LAkrns

inducing

H0pAp,clpAqrnsq Ñ HnpX ,ľp

LAkq.

The homotopy fiber of the underlying p-form map can bevery complicated (complex of keys): being closed is nota property but rather a list of coherent data.




Functoriality and gluing:

Globally we have:

the n-shifted p-forms 8-functorApp´; nq : cdgaC Ñ SSets : A ÞÑ |Ωp

QArns |, and

the n-shifted closed p-forms 8-functorAp,clp´; nq : cdgaC Ñ SSets : A ÞÑ |Ap,clpAqrns |.

Note: App´; nq and Ap,clp´; nq are derived stacks for theetale topology. underlying p-form map (of derived stacks)

Ap,clp´; nq Ñ A

pp´; nq

Notation: | ´ | denotes MapC´dgModpC,´q “ DKτď0p´q i.e.Dold-Kan applied to the ď 0-truncation.




global forms and closed forms (i)

Fix a derived Artin stack X (locally of finite presentation C)

Definition:

AppX q :“ MapdStCpX ,App´qq - space of p-forms onX ;

Ap,clpX q :“ MapdStCpX ,Ap,clp´qq - space of closedp-forms on X ;

n-shifted versions : AppX ; nq :“ MapdStCpX ,App´; nqqand Ap,clpX ; nq :“ MapdStCpX ,Ap,clp´; nqq

an n-shifted (respectively closed) p-form on X is anelement in π0A

ppX ; nq (respectively in π0Ap,clpX ; nq)




global forms and closed forms (ii)

Note:

1) If X is a smooth scheme there are no negatively shiftedforms.

2) When X “ SpecA then there are no positively shiftedforms.

3) For general X shifted forms may exist for any n P Z.





underlying p-form map (of simplicial sets)

Ap,clpX ; nq Ñ A

ppX ; nq

not a monomorphism for general X , its homotopy fiber ata given p-form ω0 is the space of keys of ω0.

If X is a smooth and proper scheme then this map isindeed a mono (homotopy fiber is either empty orcontractible) ñ no new phenomena in this case.





Theorem (PTVV): for X derived Artin, then forms satisfysmooth descent:

AppX ; nq » MapLqcohpX qpOX , p

ľpLX qrnsq.

In particular: an n-shifted p-form on X is an element inHnpX ,

ŹpLX q




global forms and closed forms (iii)

Remark: If A P cdga is quasi-free, and X “ SpecA, then

Ap,clpX ; nq “

ˇ

ˇ

ˇ

ˇ

ˇ

ź

iě0

`

Ωp`1A rn ´ i s, d ` dDR

˘

ˇ

ˇ

ˇ

ˇ

ˇ

“ˇ

ˇtotΠpF ppAqqrnsˇ

ˇ

“ |NC pAqppqrn ` ps|






Ap,clpX ; nq “

ˇ

ˇ

ˇ

ˇ

ˇ

ź

iě0

`


˘

ˇ

ˇ

ˇ

ˇ

ˇ

“ˇ


ˇ


negative cyclic com-plex of weight p






Ap,clpX ; nq “

ˇ

ˇ

ˇ

ˇ

ˇ

ź

iě0

`


˘

ˇ

ˇ

ˇ

ˇ

ˇ

“ˇ


ˇ


Hence

π0Ap,clpX ; nq “ HC

n´p´ pAqppq.




global forms and closed forms (iv)Definition: Given a higher Artin derived stack the n-th alge-braic de Rham cohomology of X is defined to be Hn

DRpX q “π0A

0,clpX ; nq.

Remark:

agrees with Illusie’s definition in the affine case.

if X is a higher Artin derived stack locally f.p., thenH‚

DRpX q – H‚DRpt0X q “ algebraic de Rham cohomology

of the underived higher stack t0X defined byHartschorne’s completion formalism.




global forms and closed forms (iv)Definition: Given a higher Artin derived stack the n-th alge-braic de Rham cohomology of X is defined to be Hn

DRpX q “π0A

0,clpX ; nq.

Remark:

agrees with Illusie’s definition in the affine case.

if X is a higher Artin derived stack locally f.p., thenH‚

DRpX q – H‚DRpt0X q “ algebraic de Rham cohomology

of the underived higher stack t0X defined byHartschorne’s completion formalism.

Corollary: Let X be a locally f.p. derived stack and let ω bean n-shifted closed p-form on X with n ă 0. Then ω is exact,i.e. rωs “ 0 P H

n`pDR pX q.




Examples (i):(1) If X “ SpecpAq is an usual (underived) smooth affinescheme, then

Ap,clpX ; nq “ pτďnp Ωp

A

dDR // Ωp`1A

dDR // ¨ ¨ ¨

0 1

qqrns,

and hence

π0Ap,clpX ; nq “

$

’

&

’

%

0, n ă 0

Ωp,clA , n “ 0

Hn`pDR pX q, n ą 0

Remark: The standard notation for forms is inadequate - ifX “ Cˆ, then dzz P π0A

1,clpX ; 0q and alsodzz P π0A

0,clpX ; 1q.Tony Pantev University of Pennsylvania



Examples (ii):

(2) If X is a smooth and proper scheme, thenπiA

p,clpX ; nq “ F pHn`píDR pX q.

(3) If X is a (underived) higher Artin stack, and X‚ Ñ X is asmooth affine simplicial groupoid presenting X , thenπ0A

ppX ; nq “ HnpΩppX‚q, δq with δ “ Cech differential.In particular if G is a complex reductive group, then

π0AppBG ; nq “

#

0, n ‰ p

pSym‚g

_qG , n “ p.




Examples (iii):

(4) Similarly

Ap,clpBG ; nq “

ˇ

ˇ

ˇ

ˇ

ˇ

ź

iě0

`

Sympìg

_˘G

rn ` p ´ 2i s

ˇ

ˇ

ˇ

ˇ

ˇ

,

and so

π0Ap,clpBG ; nq “

#

0, if n is odd

pSympg

_qG , if n is even.




Examples (iv):(5) If X “ Rzeropsq for s P H0pL,E q on a smooth L, then

Ω1X “ E_

|Z

p∇sq5

// Ω1L|Z ,

´1 0

and if we choose ∇ local flat algebraic connection on E wecan rewrite Ω1

X as a module over the Koszul complex:

¨ ¨ ¨ // ^2E_ b Ω1L

s5// E_ b Ω1

Ls5

// Ω1L

// Ω1L|Z 0

¨ ¨ ¨ // ^2E_ b E_ s5//

OO

E_ b E_ s5//

OO

E_ //

r∇,s5s

OO

E_|Z

p∇sq5

OO

´1




Examples (v):In the same way we can describe Ω2

X as a module over theKoszul complex

¨ ¨ ¨ // ^2E_ b Ω2L

// E_ b Ω2L

// Ω2L

// Ω2L|Z

¨ ¨ ¨ // ^2E_ b E_ b Ω1L

//

OO

E_ b E_ b Ω1L

//

OO

E_ b Ω1L

//

OO

pE_ b Ω1Lq|Z

OO

¨ ¨ ¨ // ^2E_ b S2E_ //

OO

E_ b S2E_ //

OO

S2E_ //

OO

S2E_|Z

OO






¨ ¨ ¨ // ^2E_ b Ω2L

// E_ b Ω2L

// Ω2L

// Ω2L|Z

¨ ¨ ¨ // ^2E_ b E_ b Ω1L

//

OO

E_ b E_ b Ω1L

//

OO

E_ b Ω1L

//

OO

pE_ b Ω1Lq|Z

OO

¨ ¨ ¨ // ^2E_ b S2E_ //

OO

E_ b S2E_ //

OO

S2E_ //

OO

S2E_|Z

OO

2 forms of degree´1






¨ ¨ ¨ // ^2E_ b Ω2L

// E_ b Ω2L

// Ω2L

// Ω2L|Z

¨ ¨ ¨ // ^2E_ b E_ b Ω1L

//

OO

E_ b E_ b Ω1L

//

OO

E_ b Ω1L

//

OO

pE_ b Ω1Lq|Z

OO

¨ ¨ ¨ // ^2E_ b S2E_ //

OO

E_ b S2E_ //

OO

S2E_ //

OO

S2E_|Z

OO

Note: The de Rham differnetial dDR : Ω1X Ñ Ω2

X is the sumdDR “ ∇ ` κ, where κ is the Koszul contraction

κ : âE_ b SbE_ Ñ â´1E_ b Sb`1E_.




Examples (vi):

Lemma [Behrend] If E “ Ω1L and so s is a 1-form, then a

2-form of degree ´1 corresponds to a pair of elements

α P pΩ1Lq_bΩ2

L and φ P pΩ1Lq_bΩ1

L such that r∇, s5spφq “ s5pαq.

Take φ “ id P pΩ1Lq_ b Ω1

L. Suppose the local ∇ is chosen sothat ∇pidq “ 0 (i.e. ∇ is torsion free). Then r∇, s5spidq “ ds.

Conclusion: The pair pα, idq gives a 2-form of degree ´1 iffds “ s5pαq, or equivalently ds|Z “ 0, i.e. is an almost closed1-form on L.



Shifted symplectic geometry

Symplectic structures

Recall: For X a smooth scheme/C is a symplecticstructure is an ω P H0pX ,Ω2,cl

X q such that its adjointω5 : TX Ñ Ω1

X is a sheaf isomorphism.

Note: Does not work for X singular (or stacky or derived):

TX and Ω1X are too crude as invariants and get promoted

to complexes TX and LX .

A form being closed is not just a condition but rather anextra structure.




Definition: X derived Artin stack locally of finite presentation(so that LX is perfect).

A n-shifted 2-form ω : OX Ñ LX ^ LX rns - i.e.ω P π0pA

2pX ; nqq - is nondegenerate if its adjointω5 : TX Ñ LX rns is an isomorphism (in DqcohpX q).

The space of n-shifted symplectic forms SymplpX ; nq onXC is the subspace of A2,clpX ; nq of closed 2-formswhose underlying 2-forms are nondegenerate i.e. we havea homotopy cartesian diagram of spaces

SymplpX , nq //

A2,clpX , nq

A2pX , nqnd // A2pX , nq




Shifted symplectic structures: examples (i)

Nondegeneracy: a duality between the stacky (positivedegrees) and the derived (negative degrees) parts of LX .

G “ GLn BG has a canonical 2-shifted symplectic formwhose underlying 2-shifted 2-form is

k Ñ pLBG ^ LBG qr2s » pg_r´1s ^ g_r´1sqr2s “ Sym2

g_

given by the dual of the trace map pA,Bq ÞÑ trpABq.

Same as above (with a choice of G -invariant symm bil formon g) for G reductive over k .

The n-shifted cotangent bundleT_X rns :“ SpecX pSympTX rńsqq has a canonical n-shiftedsymplectic form.




Shifted symplectic structures: examples (ii)Theorem: [PTVV] Let F be a derived Artin stack equippedwith an n-shifted symplectic form ω P SymppF , nq. Let X bean O-compact derived stack equipped with an O-orientationrX s : HpX ,OX q ÝÑ kr´ds of degree d . If the derived map-ping stack MappX , F q is a derived Artin stack locally of fi-nite presentation over k, then, MappX , F q carries a canonicalpn ´ 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 structureof dimension d ;

2) A compact oriented topological d -manifold has anO-orientation of degree d (Poincare duality).




Lagrangian structures

Let pY , ωq be a n-shifted symplectic derived stack. Alagrangian structure on a map f : X Ñ Y is

path γ in A2,clpX ; nq from f ˚ω to 0

that is ’non-degenerate’ (in a suitable sense), i.e. theinduced map θγ : Tf Ñ LX rn ´ 1s is an equivalence.

Examples:

usual smooth lagrangians L ãÑ pY , ωq where pY , ωq is asmooth (0)-symplectic scheme.

there is a bijection between lagrangian structures on thecanonical map X Ñ pSpec k, ωn`1q and n-shiftedsymplectic structures on X (thus lagrangian structuresgeneralize shifted symplectic structures)




Shifted symplectic structures: examples (iii)

Theorem: [PTVV] Let pF , ωq be n-shifted symplectic derivedArtin stack, and Li Ñ F a map of derived stacks equippedwith a Lagrangian structure, i “ 1, 2. Then the homotopyfiber product L1 ˆF L2 is canonically a pn´ 1q-shifted derivedArtin stack.

In particular, if F “ Y is a smooth symplecticDeligne-Mumford stack (e.g. a smooth symplectic variety),and Li ãÑ Y is a smooth closed lagrangian substack, i “ 1, 2,then the derived intersection L1 ˆF L2 is canonicallyp´1q-shifted symplectic.




Remark: An interesting case is the derived critical locusRCritpf q for f a global function on a smooth symplecticDeligne-Mumford stack Y . Here

RCritpf q //

Y

df

Y

0// T_Y



Diagrams, sheaves, and forms

Sheaves of derived stacks

Problem: Define and construct symplectic structures on thestalks of a sheaf of derived stacks F over a space S .

Note:

For this to make sense the sections of the sheaf F willhave to satisfy representability conditions.

The non-degeneracy condition on a stalkwise symplecticform will have to involve some notion of duality forcomplexes of sheaves of C-vector spaces on S .




Diagrams of sheaves and stacksFix

‚ I - a finitely presentable 8-category;‚ C - any category with finite limits.

Notation:

F‚ - a diagram of shape I in C .






Notation:

F‚ - a diagram of shape I in C .

a functor F‚ : I Ñ C ,F‚ “ tFα|α P Iu






Notation:

F‚ - a diagram of shape I in C .FI “ lim

αPIFα - global sections of the diagram F‚.






Notation:



Itw - the category of twisted arrows in I.

obpItw q: maps xγ //y P morpI q;

HomItw

˜

x1γ1

y1

, x2γ2

y2

¸

: commutative diagramsx1

γ1

x2uoo

γ2y1 v

// y2






Notation:



Itw - the category of twisted arrows in I.pt, sq : Itw Ñ I ˆ Iop - the natural functor.




Closed forms on diagrams of stacks (i)

Fix E‚ : I //VectC , and F‚ : I //dStC .

Consider the functor

Ap,clI pF‚qE‚ : Itw // VectC

γ // Ap,clpFspγqq b Etpγq,

and define the complex of closed p-forms on F‚ withvalues in E‚ as the complex

Ap,clI pF‚qE‚

“ limγPItw

Ap,clpFspγqq b Etpγq.




Closed forms on diagrams of stacks (iii)

The space of closed p-forms on F‚ with values in E‚ isdefined as

Ap,clI pF‚qE‚ “

ˇ

ˇ

ˇDK

´

τď0A

p,clI pF‚qE‚

¯ˇ

ˇ

ˇ

and an E‚-valued closed p-form on F‚ is an element inπ0A

p,clI pF‚qE‚ “ H0pAp,cl

I pF‚qE‚q.

Note: The space of forms comes equipped with a naturalglobal sections morphism

Γ : Ap,clI pF‚qE‚

Ñ Ap,clpFIq b EI.




Cospecialization and gluing (i)

Suppose

X - nice topological space (e.g. a CW complex);

ı : Z ãÑ X - closed subspace;

: U ãÑ X - the complementary open subspace.

C -an 8-category with all small limits and colimits.

For any F P ShpX ,C q write F|Z “ ı˚F and F|U “ ˚F .Applying ı˚ to the unit of the adjunction ˚ % ˚ yields acospecialization map in ShpZ ,C q:

cospZ : F|Z Ñ ı˚˚

`

F|U

˘

.




Cospecialization and gluing (ii)

The assignment F ÝÑ`

F|U , F|Z , cospZ˘

provides anequivalence of 8-categories:

ShpX ,C q–

ÝÑ laxop lim

„

ShpU,C qı˚˚ //ShpZ ,C q

,

i.e. ShpX ,C q can be viewed as the laxop limit of the functorı˚˚.

Key observation: Applying this gluing description to stratain a stratification leads to a combinatorial picture for closedforms and symplectic structures on constructible sheaves ofstacks over a space.




Sheaves on a stratified space (i)

Suppose

‚ X is a good stratified space .

‚ I is the finite poset labeling the strata of X .

‚ Xα Ă X - the stratum labeled by α P I .

‚ ShstrpX q - sheaves F of spaces, constructible for the givenstratification.

‚ Fα P ShstrpXαq - the restriction of F to Xα.




Sheaves on a stratified space (ii)

Construction: Fix α P I , F P ShstrpX q, and let

‚ Z Ă X - a closed subset s.t. Xα Ă Z is open.

‚ U Ă X - the complementary open to Z .

‚˝Fα :“

`

ı˚˚

`

F|U

˘˘

|Xα

.

‚ cospα : Fα Ñ˝Fα - the restriction of cospZ to Xα.

Note:˝Fα and cospα depend only on α and not on Z .˝Fα is the sheaf of nearby (co) cycles of F along Xα, andcospα is the integral of cospecialization maps over nearbypoints.




Sheaves on a stratified space (iii)

Note: As with gluing, nearby cycles, and integratedcospecializations make sense for constructible sheaves withvalues in any category C that admits finite limits.

Let I be the 8-category of exit paths for the stratificationon X . Then

If I is good, then I is finitely presentable.

ShstrpX q “ FunpI, SSetsq.

For any category C with finite products we define

ShstrpX ,C q “ FunpI,C q,

i.e. C -valued constructible sheaves on X are I-shapeddiagrams in C .




Sheaves on a stratified space (iii)

Notation: Fix F P ShstrpX ,C q. Then:

Every x P X gives an object ιpxq P I. The valueF pιpxqq P C is called the stalk of F at x and is denotedby Fx .

ΓpX , F q :“ limσPI Fσ P C is the global sections object ofF . We have a natural evaluation mapevx : ΓpX , F q Ñ Fx for every x P X .

For α P I the fundamental groupoid Π1pXαq of thestratum Xα embeds in I as the full subcategory Iα Ă Ispanned by ιpxq for all x P Xα. We define the value of Fon Xα as Fα :“ F|Iα P ShstrpXα,C q.




Sheaves on a stratified space (iv)

Notation: Fix α P I , and F P ShstrpX ,C q. Then consider:

Iěα “ ta1 Ñ a | a1 P I, a P Iα u

Iąα Ă Iěα - the full subcategory consisting of all a1 Ñ a

which are not isomorphisms.˝Fα P ShstrpXα,C q - the right Kan extension of F|Ią

α

alongthe natural functor Iąα Ñ Iα.

cospα : Fα Ñ˝Fα - the map guaranteed by the universal

property of the right Kan extension.

Terminology:˝Fα is the sheaf of nearby cycles of F at Xα,

and cospα is the (integrated) cospecialization map.




Sheaves on a stratified space (v)

Given a nice stratified space X with strata labeled by a posetI , and an exit path category I, andE P ShstrpX ,VectCq “ FunpI,VectCq,F P ShstrpX , dStCq “ FunpI, dStCq, we get

a complex Ap,clX pF qE :“ A

p,clI pF‚qE‚ and a space

Ap,clX pF qE :“ Ap,cl

I pF‚qE‚ of global closed E -valuedp-forms on F ;

a natural global sections map of complexes

Γ : Ap,clX pF qE Ñ A

p,clpΓpX ,F qq b ΓpX ,E q.




Sheaves on a stratified space (vi)

Let F be a constructible sheaf of locally f.p. derived Artinstacks. For any point z P ΓpX ,F q the relative tangentcomplex TF ,z P ShstrpX ,VectCq is a constructible complex ofvector spaces on X .

Given a point z P ΓpX ,F q, any closed form ω P Ap,clX pF qE

defines a map of constructible complexes on X

ω5z :

pľ

TF ,z Ñ E .

Definition: The induced map Γpω5zq on global sections is

the value of ω at z .




Sheaves on a stratified space (vi)

Note: All of this readily sheafifies:

For reasonable open sets U Ă X the assignment

U ÝÑ Ap,clU pF|UqE|U

is a presheaf of complexes. Its sheafification Ap,clX pF qE is

a sheaf of complexes on X equipped with a natural map

Γ : ΓpX ,Ap,clX pF qE q Ñ A

p,clpΓpX ,F qq b ΓpX ,E q.

In terms of diagrams Ap,clX pF qE P ShstrpX ,VectCq is

constructed as the right Kan extension ofA

p,clI : Itw Ñ VectC along the functor t : Itw Ñ I.




Non-degeneracy (v)

Setup:

X - nicely stratified space with equidimensional strata.

F P ShstrpX , dStCq is a constructible sheaf of locally f.p.derived Artin stacks (or just locally formally representablederived stacks).

E “ KX rns P ShstrpX ,VectCq, where n P Z and KX is theVerdier’s dualizing complex of X .




Non-degeneracy (vi)Definition:

(a) The complex and space of relative n-shifted closedp-forms on F are defined to be

Ap,clX pF , nq :“ A

p,clX pF qKX rns,

Ap,clX pF , nq :“ Ap,cl

X pF qKX rns.

(b) A closed relative n-shifted 2-form ω P Ap,clX pF , nq is

symplectic if for every U Ă X and any pointz P ΓpU,F q, the map

ω5z : TF|U ,z

//LF|U ,z b KUrns

is a quasi-isomorphism.




Non-degeneracy and nearby cycles (i)

Let F P ShstrpX , dStCq, E “ KX rns, α P I .

Fα and˝

Fα are local systems of derived stacks on Xα, i.e.are derived stacks equipped with an action ofΠ1pXαq “ Iα.

The fiber of cospEα : Eα Ñ˝Eα is equal to the !-restriction

of KX rns to Xα and so we have an exact triangle

(˚) Eα

cospEα //

˝Eα

//C rn ` 1 ` dimXαs .

Note: When α P I is maximal we have˝Eα “ 0 and so

Eα “ C rn ` dimXαs.




Non-degeneracy and nearby cycles (ii)

Let ω P A2,clX pF , nq “ A2,cl

X pF qE be a relative n-shifted closed2-form on F . Then ω induces

ωα a relative closed Eα-valued 2-form on Fα, i.e.ωα P A2,cl

Xα

pFαqEα.

˝ωα a closed

˝Eα-valued 2-form on

˝Fα, i.e.

˝ωα P A2,cl

Xα

p˝

Fαq ˝Eα

.

ωα a closed pn ` 1q-shifted 2-form on˝

Fα, i.e.

ωα P A2,clXα

p˝

Fα, n ` 1q “ A2,clXα

p˝

FαqCrn`1`dimXαs.




Non-degeneracy and nearby cycles (iii)

Note:

ωα is the pushout of ˝ωα by the map

˝Eα Ñ C rn ` 1 ` dimXαs .

Viewing˝

Fα as a constant derived Artin stack equippedwith a Π1pXαq action, then we can view ωα equivalentlyas an absolute pn ` 1 ` dimXαq-shifted 2-form, i.e.

ωα P A2,clp

˝Fα, n ` 1 ` dimXαq.




Non-degeneracy and nearby cycles (iv)

Key observation: Consider the Π1pXαq-equivariantcospecialization map

cospF

α : Fα Ñ˝

Fα

of derived Artin stacks.The exact triangle

Eα

cospEα //

˝Eα

//C rn ` 1 ` dimXαs .

yields a natural path hα between cospF˚α pωαq and 0 in the

space A2,clXα

pFα, n ` 1q or equivalently a path betweencospF˚

α pωαq and 0 in the space A2,clpFα, n ` 1 ` dimXαq.




Non-degeneracy and nearby cycles (v)

Theorem: [Arinkin-P-Toen] Suppose

F P ShstrpX , dStCq - a constructible sheaf of derived Artinstacks, locally of f.p.;

ω P A2,clX pF , nq - a relative closed n-shifted 2-form on F .

Then ω is symplectic if and only if for every α P I the followingtwo conditions hold:

(a) ωα is symplectic.

(b) cospFα : pFα, hαq Ñ p

˝Fα, ωαq is Lagrangian.

Claim: [Arinkin-P-Toen] Fix β P I and assume that(a) and (b) hold for all α ą β. Then ωβ is a shifted sym-plectic form on

˝Fβ.




Sheaves and functors (i)

X - locally compact Hausdorff space;U pX q - the poset of opens in X ;C - an 8-category which admits all small limits.

Definition: A C -valued sheaf on X is a functorF : U pX qop Ñ C satisfying the sheaf condition: for everyopen cover tUαu of an open set U the natural map

F pUq ÝÑ limÐÝV

F pV q

is an equivalence in C . Here the limit is taken over all opensubsets V Ă U which are contained in one of the Uα.




Sheaves and functors (ii)Remark: Equivalently a C -valued functor F : U pX qop Ñ C

is a sheaf if

For every sequence U0 Ă U1 Ă ¨ ¨ ¨ Ă Uk Ă ¨ ¨ ¨ of opensin X the natural map F pYUi q Ñ lim

ÐÝi

F pUi q is an

equivalence in C .The object F p∅q is terminal in C .For every pair of opens U,V Ă X the diagram

F pU Y V q //

F pUq

F pV q // F pU X V q

is a pullback square in C .

BackTony Pantev University of Pennsylvania


Stratifications

Stratifications (i)

Let I be a poset viewed as a topological space for the upwardclosed topology: U Ă I is open if and only if x P U impliesy P U for all y ě x in I .

Definition: An I -stratification of a topological space X is acontinuous map a : X Ñ I . The stratum in X correspondingto α P I is the subset Xα “ a´1pαq.

Note: We will only be working with stratifications that satisfya regularity condition.



Stratifications

Stratifications (ii)

Notation: If I is a poset, write IŸ for the poset obtainedfrom I by adjoining a new smallest element ´8.

Definition: Let a : X Ñ I be a stratified space. Thecone over X is the IŸ-stratified space C pX q defined as follows:

As a set C pX q “ t˚uŮ

pX ˆ Rą0q.

A subset U Ă C pX q is open if and only if U X pX ˆ Rą0qis open, and if ˚ P U, then X ˆ p0, ǫq Ă U for somepositive real ǫ.

C pX q is stratified by the map aŸ : C pX q Ñ IŸ given byaŸp˚q “ ´8, and aŸpx , tq “ apxq for px , tq P X ˆ Rą0.



Stratifications

Stratifications (iii)

Definition: [Lurie] An I -stratification on X is conical if everypoint x P Xα Ă X has a stratified neighborhood Ux which isstratified homeomorphic to a product Z ˆ C pY q where Z is atopological space, and Y is a Iăα-stratified space.

Definition: An I -stratification on X is good if I is finite andevery point x P Xα Ă X has a stratified neighborhood Ux whichis stratified homeomorphic to a product Rk ˆ C pY q for some k

and some compact Iăα-stratified space Y .

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Exit paths

Exit paths (i)

Setup:

X be a stratified space with strata labeled by a poset I .

|∆n| “!

pt0, . . . , tnq P r0, 1sˆn

ˇ

ˇ

ˇ

ř1

i“0 “ 1)

is the

standard simplex.

Definition: The simplicial set of exit paths of X is thesimplicial subset I Ă SingpX q consisting of those simplicesσ : |∆n| Ñ X that satisfy the condition

(˚)

˜ There exists a chain of elements α0 ă α1 ă ¨ ¨ ¨ ă αn P I

so that for every point pt0, t1, . . . , ti , 0, . . . , 0q P |∆n|

with ti ‰ 0 we have that σpt0, t1, . . . , ti , 0, . . . , 0q P Xαi.

¸



Exit paths

Exit paths (ii)

Theorem: [Lurie]

(a) If the stratification on X is conical, then I is an8-category.

(b) Let X be a paracompact topological space which is locallyof a singular shape, and is equipped with a conicalI -startification. Then the 8-category of I -constructiblesheaves of spaces on X is equivalent to the 8-categoryFunpI, SSetsq.

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Constructing shifted symplectic structures part 1Derived algebraic geometry Derived stacks (i) There is a category of derived schemes such that: It contains the category of schemes.

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