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Home , Sheaf of modules

COLUMBIA SEMINAR FALL 2009

Abstract. These are notes on non-commutative theory which plagiarize, to a large extent, the lectures of Ginzburg [G]. The original aim was to study D-modules.

Contents 1. Introduction 1 1.1. Differential Operators 3 2. Sheaves of non-commutative rings 3 2.1. Rees ring 4 2.2. Characteristic variety 4 2.3. Gabber filtration 5 2.4. Poisson structures 6 2.5. Some deformation theory 6 2.6. Hochschild cohomology 7 2.7. (super) Gerstenhaber algebras 8 3. Fedosov quantization 9 3.1. Notation and assumptions 10 3.2. Symplectic deformations 10 3.3. Non-commutative deformations 11 4. More sheaves of non-commutative rings 11 4.1. Gabber’s Theorem 11 4.2. Commutative localization 12 4.3. Non-commutative localization 12 4.4. Duality and the Sato-Kashiwara filtration 13 5. Twisted Differential Operators 16 References 18

1. Introduction Let X be a smooth algebraic variety over C. An algebraic geometer would commonly study (quasi- coherent) sheaves of X -modules where X is the sheaf of algebraic functions on X (notice that X is a sheaf of rings). O O O A non-commutative algebraic geometer might be inclined to study sheaves of modules over a sheaf of non-commutative rings on X. One natural non-commutative sheaf of rings is X which is the sheaf of differential operators on X. Notice that is always a subring so anyD - is naturally OX ⊂DX DX an X -module. Example.O Let’s consider the variety X = A1 = Spec(C[x]). The ring of global sections Γ( ) is DX generated by x and ∂ := d . Now is naturally a -module so taking global sections Γ( )= C[x] x dx OX DX OX

Date: December 20, 2009. 1 2 D-MODULES NOTES is naturally a Γ( )-module. For f C[x] we see that DX ∈ (∂ x) f =(∂ )(xf)= f + x∂ f. x · · x x So we get the relation ∂xx =1+ x∂x. This is the only relation so we get = C(x,∂ )/(∂ x =1+ x∂ ) DX ∼ x x x which is the classical Weyl algebra W1. Example. Let’s consider the variety Y = P1. We think of A1 = P1 and use the same local coordinate x on A1. Then via restriction we have \ ∞ . Γ( ) ֒ Γ( ) = C(x,∂ )/(∂ x =1+ x∂ )= W DY → DX ∼ x x x 1 Now if we change local coordinates x 1/x so that we around looking locally around P1 then 2 7→ ∞ ∈ ∂x x ∂x then the subalgebra of the Weyl algebra which extend to regular is generated (as an algebra) by 7→ 2 (1) E := ∂x,F := x ∂x and H := 2x∂x. Exercise. Check that [E,F ]= H, [H,E] = 2E and [H,F ]= 2F . − We conclude that there is a map of algebras U(sl2) Γ( P1 ) where U(sl2) is the universal enveloping → D P1 C algebra of sl2. Why does this map exist? The reason is that ∼= G/B where G = SL2( ) and B is the standard Borel subgroup (upper triangular invertible matrices). So G acts transitively on P1 and given a vector v sl2 = TeG we can use the action of G to translate v and obtain a global vector field. In other words, we∈ have a map sl Γ(TP1 ) 2 → of Lie algebras which induces the map (1) above. Exercise. Find an element in the kernel of this map. Exercise. Show that if G is a group acting on X then it induces a map g Γ(TX ) of Lie algebras. Question. So why study modules over a sheaf of non-commutative rings?→

One reason is the construction above which generalizes to give a map U(g) Γ( G/P ) for • any reductive Lie algebra G and parabolic subgroup P . Consequently one gets→ aD map from G/P -modules to U(g) which is a very useful tool in geometric representation theory. DAnother reason is its relation to perverse sheaves. More precisely, inside the (derived) category • Dcon(X) of constructible sheaves on X there is a special t-structure. The heart of this t- structure is (by definition) the of perverse sheaves. The advantage of this abelian category (over the standard abelian category of constructible sheaves) is that it’s much better behaved under standard operations like Verdier duality and pushforwards. In particular, the decomposition theorem of Beilinson-Bernstein-Deligne says that the direct image of a perverse sheaf decomposes into a direct sum of perverse sheaves (with shifts). rh It turns out that there is a map, the DeRham functor DR : X mod PervX which is an equivalence of categories between (regular holonomic) -modulesD − and→ perverse sheaves. DX In particular, this means that (regular holonomic) X -modules are very well behaved under standard operations. This is remarkable since D-modules are not so well behaved – the OX pushforward of an X -sheaf under an arbitrary map can be an arbitrarily nasty complex of sheaves. O The simplest examples of -modules are locally free sheaves with a flat connection (E, ). • DX ∇ The connection is a map : E TX E which is the same as an action of TX on E. Flatness implies that ∇ =[ ⊗ , →] which is the necessary condition for to induce a ∇[θ1,θ2] ∇θ1 ∇θ2 ∇ X -module structure on E. D So, if V is a local system (for example, one that comes from a variation of Hodge structures) then V C is a -module. ⊗ OX DX D-MODULES NOTES 3

If one studies perverse sheaves in positive characteristic (using ´etale cohomology and so on) • then one there is also Deligne’s theory of weights. This theory does not exist over C. However, one can enhance the category of X -modules to the category of Hodge X -modules where such a theory of weights has beenD developed by Saito. Understanding someD of Saito’s theory is one (long term) hope for this seminar. 1.1. Differential Operators. Let U X an open subset of X. One can recursively define the sheaf (U) of differential operators on U as⊂ follows. A C-linear map D : (U) (U) is a differential DX OX → OX operator of order k if and only if [D,f] is a linear operator of order k 1 for any function f X (U). We denote by i the linear operators of order i. − ∈ O DX ≤ One can check that if D1 and D2 are differential operators of orders k1 and k2 then D1D2 has order at most k1 + k2 while [D1,D2] has order at most k1 + k2 1. This gives X the structure of a sheaf of filtered rings where the associated graded gr( ) is a sheaf− of graded commutativeD -algebras. DX OX For example, multiplication by any f X (U) is a linear operator of order zero, while any vector field induces a linear operator of order one.∈ O Proposition 1.1. 1 = T while gr( ) = Sym (T ). DX ∼ OX ⊕ X DX ∼ OX X Proof. (sketch) We have a natural map Sym (T ) gr( ) induced by T 1 . One can then OX X → DX X → DX show by induction that gri( ) = Symi (T ). (see Prop 5.2 of [GA]).  DX ∼ OX X Proposition 1.2. is generated as a ring by f and ζ T subject to the relations DX ∈ OX ∈ X f f = f f ,f ζ = fζ,ζ ζ ζ ζ =[ζ ,ζ ],ζ f f ζ = ζ(f) 1 ∗ 2 1 2 ∗ 1 ∗ 2 − 2 ∗ 1 1 2 ∗ − ∗ where is the multiplication in . ∗ DX Proof. (sketch) The idea is to construct a map of filtered algebras AX X where AX is the algebra described above. Then to show this map is an isomorphism it suffices→D to show that the induced map gr(A ) gr(D ) = Sym (T ) is an isomorphism of graded algebras (see Prop 5.3 of [GA]).  X → X ∼ OX X Remark 1.3. Notice that the definition of X from X works equally well to give a sheaf of differential operators starting from any sheaf of commutativeD rings.O For example, you can start from the sheaf of smooth functions to get the sheaf of smooth differential operators. One can also define X if X is not smooth. D 2 Talk. Describe the category of X -modules on X = A supported on the line x = 0. This should be D 2 the same as the category of DA1 -modules. Now describe the category of -modules on A supported DX on xy = 0 and compare with the category of Y -modules where Y is the variety xy = 0. Do these categories differ? D

2. Sheaves of non-commutative rings

Inspired by the description of X above we begin by studying general (sheaves of) non-commutative rings following closely the first partD of [G]. A· is a filtered ring if ...Ai Ai+1 ... with AiAj Ai+j and 1 A0 (also iAi = A). For simplicity we assume i N (more⊂ generally⊂ we can let i ⊂Z). gr(A ) denotes∈ the associated∪ graded ∈ ∈ · ring i(Ai/Ai−1). A· is called almost commutative if gr(A·) is commutative. Example.⊕ Let g be a Lie algebra. Then the universal enveloping algebra Ug = T g/(x y y x =[x,y]) ⊗ − ⊗ is a filtered algebra where g has degree one. The Poincare-Birkhoff-Witt theorem states that gr(Ug) ∼= Symg. So Ug is almost commutative. This can be seen quite directly since in gr(Ug) the relation x y y x =[x,y] becomes x y y x = 0 because [x,y] has degree one. ⊗ − ⊗ ⊗ − ⊗ 4 D-MODULES NOTES

Proposition 2.1. Let A be an almost commutative algebra. Suppose A0 = C and A is generated as a ring by A1. Then A ∼= Ug/I for some Lie algebra g and I.

Proof. Almost commutativity means that ab ba A1 if a,b A1. Thus A1 becomes a Lie algebra under the bracket [a,b] = ab ba. By the univeral− ∈ property of∈UA we have a map UA A which − 1 1 → is surjective since A is generated by A1 (see Prop 1.1.5 of [G]).  Proposition 2.2. If gr(A) is Noetherian then so is A (but not vice-versa). If gr(A) has no zero-divisors then A has no zero-divisors.

A filtered module M· over a filtered ring A· is a sequence of modules Mi Mi+1 ... such that A M M . We require that M = 0 for i 0. Define gr(M···⊂) := M⊂ /M ⊂to be the i · j ⊂ i+j i ≪ · ⊕i i i−1 associated graded gr(A·)-module. ˆ i 2.1. Rees ring. Define the Rees ring of A by A := P t Ai. If one thinks of this as a family of rings over A1 parametrized by t then the fibre over t = 0 is isomorphic to A while the fibre over t = 0 is isomorphic to gr(A). To see the latter note that there6 is a map gr(A) A/tˆ Aˆ → i i given by Ai/Ai−1 a¯i t ai. This map is well defined because t ai−1 = 0 on the right hand side if ∋ 7→ i ai−1 Ai−1. The inverse map is t ai a¯i. Example.∈ Consider A = Ug where7→g is some Lie algebra. Then Aˆ = T g/(x y y x = t[x,y]). ∼ ⊗ − ⊗ ˆ i Similarly, define M := P t Mi for any filtered A-module M. Again we have that the fibre over t = 0 ˆ ˆ is isomorphic to M while the fibre over t = 0 is isomorphic to M/tM ∼= gr(M). Thus one can think of filtered rings and modules as natural deformations of their associated graded. Remark 2.3. In what way is Aˆ a flat family of rings over A1? Aˆ is a C[t]-module via the natural map C[t] Aˆ. The module structure does not see the multiplicative structure on Aˆ and since all fibres of → Aˆ are isomorphic once you pass to the associated graded this should mean the family is flat. More precisely, Aˆ is a free C[t] module. Namely, it is isomorphic to C[t] C gr(A). To see this we i i ⊗ \ have an inclusion A0 C C[t] Aˆ given by t A0 ֒ t Ai. The cokernel is isomorphic to A/A0. Since ⊗ → → \ short exact sequence of C[t]-modules split we have Aˆ = A C C[t] A/A . Now you repeat. ∼ 0 ⊗ ⊕ 0 2.2. Characteristic variety. A filtration of M is good if gr(M) is a finitely generated gr(A)-module. From now on assume that A is almost commutative and that gr(A) is finitely generated as a C-algebra. Let M be an A-module equipped with a good filtration. Then gr(M) is a finitely generated gr(A)- module which means that gr(M) is a on Spec(gr(A)). We define the characteristic variety (or singular support) of M to be the support of gr(M) with its reduced structure. We denote this CV (M). For each component S of CV (M) we denote by mult(M,S) the multiplicity of S in the support of gr(M). Proposition 2.4 (J. Bernstein). Neither CV (M) nor mult(M,S) depend on the choice of a good filtration for M. Moreover, if 0 M ′ M M ′′ 0 → → → → is a short exact sequence of A-modules then mult(M,S)= mult(M ′,S)+ mult(M ′′,S). It is possible that you choose different good filtrations and obtain different non-reduced structures. Some properties of good filtrations: D-MODULES NOTES 5

suppose M ′ M is an A-submodule. Then a good filtration on M induces a good filtration • on M ′. ⊂ the induced filtration on M/M ′ is also good and there is a short exact sequence • 0 gr(M ′) gr(M) gr(M/M ′) 0 → → → → which implies CV (M)= CV (M ′) CV (M/M ′) ∪ if M = A/J then CV (M) is the zero variety of gr(J) • ∼ A1 C C Example. Let X = . Then X ∼= (x,∂x)/(∂xx = x∂x + 1) with grDX ∼= [x,∂x] where ∂x has degree one and x degree zero. ThenD C CV ( X ) ∼= Spec( [x,y]) where y has degree one and x degree zero. • Let UD = / (x λ)). Then U as a left -module is spanned over C by elements ∂i . • λ DX DX − λ DX x Now x ∂x = ∂xx = λ∂x. Thus CV (Uλ) is defined by x λ = 0. · ∼ − i Let Vλ = X / X (∂x λ). Then Vλ as a left X -module is spanned by elements x . Now • ∂ 1= ∂ D= λDso the− associated graded relationD is y = 0. Thus CV (V ) is defined by y = 0. x · x λ Notice that when λ = 0 X / X ∂x is the standard representation of the Weyl algebra (namely, the Weyl algebra actingD D on C[x]). For instance, ∂ (x2)=(x∂ + 1) x = x2∂ + x + x = 2x. x · x · x For general λ we get a deformed version of the standard representation where ∂x 1= λ and ∂ x = λx +1 and ∂ x2 = λx2 + 2x and so on. This action is the same as the action· induced x · x · by X X where x x and ∂x ∂x + λ. D →D 7→ 7→ i Let Mλ = X / X (x∂x λ). Then CV (Mλ) is given by xy = 0. The module is spanned by x • and ∂i (i DN).D Notice that− 1 generates the module so it is indecomposable. If λ Z then M x ∈ 6∈ λ is irreducible because one can take any element of Mλ and multiplying by x and ∂x and using relations x∂x = λ and ∂xx = 1 λ one can obtain 1. In particular, this gives an example of an irreducible A-module whose− CV is reducible. −λ −λ+1 Now if λ Z and λ 1 then x ,x ,... is invariant under the action of X . This is because ∂∈ x−λ = 0.≤− Thus we get{ a short exact} sequence D x · 0 V M Q 0. → 0 → λ → λ → −λ−1 i The quotient module Qλ is spanned by 1,...,x and ∂x with i 1. If λ 0 then ∂λ,∂λ+1,... is invariant and we get a short exact≥ sequence ≥ { x x } 0 U M Q 0 → 0 → λ → λ → where the quotient Q is spanned by 1,...,∂λ and xi with i 1. λ x ≥ 2.3. Gabber filtration. We assume Spec(gr(A)) is smooth and connected. For any finite generated (and hence Noetherian) A-module M define the Gabber filtration by G (M)= m M : dim(CV (A m)) i . i { ∈ · ≤ } This is the same as the largest A-submodule of M whose CV-support has dimension i. To see that G (M) M is indeed a submodule suppose m ,m G (M). Then A (m + m ) ≤A m + A m i ⊂ 1 2 ∈ i · 1 2 ⊂ · 1 · 2 so it suffices to show that dimCV (A m1 + A m2) i. But from the surjective map A m1 A m2 A m + A m it suffices to show dim· CV (A· m ≤A m ) i which follows by assumption.· ⊕ · → · 1 · 2 · 1 ⊕ · 2 ≤ Theorem 2.5. CV (Gi(M)/Gi−1(M)) is a variety of pure dimension i. 6 D-MODULES NOTES

2.4. Poisson structures. Recall that a Poisson algebra consists of a commutative, associative algebra with unit • a Lie bracket , • subject to the{· Leibniz·} identity ab,c = a b,c + b a,c . • { } { } { } If the algebra is graded then we allow , to have some fixed degree d. We then speak of graded Poisson algebras with bracket of degree {·d. ·} Proposition 2.6. If A is filtered, almost commutative then gr(A) has a canonical structure of a graded Poisson algebra with bracket of degree 1. − Proof. Define a,¯ ¯b := ab ba. One can easily check this is well defined (the fact that gr(A) is commutative is{ used} here).− Checking the Lie bracket identity and Leibniz identity is immediate. 

Notice that if A is commutative then the Poisson bracket is trivial. So the Poisson bracket measures the non-commutativity of A. Example. If g is a Lie algebra then U(g) is an almost commutative filtered algebra. This induces C ∨ a Poisson structure on Sym(g) ∼= [g ].

2.5. Some deformation theory. Consider the following diagram of relations. Y denotes Spec(gr(A)) although we can certainly sheafify everything and work on some algebraic Poisson manifold Y where A is a sheaf of filtered almost commutative associative algebras whose associated graded is . OY filtered almost commutative / one parameter associative algebra A deformations of gr(A)

  0 2 ( 1 graded) Poisson P H ( TY ) flat first order deformation − ∈ ∧ algebra structure o / (of weight 1) such o / of gr(A) which extends on gr(A) that [P,P−] = 0 to a second order deformation i The arrow in the top row associates to A the Rees ring Aˆ = t Ai which is a one parameter deformation of gr(A). ⊕ The leftmost vertical arrow associates to A the graded Poisson algebra given by a,¯ ¯b = ab ba. The rightmost vertical arrow associates to a genuine one parameter deformation{ } its first− order deformation. The leftmost bottom arrow associates to any P H0( 2T ) a bracket via ∈ ∧ Y f,g = P (df dg). { } ∧ This bracket immediately satisfies the Leibniz condition while [P,P ] = 0 ensures that it also satisfies the Jacobi identity. Here [P,P ] denotes the Schouten-Nijenhuis bracket on polyvector fields. If P is non-degenerate then P corresponds to a non-degenerate 2-form ω and the condition [P,P ] = 0 translates to dω = 0. This follows from the identity [v1,v2]= Xω(v1,v2) for any two vector fields v1,v2 [Prop 3.1. [Mein]]. − Conversely, given any Poisson structure, the map g f,g is a derivation and induces a vector 7→ { } ∨ field Xf . This vector field only depends on df so we get a map TY TY given by df Xf . The fact → 0 2 7→ that f,g = g,f means that this map is equivalent to a section P H ( TY ). In{ the} graded−{ picture} we have a graded Poisson algebra structure on∈ gr(A∧). This correponds to the fact that P has weight 1. Geometrically this means Y should have a C×-action under which − × P has weight 1. The standard example is Y = Spec C[x,y] where P = ∂x ∂y and C acts by (x,y) (x,ty).− ∧ 7→ D-MODULES NOTES 7

Example. Let us take X = A1 = Spec C[x] and Y = T ∨A1 = Spec C[x,y] where y has degree 1. k l We take A = X = C(x,∂x)/∂xx = x∂x + 1. Let us compute x ,y . We lift x to x and y to ∂x. Then D { } xk∂l ∂l xk = klxk−1∂l−1 + lower order terms x − x − x so that xk,yl = klxk−1yl−1. { } − On the other hand, consider the standard symplectic form ω = dx dy or equivalently P = ∂x py (notice this has weight 1). Then ∧ ∧ − xk,yl = P (klxk−1yl−1dx dy)= klxk−1yl−1. { } ∧ So the two graded Poisson structures we get should agree (up to a sign). 2 ′ ′ A first order deformation of gr(A) is a map C[t]/t A such that A C[t]/t2 C[t]/t = gr(A). This gives us a short exact sequence → ⊗ 0 tA′ A′ gr(A) 0 → → → → and subsequently an isomorphism A′ = A tA′ as C[t]/t2-modules via f (f,tf¯ ). The fact that ′ ′ ∼ ⊕ 7→ A is flat means tA ∼= A. Thus we get that a flat first order deformation of gr(A) is of the form gr(A) tgr(A) with some funny multiplication. The⊕ rightmost bottom arrow associates to P the first order deformation given by (f ,g ) (f ,g )=(f f ,f g + g f + P (df df )). 1 1 ∗ 2 2 1 2 1 2 1 2 1 ∧ 2 Exercise. Check that this defines an associative product. Conversely, any such deformation is of the form (f ,g ) (f ,g )=(f f ,f g + g f + α(f ,f )). 1 1 ∗ 2 2 1 2 1 2 1 2 1 2 The associativity condition translates to (2) α(f f ,f )= f α(f ,f ) f α(f ,f )+ α(f ,f f ). 1 2 3 1 2 3 − 3 1 2 1 2 3 In this context, the condition [P,P ] = 0 corresponds to the fact that this deformation can be lifted to second order.

2.6. Hochschild cohomology. The local version of Hochschild cohomology consists of a (graded) algebra A and the chain complex

d ⊗n d ⊗n+1 d ... HomC(A , A) HomC(A , A) ... −→ −→ −→ where d(φ)(a ,...,a )= a φ(a ,...,a ) φ(a a ,a ,...,a )+ +( 1)n+1φ(a ,...,a )a . 0 n 0 1 n − 0 1 2 n ··· − 0 n−1 n Example. Z0(A, A) consists of maps φ : C A such that dφ(a) = aφ(1) φ(1)a = 0. So HH0(A, A)= AA (i.e. elements h = φ(1) in the centre→ of A). If A is commutative then− HH0(A, A)= A. 1 Z (A, A) consists of maps φ : A A such that φ(f1f2) = f1φ(f2) φ(f1)f2. If A is commutative then these are derivations of A and the→ image dC0(A) is zero so HH1(A,− A) consists of vector fields on Spec(A). If A is commutative then the same argument shows that HH∗(A, A) consists of polyvector fields on Spec(A). Now there is a cup product at the level of cochains which descends to a product on HH∗(A, A). Moreover, there is a bracket on HH1(A, A) which is given by the commutator: recall that C1(A, A)= ∗ HomC(A, A). This bracket extends via an equation just like (3) to a bracket on HH (A, A). This gives HH∗(A, A) the structure of a Gerstenhaber algebra (see the next section for a definition). If A is commutative then this gives the usual Gerstenhaber structure on polyvector fields. 8 D-MODULES NOTES

The non-local version consists of a (smooth) variety Y (not-necessarily affine) where the aim is to deform the structure sheaf Y . One parameter deformations of Y are parametrized by the Hochschild cohomology group HH2(CohO (Y )) of the category of coherentO sheaves on Y . This is the same as 2 ExtY ×Y ( ∆, ∆) where ∆ Y Y denotes the diagonal. By the Hochschild-Kostant-Rosenberg we have O O ⊂ × Ext2 ( , ) = H0( 2T ) H1(T ) H2( ) Y ×Y O∆ O∆ ∼ ∧ Y ⊕ Y ⊕ OY (the content of the theorem is that the spectral sequence computing the left hand side degenerates). If 0 2 Y is affine then the higher cohomology groups vanish so we are left with H ( TY ) as above. 1 ∧ In general H (TY ) correspond to first order deformations of Y . To be precise, if Y˜ Spec C[t] is a deformation of Y then we get the standard short exact sequence →

0 T T ˜ 0. → Y → Y |Y → OY → The extension class of this sequence is an element in Ext1( ,T )= H1(T ). This corresponds to a OY Y Y deformation of Y which does not split as Y t Y . O 2 O ⊕ O The third term H ( Y ) corresponds to looking at the category of twisted sheaves. Fix a 2-cocycle α with respect to someO open cover U of Y . Then a twisted sheaf is a pair ( ,φ ) where is a ijk { i} Fi ij Fi sheaf on Ui and φ : ∼ ij Fi|Ui∩Uj −→Fj|Ui∩Uj are isomorphisms such that

φii = 1 • φ = φ−1 • ij ji φki φjk φij = αijk. • ◦ ◦ 2 ∗ If all αijk = 1 this is just the usual sheaf. If αijk H ( Y ) (i.e. does not vanish) then one obtains the category of coherent twisted sheaves Coh(Y,α).∈ O 2.6.1. Higher order deformations. If A is a (graded) algebra then a higher order deformation of A corresponds to a deformation of the product m and can be written as 2 mt = m + tβ1 + t β2 + ... 2 where β HC (A, A) are Hochschild 2-cochains. In other words words β : A C A A where i ∈ 1 ⊗ → dβ1 = 0. The condition that m + tβ1 is associative is precisely equation (2) which is equivalent to the condition that dβ1 = 0 – recall that dβ(f ,f ,f )= f β(f ,f ) β(f f ,f )+ β(f ,f f ) β(f ,f )f . 1 2 3 1 2 3 − 1 2 3 1 2 3 − 1 2 3 2 The condition that m+tβ1 +t β2 is associative is given by the Maurer-Cartan relation dβ2 +[β1,β1]=0. 3 This condition is equivalent to saying that [β1,β1] vanishes in HH (A, A). If A is commutative then 3 0 3 HH (A, A) is the same as H ( TY ) where Y = Spec A. 3 ∧ If HH (A, A) = 0 then all the associativity conditions automatically hold and one can lift m + tβ1 to arbitrary order. The higher Hochschild cohomology groups come in if one considers deformations of an A∞ algebra ⊗n A. Recall that an A∞ algebra is a graded vector space A with higher multiplications mn : A A (satisfying some relations). If m = 0 for n 3 then A is just a differential graded algebra and→ if n ≥ m1 = 0 as well then one obtains a graded algebra. See [K] for a brief introduction to A∞ algebras. 2.7. (super) Gerstenhaber algebras. A reference for this subsection is [KS]. A Gerstenhaber algebra differs from a super graded Poisson algebra in that , has degree 1 instead of degree 0. To be precise, a Gerstenhaber algebra is a (differential) graded,{· associative·} algebra− with a bracket , satisfying: {· ·} ab =( 1)|a||b|ba • − D-MODULES NOTES 9

a,bc = a,b c +( 1)(|a|−1)|b|b a,c • { } { }} − { } a,b = ( 1)(|a|−1)(|b|−1) b,a • { } − − { } a,b ,c = a, b,c +( 1)(|a|−1)(|b|−1) b, a,c • {{ } } { { }} − { { }} Standard examples of Gerstenhaber algebras include the algebra of polyvector fields ∗T on a manifold Y where multiplication is given by wedging • ∧ Y – the Lie bracket is given by extending the usual Lie bracket on TY . This gives the Schouten- Nijenhuis bracket and is defined by

i+j (3) [a1 ...am,b1 ...bn]= X( 1) [ai,bj ]a1 ...ai−1ai+1 ...amb1 ...bj−1bj+1 ...bn. − i,j the Hochschild cohomology HH∗(A, A) of a graded algebra A • Let A be an associative, graded commutative algebra. Let f : A End(A) be a map of degree f . If u,v End(A) we denote → | | ∈ [u,v] := uv ( 1)|u||v|vu. − − For a,b A we set [a,b]= f(a)b. This means [Ai, Aj ] Ai+j+|f|. We impose that ∈ ⊂ (i) graded Jacobi identity: [f(a),f(b)] = f(f(a)b) (ii) graded skew-symmetry indentity: f(a)b = ( 1)(|a|+|f|)(|b|+|f|)f(b)a (iii) f(a) is a graded derivation of A of degree −a −+ f | | | | Then if f is even then by definition we have a super graded Poisson algebra, if f is odd we get a Gerstenhaber| | algebra. Notice that if you ignore the grading (i.e. make everything| degree| zero) then you get the usual definition of a Poisson algebra. Example. Suppose we start with a graded associative, almost super-commutative algebra B and consider A = gr(B). Then we can take f : A End(A) to be → f(¯a)(¯b)= ab ( 1)(|a|−1)(|b|−1)ba − − where a,b B. In this case f = 1. Exercise.∈ Check that this| | definition− leads to a Gerstenhaber algebra. If we start with then we do not get a Gerstenhaber algebra (a.k.a. a graded Poisson super algebra DX of odd degree) because X is almost commutative (not almost super-commutative). Consequently we get only a graded PoissonD algebra with bracket of degree 1. On the other hand, recall (Proposition 1.2) that is− generated as a ring by f and ζ T DX ∈ OX ∈ X subject to some relations. One of these relations is ζ1 ζ2 ζ2 ζ1 =[ζ1,ζ2]. It is this relation that leads to an almost commutative algebra (instead of an almost∗ super− ∗-commutative algebra). If we change this relation to ζ ζ + ζ ζ =[ζ ,ζ ] 1 ∗ 2 2 ∗ 1 1 2 ∗ then we get an almost super-commutative algebra. The associated graded is just TX (instead of ∗ ∧ Sym TX ). The construction above then leads to the Gerstenhaber structure on the wedge algebra of polyvector fields.

3. Fedosov quantization Basic question: given a (holomorphic) Poisson structure P on a complex variety X can you deform along direction P ? OX More precisely, as we saw P defines a first order deformation of X . We would like to know if this first order deformation extends to a formal deformation and, ifO it does, then how many such deformations are there. 10 D-MODULES NOTES

This question was studied in the smooth case by De Wilde-Lecomte and Fedosov when P is a sym- plectic structure (Deligne has a classic exposition of these results). We present an algebraic approach to this problem when X is (holomorphic) symplectic following [BK]. Definition. A quantization of is a sheaf of associative, flat C[[h]] algebras on X (complete D OX in the h-adic topology) and equipped with an isomorphism /h ∼= X . As we learned already a quantization leads to a Poisson strucD DtureO with associated bivector field P . Conversely, any first order quantization which splits (i.e. the deformation is purely non-commutative) is given by a bivector field P . We say is non-degenerate if P is non-degenerate (i.e. P induces an ∼ D isomorphism T Ω ). X −→ X Question. Classify non-degenerate quantizations of X . For example, we have the following classificationD of non-degO enerate quantizations of the formal polydisc.

Lemma 3.1. Let X = Spec C[[x1,y1 ...,xn,yn]] denote the formal polydisc. Then a non-degenerate quantization is isomorphic to the (formal) Weyl algebra

[xi,xj ]=[yi,yj ]=[h,xi]=[h,xj ] = 0 and [xi,yj ]= δij h. 2 Notice that HH (C[[x1,y1,...,xn,yn]]) has basis given by 2-vector fields. Those corresponding to non-degenerate first order deformations are the non-degenerate 2-vector fields. The result above basically says that any such non-degenerate 2-vector field is isomorphic (after changing coordinates) to ∂ ∂ (which is a Darboux-type result). Pi xi ∧ yi • 3.1. Notation and assumptions. Let X be a smooth variety and denote by ΩX the DeRham complex 0 ∂ Ω1 ∂ Ω2 ∂ .... → OX −→ X −→ X −→ ∗ The (hyper)cohomology of this complex is denoted HDR(X). Now consider the exact sequence Ω≥1 Ω• X → X → OX ∗ where the left hand term is the kernel of the second map. We denote by HF (X) the (hyper)cohomology ≥1 of ΩX . i i We say that X is admissible if the induced maps HDR(X) H ( X ) are surjective for i = 1, 2. i → O This is the case if X is affine (since H ( X )=0 for i> 0) or if X is projective (Hodge theory). If X is admissible then O 2 2 2 HDR(X) ∼= HF (X) H ( X ). 2 2 ⊕ O 2 2 We have a canonical map HF (X) HDR(X) but no canonical map pr : HDR(X) HF (X). → • ≤n → Notice also that for each n there is a map ΩX ΩX which induces an isomorphism on (hy- ≤n → n ≤n per)cohomology groups H . On the other hand there is a map ΩX [ n] ΩX which induces a map − → H0(Ωn ) Hn(Ω≤n)= Hn (X). X → X DR 3.2. Symplectic deformations. We begin by considering the following commutative deformation problem: classify (commutative) deformations of the pair (X, Ω). Theorem 3.2 ([KV] 2.6). Let X be an admissible holomorphic symplectic variety with form Ω. Denote 2 by [Ω] HDR(X) the corresponding class. Let Def(X, Ω) denote the universal formal deformation space of∈(X, Ω). Then there exists a natural embedding Per : Def(X, Ω) H2 (X) → DR 2 called the period map which remembers the associated cohomology class [Ωh] HDR(X). Composed 2 2 ∈ with any splitting pr : HDR(X) HF (X) this identifies Def(X, Ω) with the formal completion of H2 (X) at the point [Ω] H2 (X).→ F ∈ F D-MODULES NOTES 11

2 In particular, there exists a formal symplectic deformation over the formal completion of HF (X) at [Ω] such that any deformation over some Artin scheme S is the pullback of this universal deformation 2 via the period map S HF (X). When X is compact,→ Bogomolov and Beauville show that the period map immerses the coarse 2 2 marked moduli space of (X, Ω) into HDR(X). The image is the quadric in HDR(X) cut out by the Bogomolov-Beauville form.M Moreover, the period map H2 (X) is locally an isomorphism. M → DR 3.3. Non-commutative deformations. The following theorem classifies non-degenerate quantiza- tions of an admissible holomorphic symplectic variety. Theorem 3.3 ([BK] 1.8). Let X be an admissible holomorphic symplectic variety with form Ω. Denote 2 by [Ω] HDR(X) the corresponding class. Let Q(X, Ω) denote the set of isomorphism classes of quantizations∈ of X extending the first order quantization induced by Ω. Then there exists a natural injection Per : Q(X, Ω) H2 (X)[[h]] → DR called the non-commutative period map. For q Q(X, Ω) the constant term of Per(q) is [Ω]. Moreover, 2 2 ∈ 2 2 any splitting pr : HDR(X) HF (X) of the canonical embedding HF (X) HDR(X) induces an isomorphism → → pr Per : Q(X, Ω) ∼ pr([Ω]) + hH2 (X)[[h]] H2 (X)[[h]]. ◦ −→ F ⊂ F If X is projective then Hodge theory provides a canonical splitting pr. A quantization is canonical if its period map is the constant power series [Ω]. A canonical quantization need not exist,D even if X is admissible. Why do these theorems 3.2 and 3.3 look so similar? Consider the universal family (X,˜ Ω)˜ over Def(X, Ω). By theorem 3.3 (which also holds in families) this has a quantization which gives a family over ∆ Def(X, Ω) where ∆ is the formal disc. Now for any section s : ∆ ∆ Def(X, Ω) the pullback× of the universal deformation via s gives a quantization of X. Algebraically,→ × any such section s is given by a power series P H2 (X)[[h]]. s ∈ DR Lemma 3.4 ([BK], Lemma 6.4). If the quantization of (X,˜ Ω)˜ Def(X, Ω) is canonical then the → non-commutative period map sends the quantization given by section s to the formal power series Ps. Remark: If X is compact it would be interesting to obtain a Bogomolov-Beauville type result. In other words, one would like to identity explicitly the image of the moduli space of non-commutative 2 deformations inside HDR(X). 4. More sheaves of non-commutative rings

4.1. Gabber’s Theorem. A subvariety V Spec(B) is coisotropic if for any f,g IV (where IV denotes the ideal defining V ) one has f,g ⊂I . ∈ { }∈ V Theorem 4.1. Suppose A is almost commutative and gr(A) Noetherian. Let M be a finite generated A-module. Then CV (M) Spec(gr(A)) is coisotropic (with respect to the natural Poisson structure on gr(A)). ⊂ Proof. The proof is not so easy and originally due to Gabber. In section 1.2.7 of [G] a simpler proof by Knop is presented. If M = A/J then ICV (M) = pgr(J). Proving this special case implies the Theorem. It is actually easy to show that f,g gr(J) if f,g gr(J). But in general it is not true that I,I I implies { } ∈ ∈ { } ⊂ √I, √I √I in a Poisson algebra.  { }⊂ Remark 4.2. If one translates the isotropic condition into the language of symplectic geometry then it’s equivalent to asking that the tangent space to CV (M) be coisotropic with respect to the holomorphic symplectic form on X. 12 D-MODULES NOTES

Example. is a sheaf of almost commutative filtered algebras. The associated graded is the DX sheaf Sym(TX ). Sections of this sheaf are the same as sections of the structure sheaf of the cotangent ∨ ∨ bundle TX of X (in general sections of Sym(E) are the same as functions on the total space E ). Then ∨ by the above TX carries a Poisson structure. This Poisson structure is induced by the natural 2-form ∨ on TX . ∗ ∨ More precisely, if π : E X is a vector bundle then there is a natural map π (E ) E. ∨ → ∗ → O Applying this to π : T X we get a natural map π (TX ) T ∨ . Precomposing with the natural X → → O X map T ∨ π∗(T ) we get a map T ∨ ∨ (in other words, T ∨ carries a natural 1-form γ). Let TX X TX TX X → → O ∼ ∨ ω = dγ. Then ω is non-degenerate and hence induces an isomorphism TX TX . In particular, we 2 −→ also have a natural section α Γ( TT ∨ ) which induces the Poisson structure mentioned above. ∈ ∧ X 4.2. Commutative localization. Recall that if A is a then Spec A carries a sheaf −1 of algebras. Namely to any open U Spec A we associate the localization AU := S A of A at the multiplicative set S made up of functions⊂ invertible on U. Moreover, to an A-module M we associate a sheaf of modules via U M A AU . This gives us the usual equivalence between finitely generated A-modules and coherent sheaves7→ ⊗ on Spec A. Next we would like to generalize this point of view to when A is a filtered almost commutative algebra. 4.3. Non-commutative localization. Let A be an associative ring with unit. S A is multiplicative if 1 S, 0 S and it is closed under multiplication. If A were commutative then⊂ we would define the ∈ 6∈ localization AS of A by formally inverting elements in S. When A is non-commutative there are problems with doing this. For instance, it is not clear if we should throw in elements of the form s−1a or as−1 and then it seems difficult to multiply two elements of the form s−1a. To fix this we impose the Ore conditions on S: −1 −1 any s1 a1 can be written as a2s2 (and vice versa) • if sa = 0 then as′ = 0 for some s′ (and vice versa) • Usually one avoids the second condition by asking that S contain no zero-divisors.

Theorem 4.3 (Ore). If Ore’s condition is satisfied then it makes sense to define AS as the ring of −1 elements of the form s a (and this AS satisfies the universal property of localization). Proof. For instance, multiplication is defined in the obvious way as s−1a s−1a = s−1s−1a a =(s s )−1a a . 1 1 · 2 2 1 3 3 2 3 1 3 2 

Remark 4.4. One can localize in a similar manner additive categories (note that a ring is an additive category with one object). In this setting A is an additive category, S is a subcategory where the objects are the same but we take a subset of morphisms which is closed under composition and contains all identity morphisms. Then the localization AS is the category where the objects are the same and maps X Z are diagrams of the form X fY gZ where g S. → → ← ∈ Here are two conditions for verifying that S satisfies the Ore condition. Proposition 4.5. If [s, ] is a (locally) nilpotent operator for any s S then S satisfies the Ore condition. · ∈ Proof. Follows directly by writing out the nilpotency condition [s, [s,... [s,a] ... ]] = 0.  Proposition 4.6. Suppose gr(A) is finitely generated and commutative. Given multiplicative S¯ gr(A) with no zero divisors let S := s A :s ¯ S¯ . Then S A satisfies the Ore conditions. ⊂ { ∈ ∈ } ⊂ D-MODULES NOTES 13

Proof. See Prop. 1.3.9 of [G]. 

We now assume A has no zero divisors. So then AS exists for any multiplicative subset S¯ gr(A). We say that s−1a has degree j i if deg(¯a)= j and deg(¯s)= i. Notice that this is a Z-filtration⊂ on S−1A since there are non-trivial≤ terms− in all negative degrees. Exercise. Check that this induces a well defined filtration on AS. −1 We call the completion AS of S A in the topology defined by this filtration the formal microlocal- ization of A at S. Given an A-module M we define its formal microlocalization by MS := AS A M. Properties: ⊗

(have A֒ AS where AS is flat over A (since localization and completion are exact functors • gr(A )=(→ S¯)−1gr(A) (because taking associated grading does not see the completion) • S Now, for any C∗-equivariant open subset U Spec gr(A) (we think of Specgr(A) as having the ⊂ Zariski cone-topology – i.e. open sets are cone-subsets) we denote by AU the microlocalization of A with respect to the set of elements invertible on U. We define M := A M. U U ⊗A Proposition 4.7. U AU defines a on Spec grA (thought of as a cone-scheme). Also, M M gives a7→ sheaf of modules for any finitely generated A-module M. 7→ U For an X -sheaf of filtered almost commutative algebras A on X we have SpecX (grA) X (equippedO with a C× action fibrewise). The localization picture above gives us a sheaf of filtered→ almost commutative algebras on SpecX (grA) whose associated graded is the structure sheaf. In other words, via the Rees construction we get a non-commutative deformation of the structure sheaf of SpecX (grA). For example, if we start with some variety X and take A = X (the sheaf of differential operators) ∨ ∨ D then SpecgrA ∼= TX . The associated sheaf of algebras on TX induces a first order (non-commutative) deformation of T ∨ which is encoded by an element in H0( 2T ∨ ). This element is (the dual of) the X TX ∨ ∧ natural symplectic form on TX .

4.4. Duality and the Sato-Kashiwara filtration. We will assume in this section that A is almost commutative algebra with X := Specgr(A) smooth. The grading on gr(A) induces a C× action on X whose fixed point set X0 := Spec A0 must also be is also smooth. (why?) Lemma 4.8. Let M be a filtered A-module such that 0 grM gr(A)⊕k gr(A)⊕l ← ← ← is exact. Then one can lift this exact sequence to an exact sequence 0 M A⊕k A⊕l ← ← ← where the maps are strictly compatible with filtrations.

′ ′ Proof. A map f : N N is strictly compatible with filtrations if f(Ni) = f(N) Ni . See Lemma 1.4.2 of [G] for a sketch→ (you get your hands dirty a bit). ∩ 

Corollary 4.9. Given a projective resolution 0 grM N N N ... ← ← 0 ← 1 ← 2 ← one can (after possibly having to localize with respect on X0) lift this resolution to a free resolution 0 M Ak0 Ak1 Ak2 ... ← ← ← ← ← where the maps are strictly compatible with filtrations. 14 D-MODULES NOTES

Proof. Notice that if A is filtered almost commutative then for any s A the operator [s, ] is nilpotent ∈ 0 · as a direct consequence of grA being commutative. So, in this case, any multiplicative subset of A0 satisfies the Ore condition and we can localize. Thus we can shrink X0 so that the resolution for grM becomes free. Then by the Lemma above we can lift this resolution.  Corollary 4.10. The homological dimension of A is equal to dimX. Proof. We can lift any free resolution of grM to a free resolution of M. So the length of the minimal free resolution of M and grM are the same. Hence A and grA have the same homological dimension (namely dimX). 

If M is a left (resp. right) A-module then omA(M, A) is a right (resp. left) A-module where the multiplication is on the A. If M is filtered thenH om (M, A) inherits a filtration: φ : M A is in the H A → lth term if φ(Mi) Ai+l for any i. One can check that gr HomA(M, A) = HomgrA(grM, grA). More generally, this means⊂ that: j Z j Lemma 4.11. xtA(M, A) is a right (resp. left) A-module for j and gr xtA(M, A) is a subquotient j E ∈ E of ExtgrA(grM, grA).

Proof. Take a free resolution of grM and lift it to a free filtered resolution M F0 F1 ... . Then j ← ← ← ExtA(M, A) is computed using the complex 0 Hom (F , A) Hom (F , A) ... → A 0 → A 1 → Since gr HomA(M, A) = HomgrA(grM, grA) there is a spectral sequence which computes the cohomol- j  ogy of this complex from ExtgrA(grM, grA). The result follows. Applying M Hom (M, A) in the derived category induces duality functors 7→ A D : Db(mod A) Db(mod A) − ↔ − To make sure that one lands in the bounded derived category it suffices that A has finite cohomological dimension (which follows if grA is smooth). Example: DA = A while D(A/Aa)= A/(aA)[ 1] (the second example corresponds to a divisor in X). − b Lemma 4.12. There is a functorial isomorphism DD(M) ∼= M in D (A). b i Given M D (mod A) we denote by CV (M) := iCV (H (M)). We also let d(M) := dimCV (M) and j(M) :=∈ min i : xt−i (M, A) = 0 . ∪ { E A 6 } Proposition 4.13. If M is a finitely generated A-module then j(M)+ d(M)= m • d( xti (M, A)) m i • E A ≤ − d( xtj(M)(M, A)) = d(M) • E A where m = dim Spec grA. Proof. If we replace A, M and CV by grA, grM and supp then this is a standard result in commutative algebra. My personal mnemonic for this result is as follows. Let i : Y ֒ X be a subvariety where X is proper. Then for any sheaf on X we have → F Hi( ∨ ) = Hom( , [i]) = Hom( [i], ω [m]) = Hom( [i],i i∗ω [m]) OY ⊗F ∼ OY F ∼ F OY ⊗ X ∼ F ∗ X = Hom(i∗ [i],i∗ω [m]) = Hom(i∗ω [m],i∗ [i] ω [n]) ∼ F X ∼ X F ⊗ Y = Hom(ω [m],i∗i∗ [i] ω [n]) = Hom( , [i] ω [n m]) ∼ X F ⊗ Y ∼ OY F ⊗ Y/X − = Hi( ω [n m]) ∼ F ⊗ Y/X − D-MODULES NOTES 15

∨ where m = dimX and n = dimY . This means that Y = ωY/X [ codimY ]. In particular, we see that ∨ O ∼ − Y is supported in degree codim(Y ) and has support Y . The moral is that this commutative algebra resultO can be viewed as a formal consequence of . j j The result when A is almost commutative uses that grExtA(M, A) is a subquotient of ExtgrA(grM, grA) (Lemma above). 

· · · j · Given a complex K denote by τ≥iK the truncation of K so that (τ≥iK ) = 0 for j

F → OP1 ⊕ OP2 → Op we get [ 4] ω [ 2] ω [ 2] ∨ ω . Op − → P1 − ⊕ P2 − →F ⊗ X Dualizing the natural cohomological filtration we get Im(H0( [ 1]) ) which is just zero. So Op − → F the filtration is just S1( ) = 0 and S2( ) = . In particular, we see that this filtration really just captures information aboutF the dimensionF of supportsF of sections and not more subtle information like Cohen-Macaulayness.

5. Twisted Differential Operators 0 Let A be a commutative ring and M,N two A-modules. Then one can define DiffA(M,N) := k k−1 Hom (M,N) and f Diff (M,N) HomC(M,N) if [a,f] Diff (M,N) for any a A. A ∈ A ⊂ ∈ A ∈ Thus Diff (M,N) HomC(M,N) is the largest subspace on which the adjoint action of A is A ⊂ nilpotent. Another way of saying this is the following. HomC(M,N) is naturally an A A-bimodule. Since A is commutative it is a sheaf on Spec A Spec A. Since ∆ Spec A Spec A is− generated by × ⊂ × a 1 1 a this means that Diff (M,N) is the largest submodule of HomC(M,N) setwise supported on ⊗ − ⊗ A the diagonal. This in contrast to HomA(M,N) which is the largest submodule scheme-wise supported on the diagonal. A sheaf of twisted differential operators (TDO) on X is a sheaf of filtered almost commutative D algebras together with an isomorphism gr = SymTX of Poisson algebras. 0 D ∼ By definition ∼= X . Now any a defines a derivation [a, ]. Since is almost commutative this defines a mapD ≤1O T since [a,f∈] D for any a ≤1 and· f .D A priori this map might D → X ∈ OX ∈D ∈ OX be zero (for instance, if were commutative) but the fact that gr ∼= SymTX as Poisson algebras rules this out. D D To see this note this map is precisely the Poisson bracket [a,f]= a,f so if this is zero for all f it means that a 0. In particular, we get an exact sequence { } ∈D 0 ≤1 T . → OX →D → X ≤ ≤1 The fact that TX is onto follows since a, : Sym TX TX is onto. A TDO structureD → on a given sheaf of filtered{ ·} algebras can be→ reconstructed from an isomorphism 0 ∼= X and a short exact sequence as above such that the induced map SymTX is bijective (then Dit’s automaticallyO a Poisson algebra isomorphism). Moreover, two TDOs and→Dare isomorphic if D1 D2 and only if there exists an isomorphism of sheaves 1 ∼= 2 which takes X 1 to X 2. Given a TDO the commutator on 1 gives theD structureD of a Lie algebra.O ⊂D One seesO that⊂D D D there exists a short exact sequence of modules • OX 0 1 T 0 → OX →D → X → 1 [a1,f a2]=(¯a1f) a2 + f [a1,a2] wherea ¯ denotes the image of a in TX • 1 · is a central· element· of 1 ∈D • ∈ OX D The first condition is explained above. Any derivation on an algebra takes 1 0 which explains the third condition. To see the second condition note that the left side equals 7→ a f a f a a 1 · · 2 − · 2 · 1 D-MODULES NOTES 17 and the right side equals (a ¯ f) a + f a a f a a 1 · · 2 · 1 · 2 − · 2 · 1 so it suffices to show a f a f a a = (a ¯ f) a 1 · · 2 − · 1 · 2 1 · · 2 which simplifies to showing a f f a =a ¯ f. 1 · − · 1 1 · The left hand side is just a1,f (the Poisson bracket) which equals the right hand side by the properties of the Poisson bracket. { } A sheaf of -modules with a Lie bracket satisfying the conditions above is called an Atiyah algebra. OX Lemma 5.1. The set of isomorphism classes of Atiyah algebra forms a vector space AtX . There is a linear map At Ext1(T , ) by forgetting the Lie bracket. X → X OX 1 Proof. λ C acts by multiplying the connecting map Ext (TX , X ). The sum of two Atiyah algebras is defined∈ by via the Bauer sum of extensions – one needs to checOk that this explicit construction gives a Lie bracket on the sum.  1 1 1 Example. Given a L on X the image of [ X (L,L)] Ext (TX , X ) ∼= H (ΩX ) is the Chern class of L. D → O Example. Given an extension α Ext1(T , ) one can define a Lie bracket on := Cone(α) via ∈ X OX A [f + ζ ,f + ζ ]= ζ f ζ f +[ζ ,ζ ] 1 1 2 2 1 2 − 2 1 1 2 locally on X. Then one can show (need to check this) that this definition glues to give a bracket on . Even more if β is a 2-form then one can define A [f + ζ ,f + ζ ]= ζ f ζ f + β(ζ ,ζ )+[ζ ,ζ ]. 1 1 2 2 1 2 − 2 1 1 2 1 2 If β is closed then this satisfies the Jacobi identity. If it is locally exact then it turns out this is locally isomorphic to X (locally trivial Atiyah algebra). Notice that in the holomorphic category any closed two form is locallyD exact so any is locally trivial (hence adding this extra β is only interesting in the algebraic category). A 1 1 Theorem 5.2. Locally trivial Atiyah algebras are parametrized by HZar(X, Ωcl). Atiyah algebras in general are parametrized by H1 (Ω1 d Ω2 ) Zar −→ cl where this is the hypercohomology of the (truncated) DeRham complex. Proof. See [G] section 2.4. One first proves the local version by noting that .( ֒ )Ω1 = Aut cl ∼ OX →DX Then one shows that any is given by data αij ,βi for some cover Ui where αij is a 1-form and β a closed 2-form. Finally,A one has the conditions{ β } β = dα and α{ +} α + α = 0 which gives i i − j ij ij jk ki a class in H1(Ω≥1). 

Remark 5.3. Notice that the locally trivial Atiyah algebras are precisely those associated to X (L,L) where L is a line bundle (two line bundles with the same Chern class give isomorphic Atiyah algebras).D Remark 5.4. Standard maps of complexes induce the long exact sequence ...H0(Ω1) H0(Ω2 ) H1(Ω≥1) H1(Ω1) ... → cl → → → 1 1 1 ≥1 as well as a map H (Ωcl) H (Ω ) which embeds the space of locally trivial Atiyah algebras into the space of all Atiyah algebras.→ Theorem 5.5. There is a natural bijections TDO Atiyah algebras . { } ↔ { } 18 D-MODULES NOTES

Proof. The arrow is 1. The arrow is U ( ). Here U ( ) denote the quotient of → D 7→ D ← A 7→ X A X A UC( ) (the universal enveloping algebra) by the relations A 1 = 1 and f a = f a where f and a . UC(A) A ⊗ · ∈ OX ∈A This is supposed to mimic the universal enveloping algebra of over X . This is not possible directly since is not central. A O  OX ⊂A Given any TDO, or equivalently any Atiyah algebra , one can form a family of filtered, associative A1 A algebras t where t := t . Thus t ∼= for t = 0 while 0 ∼= X where X is the Atiyah algebra associatedA → with A . This·A means thatA anyA Atiyah6 algebra isA a deformationA A of . Applying DX AX UX ( ) one gets that any TDO is a deformation of X . A· good picture to keep in mind is D deformation / L X (L,L) AX ↔DX A ↔D gr S   ∨ commutative deformation / ∨ TX (TX )L ∨ ∨ where (TX )L denotes the deformation of TX corresponding to the line bundle L. The vertical map denoted S takes to SymC( ) (the sheaf of symmetric algebras) modulo the relations AL AL 1 = 1 and f a = f a for any f ,a SymC(AL) AL ⊗ · ∈ OX ∈AL which is a sheaf of filtered commutative algebras on X (a “twisted ”). Taking Spec ∨ gives (TX )L which is a “twisted cotangent bundle”. ∨ Three descriptions of (TX )L: ∨ (i) To construct (TX )L let Y denote the total space of the line bundle L which comes equipped C× ∗ C with the natural action acting on the fibres. Let µ : TY be the moment map. Then −1 C× ∨ −1 C× ∨ → µ (t)/ ∼= (TX )L if t = 0 while µ (0)/ ∼= TX . (ii) (T ∨) = Conn (L) is the6 space of connections on L. More precisely, a connection takes X L X ∇ a section s of L and a vector field v and returns another section vs of L. For any one form α one can define a new connection ′ (s) := (s)+ α(v)s.∇ This way one can show ∇v ∇v that Conn (L) is an affine Ω1 bundle. Note that Conn (L) = Ω1 since (in general) there X X X 6∼ X is no section X Conn (L). If L = then there is a section, namely the connection → X ∼ OX vs = ds(v) (this just says that vector fields naturally act on functions). ∇ ∨ ∨ (iii) As a first order deformation of TX one can see (TX )L as follows. A standard computation 1 1 1 ∨ shows that H (TT ∨ ) = H (TX ) H (T ) R where R is the rest of the cohomology. In X ∼ X ⊕ 1 ∨ ⊕ 1 1 ∨ particular, for any class in c1(L) H (TX )= H (ΩX ) we get a first order deformation (TX )L ∨ ∈ of TX . 1 Example. Take X = P . Then (up to multiple) there is a unique line bundle L := P1 (1). ∨ 2 A1 O P1 Then the deformation (TX )L is the family xy z = t over t . The projection map to is (x,y,z) [x,z]. It is easy to check the fibre{ is− isomorphic} to C1∈but it is only an affine bundle (no zero section).7→ Point of view. T ∨ has natural deformations (T ∨) parametrized by c (L) H1(Ω1 ). Just like X X L 1 ∈ X DX is a non-commutative deformation of T ∨ so is X (L,L) a non-commutative deformation of (T ∨ ) . O X D O X L References

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