Seminar on Deformation Quantisation of Algebraic Varieties Domenico Monaco January 26th , 2012

Plan of the seminar This is the second of two seminars on the topic of deformation quantisation (DQ for short). Last seminar was devoted to the illustration of Kontsevich’s formality conjecture/theorem [Ko03], concerning DQ of Poisson manifolds. In this one, instead, we will follow [Ye05] and see to what extent the results and techiniques of Kontsevich may be generalised to the algebraic-geometric setting. In the following, X will denote a smooth n-dimensional algebraic variety over a field K of characteristic 0. At some point we will assume K ⊃ R, so one can think of K = C for simplicity. I plan on spending a few words concerning the definition and basic properties of the sheaf of differential operators on X, and point out how it can be viewed as the non-commutative analogue of the structure sheaf OX (the sheaf of regular functions). A possible reference for this first part is [Mi99]. In the second part of the seminar, we will go through Yekutieli’s result in the affine case and give a geometric interpretation of how one proceeds to generalise it to arbitrary algebraic varieties (which traces back to [Fe94]); this will shed some light on the motivation for the use of the sheaf of differential operators. If time allows (and I’m pretty sure it won’t), I will illustrate what are the analogies and differences between Kontsevich’s (Poisson C ∞ case) and Yekutieli’s (algebraic case) results.

1 D-modules In order to motivate the study of differential operators on algebraic varieties, let me first give a glimpse of what is non-commutative algebraic geometry – this was the original purpose of these two seminars on DQ.

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In “usual” algebraic geometry, one of the basis results in the equivalence of categories1 Γ

'

CommAlgK

AffVarg K Spec

where the l.h.s. is the category of affine varieties over K (morphisms are regular functions) and the r.h.s. is the category of (reduced, unital) commutative K-algebras. This establishes a “dictionary” between geometry and commutative algebra. One would want to take this as a paradigm and define geometric objects, starting from non-commutative algebras. Two (related) possible approaches to the deformation of a (sheaf of) commutative algebras OX (or equivalently of its associated algebraic variety X) into a noncommutative one are the following: • change the product of the algebra, keeping associativity but forgetting commutativity: this leads to the definition of star products, on which I will come back in a while; • in analogy with what happens in quantum mechanics, change coordinates on X (which are regular functions on it) with some “non-commutative coordinates”; this should have “something to do” with derivatives. I will now be more precise regarding the latter point. First I will review some algebraic definitions and then geometrise them. Let A be a (unital) commutative K-algebra. Inside the A-module EndK (A) of K-linear endomorphisms of A we have two interesting K-submodules: • the A-linear endomorphisms EndA (A). Notice that we have a map b: A → EndA (A),

b a(b) := a · b,

and that if T ∈ EndA (A) then by A-linearity and commutativity of A T (a) = T (a · 1) = a · T (1) = T (1) · a,

[ i.e. T = T (1).

We conclude that A ' EndA (A); in the following, we will consequently drop b and identify a ∈ A with the operator “multiplication by a”; • the K-derivations DerK (A), namely those K-linear D : A → A satisfying Leibnitz rule D(a · b) = D(a) · b + a · D(b). We let D(A) be the K-submodule of EndK (A) generated by A ' EndA (A) and DerK (A), and we call it the algebra of differential operators on A. 1

Here and in the following, K will always denote a field of characteristic zero.

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An equivalent definition is the following. Recall that EndK (A) is naturally a Lie algebra with [S, T ] := S ◦ T − T ◦ S, S, T ∈ EndK (A). One also checks that DerK (A) is a Lie subalgebra (clearly so is A ' EndA (A) because, by commutativity of A, the commutator of two multiplication operators vanishes). We say that T ∈ EndK (A) is a differential operator of order ≤ p if [[· · · [[T, a0 ] , a1 ] , · · · ] , ap ] = 0 for all a0 , a1 , . . . , ap ∈ A. The space of such endomorphisms will be denoted by D≤p (A). Then D(A) is filtered by D≤p (A), p ∈ Z (we set D≤p (A) = 0 if p < 0). One can check for example that D≤0 (A) = A,

D≤1 (A) = A ⊕ DerK (A).

Notice that if D ∈ DerK (A) and a ∈ A then [D, a](b) = D(a · b) − a · D(b) = D(a) · b,

i.e. [D, a] = D(a).

Define also Dp (A) = D≤p (A)/D≤p−1 (A), the space of differential operators of order p. We call elements of M Dnor (A) := Dp (A) p≥1

normalised differential operators. In general, D(A) is non-commutative, as in the following Example 1. Consider A = K [x1 , . . . , xn ], the algebra of polynomials in n variables. It is well known that DerK (K [x1 , . . . , xn ]) is generated by the usual derivatives of polynomials. Consequently, any differential operator on K [x1 , . . . , xn ] can be written as X D= PI (x1 , . . . , xn )∂I , I

where I = (i1 , . . . , ip ) is a multiindex, PI is a polynomial, and ∂I = ∂i1 ◦ · · · ◦ ∂ip =

∂p . ∂xi1 · · · ∂xip

We call D (K [x1 , . . . , xn ]) =: Wn the n-th Weyl algebra. To show that Wn is non-commutative, fix i, j ∈ {1, . . . , n} and let f ∈ K [x1 , . . . , xn ]. Then by Leibnitz rule ∂i (xj f ) = (∂i xj ) f + xj (∂i f ) = δi,j f + xj (∂i f ) i.e. [∂i , xj ] = δi,j I,

I = identity of K [x1 , . . . , xn ] .

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One similarly checks that [xi , xj ] = 0, [∂i , ∂j ] = 0. Moreover, it can be shown that Wn is isomorphic to the quotient of the free K-algebra in 2n generators x1 , . . . , xn , ∂1 , . . . , ∂n modulo these commutation relations. Let now A = K [x1 , . . . , xn ] /I, where I ⊂ K [x1 , . . . , xn ] is an ideal. It can be proved that in this case {T ∈ Wn : T (I) ⊂ I} . D(A) ' {T ∈ Wn : T (K [x1 , . . . , xn ]) ⊂ I} We will also need a slight generalisation of the concepts introduced so far. Define ^i+1 M Deripoly,K (A) := DerK (A), Derpoly,K (A) := Deripoly,K (A). i≥−1

We call Derpoly,K (A) the space of K-polyderivations of A. Notice that DerK (A) = Der0poly,K (A) and that A = Der−1 poly,K (A). There is a natural way to define a Lie bracket on Derpoly,K (A), which generalises the commutator of DerK (A): this is the SchoutenNijenhuis bracket i+j (A). [−, −]SN : Deripoly,K (A) × Derjpoly,K (A) → Derpoly,K

Together with the trivial differential d0 := 0, this turns Derpoly,K (A) into a differential graded Lie algebra (DGLA for short). We also define the space of polydifferential operators (respectively of order ≤ p, of order ≤p,i p,i nor,i i (A) (respectively Dpoly p, normalised) as follows: Let Dpoly (A), Dpoly (A), Dpoly (A)) be i+1 the space of morphism A → A which are differential operators (respectively of order ≤ p, of order p, normalised) in each variable separately. Then put M M ≤p,i ≤p i Dpoly (A) = Dpoly (A), Dpoly (A) = Dpoly (A), i≥−1 p Dpoly (A) =

M

i≥−1 p,i Dpoly (A),

nor Dpoly (A) =

i≥−1

M

nor,i Dpoly (A).

i≥−1

The Hochschild differential dHH and the Gerstenhaber bracket j i+j i [−, −]G : Dpoly (A) × Dpoly (A) → Dpoly (A) nor endow Dpoly (A) with a DGLA structure, and Dpoly (A) is a subDGLA. We can now geometrise all the above definition, especially in view of Example 1. Let X be an affine variety: we define the ring of differential operators on X as

D(X) := D(Γ(X, OX )). Similarly, one defines D≤p (X), Dp (X) and Dnor (X), and their analogues with polydifferential operators. We are now able to define the sheaf of differential operators DX of an arbitrary algebraic variety X: if U ⊂ X is an open subset, we set DX (U ) := lim D(V ). ←− V ⊂U V affine

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≤p p nor Similar definition yield to the sheaves DX , DX , DX , and the same holds for polydifferential operators (notice that the latter are sheaves of DGLAs).

Remark 1. Recall that when X is a smooth algebraic variety, one can define the tangent sheaf TX . Explicitly, if U = Spec A is an affine open subset of X, then ^ TX U = Der K (A), where g denotes localisation of a module. We shall also be interested in the sheaf M ^i+1 i i := TX . , where Tpoly,X Tpoly,X := Tpoly,X i≥−1

Also Tpoly,X is a sheaf of DGLAs: sections of Tpoly,X over an affine open U = Spec A ⊂ X are given by K-polyderivations of A. In general, global sections of TX are called vector fields over X, and hence global sections of Tpoly,X are called polyvector fields over X. One can check that DX is a quasi-coherent OX -module, filtered by the coherent sheaves ≤p p DX ; also the sheaves DX are coherent for all p. Moreover, whenever the open subset U ⊂ X is affine, one has that Γ(U, DX ) = D(U ) and the same holds for all the sheaves we have just defined. When, in addition, X is smooth, then DX is locally free: this is intuitively clear since DX is “generated” by OX and the tangent sheaf TX , which is locally free. As we already stated, we wish to consider DX as a non-commutative analogue of OX . In particular, we will consider sheaves which “behave well” with respect to DX , namely quasi-coherent and coherent DX -modules. We will say that a sheaf of DX -modules is (quasi-)coherent if it is so as an OX -module (every DX -module inherits an OX -module structure by the natural morphism OX → DX ). One can show that every quasi-coherent DX -module is the localisation of a D(X)-module when X is affine. Surprisingly enough, the same holds true also for X = PnK . Remark 2. Let F be an OX -module. Giving F a DX -module structure means that we have to define a way of multiplying sections of F by vector fields (i.e. sections of the tangent sheaf), in a way which is compatible with the OX -module structures of F and TX . Explicitly, for any open U ⊂ X we need to define v · s, for v ∈ Γ(U, TX ) and s ∈ Γ(U, F), so that for all f ∈ Γ(U, OX ) (a) (f v) · s = f (v · s) (OX -linearity); (b) v · (f s) = v(f )s + f (v · s) (Leibnitz rule); (c) [v1 , v2 ] · s = v1 · (v2 · s) − v2 · (v1 · s) (compatibility with the Lie algebra structure of Γ(U, TX )).

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Suppose now that F is locally free, i.e. it is the sheaf of sections of a vector bundle over X. Then, writing v · s =: ∇v s, we immediately realise that ∇ : TX → EndOX (F) is a connection, because it satisfies (a) and (b); moreover, condition (c) says that this connection is flat. Keep this remark in mind: it should ring a bell in a while, when we will see a geometric interpretation of DQ in terms of flat connections.

2 DQ of smooth algebraic varieties We now move on to DQ of smooth algebraic varieties. As was already pointed out, our purpose is to modify the product of the sheaf of rings OX into a non-commutative (but still associative) product, in analogy with what was done for Poisson manifolds last time by Francesca. It turns out that the theory of DQs of affine varieties, as one should expect and as we shall see in a moment, is simpler. As any scheme is covered by affine open subsets (by definition), it will thus be convenient to have a definition of DQ which is “local”. In the following definitions, U will be an open subset of X. Definition 1 (Star product). A star product on OU [[~]] is an associative product ? : OU [[~]] × OU [[~]] → OU [[~]] (with unit 1 ∈ Γ(U, OU ) ⊂ Γ(U, OU [[~]])) such that for f, g ∈ Γ(U, OU ) f ? g = fg +

∞ X

βj (f, g)~j ,

j=1


nor,1 where each βj is a normalised bidifferential operator on U (i.e. βj ∈ Γ U, Dpoly,U ). Thus, for a star product we have f ? g ≡ f g mod ~. The difference between the two is a section β of the sheaf nor,1 nor,1 Dpoly,U [[~]]+ := ~Dpoly,U [[~]].

Definition 2 (Gauge equivalences of OU [[~]]). We say γ ∈ AutK[[~]] (OU [[~]]) is a gauge equivalence if for f ∈ Γ(U, OU ) γ(f ) = f +

∞ X

γj (f )~j

j=1

where each γj is a normalised differential operator on U (i.e. γj ∈ Γ(U, DUnor )).

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It is easy to see that gauge equivalences of OU [[~]] form a group. We now come to the “global” definition. Definition 3 (DQ of OX , and gauge equivalence). A sheaf A of (~-adically complete, flat, unital) K[[~]]-algebras on X is called a deformation quantisation of OX if the following holds: (DQ1) there exists an isomorphism of sheaves of K-algebras ∼

ψ : A/~A − → OX ; (DQ2) there exist an open cover {Ui }i∈I of X and isomorphisms of sheaves of K[[~]]modules ∼ τi : OUi [[~]] − → A Ui , called differential trivialisations, satisfying the following two properties: (i) if ? is the product of A, then f ?τi g := τi−1 [τi (f ) ? τi (g)] defines a star product ?τi on OUi [[~]], and (ii) for f ∈ Γ (Ui , OUi ) ψ ◦ τi (f ) = f ; (DQ3) for any i, j ∈ I, the automorphism γj,i := τj−1 ◦ τi is a gauge equivalence of OUi ∩Uj [[~]]. Two DQs A and A0 of OX are said to be gauge equivalent if there exists an isomorphism of sheaves of K[[~]]-algebras γ : A0 → A such that (GE1) the diagram

γ mod ~



ψ0

/O X v; v v vv vv vv ψ

A0 /~A0

A/~A commutes;

 (GE2) let {Ui , τi }i∈I and Uj0 , τj0 j∈J be the differential trivialisations of A and A0 , respectively; then for all i ∈ I, j ∈ J the automorphism γ ei,j := τi−1 ◦ γ ◦ τj0 is a gauge equivalence of OUi ∩Uj0 [[~]]. If a DQ A of OX has a single differential trivialisation τ , then A is said to be globally trivialised. These DQs are in bijection with star products on OX [[~]]: the correspondence is ?τ ↔ ?. The cohomological obstruction for a DQ to be globally trivialised lies in H 1 (X, DX ), namely if this cohomology group vanishes then any DQ A of OX is gauge

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equivalent to a star product on OX (the gauge equivalence is constructed via DX -valued ˇ Cech 1-cocycles). One can further characterise a star product in terms of the section
nor,1 [[~]]+ β ∈ Γ X, Dpoly,U which appears in its definition as follows. Associativity of the star product can be rephrased in terms of β as 1 dHH β + [β, β]G = 0, 2 nor [[~]]+ . Moreover, gauge equivalent i.e. β satisfies the Maurer-Cartan equation in Dpoly,X star products lead to gauge equivalent (in a suitable sense) solutions of the MaurerCartan equation. Thus we have a bijection
1:1 nor M C Dpoly,X [[~]]+ /gauge ←→ DQ(OX )/gauge

(1)

where M C(−) is the set of solutions of the Maurer-Cartan equation in a certain sheaf of DGLAs, while DQ(OX ) denotes the set of DQs of OX . What measures the non-commutativity of a star product ? on OX ? Let f, g ∈ Γ(X, OX ); compute   f ? g − g ? f = f g + β1 (f, g)~ + O(~2 ) − gf + β1 (g, f )~ + O(~2 ) = = ~ [β1 (f, g) − β1 (g, f )] + O(~2 ). If we define {f, g}? := β1 (f, g) − β1 (g, f ) ∈ Γ(X, OX ), then one can easily verify that {−, −}? is a Poisson bracket on OX , namely a biderivation which makes OX into a sheaf of Lie algebras. Equivalently, there exists a section  ^2 
1 α ∈ Γ X, TX = Γ X, Tpoly,X such that

1 {f, g}? = hα, d f ∧ d gi =: {f, g}α , 2 where h−, −i denotes the natural pairing  ^2  h−, −i : TX × Ω2X → OX . One can check that this implies [α, α]SN = 0,

or equivalently

1 d0 α + [α, α]SN = 0, 2

i.e. α satisfies the Maurer-Cartan equation in Tpoly,X .

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More in general, we can deform Poisson brackets as follows: We say that
1 [[~]]+ α ∈ Γ X, Tpoly,X is a formal Poisson structure on OX if [α, α]SN = 0. There is a natural notion of gauge equivalence of formal Poisson structures, and moreover
1:1 M C Tpoly,X [[~]]+ /gauge ←→ F P S(OX )/gauge (2) where F P S(OX ) denotes the set of formal Poisson structures of OX . Yekutieli’s result [Ye05, Theorem 0.1], resembling Kontsevich’s formality conjecture, is as follows. Theorem 1. Let X be a smooth algebraic variety over K, where K ⊃ R. Assume that X is D-affine, i.e. H k (X, F) = 0 for all F sheaf of DX -modules, for all k > 0. Then there exists a quatisation map Q : F P S(OX )/gauge → DQ(OX )/gauge such that 1. if α ∈ F P S(OX ), α = α1 ~ + O(~2 ), and Q(α) = ?, then 1 {−, −}? = {−, −}α1 ; 2 2. Q commutes with pullback by ´etale morphisms f : X 0 → X; 3. Q is bijective when X is affine. Notice that, in Konstevich’s analogue, the map Q is always bijective. Bijectivity fails for general smooth algebraic varieties because, in view of (1) and (2) above, one tries to construct an L∞ -quasi-isomorphism between the two sheaves of DGLAs Tpoly,X and Dpoly,X , and this may fail to exists due to cohomological obstructions. The proof for X = AnK can be done in a rather simple way, and is actually based on Kontsevich’s result (hence the assumption K ⊃ R). It can also be easily extended to any affine X having an ´etale morphism X → AnK (i.e. “locally diffeomorphic” to AnK ). Any scheme can be covered by such varieties, so the problem may rise in gluing together these data. We give a differential-geometric interpretation of this gluing procedure, which will also explain heuristically why a lot of Maurer-Cartan equations appeared. The typical and simplest example of Poisson bracket on a “local chart” AnK ⊂ X is given by α1 =

n X

αkl ∂k ∧ ∂l

k,l=1

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(3)

where αkl = −αlk are constants in K. Notice that a theorem by Lie gives that, if the rank of the Poisson tensor in constant, then there always exist local coordinates such that in these coordinates the Poisson tensor is of the form (3). The star product ? associated to the formal Poisson structure α := α1 ~ is called the Moyal product, and it is given on sections of OAnK (that is, on polynomials in n variables) by f ?g =

∞ X ~j X j=0

j!

αKL ∂K f ∂L g,

K,L∈Nj

where if K = (k1 , . . . , kj ) and L = (l1 , . . . , lj ) then αKL := αk1 l1 · · · αkj lj . The Moyal product is clearly invariant under affine changes of coordinates. This suggests that we can “glue together” Moyal products, each V2 on a coordinate chart, provided transition functions of the bundle TX (or rather of TX ) are all affine. This is equivalent to having a connection on TX which is flat, i.e. having null curvature. In general, a connection ∇ is specified by a Lie-algebra valued 1-form ω, and its curvature is given by 1 Ω = ∇ω = d ω + [ω, ω]. 2 So the Maurer-Cartan equation is exactly the one satisfied by a flat connection 1-form.

References [Ko03] Maxim Kontsevich. Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66 (2003), no. 3, 157-216. [Ye05] Amnon Yekutieli. Deformation quantization in algebraic geometry. arXiv:math/0310399v5 [math.AG]. . Erratum to: Deformation quantization in algebraic geometry. arXiv:0708.1654v1 [math.AG]. [Mi99] Dragan Miliˇci´c. Lectures on Algebraic Theory of D-Modules. http://www.math.utah.edu/~milicic/Eprints/dmodules.pdf. [Fe94] Boris V. Fedosov. A simple geometrical construction of deformation quantization. J. Diff. Geom. 40 (1994), 213-238.

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