Notes on Formal Neighborhoods and Jet Bundles

NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES

SHILIN YU

ABSTRACT. The purpose of this note is to review the construction of smooth and holomorphic jet bundles and its relation to formal neighborhood of the diagonal embedding. I will show that there is a natural notion of “Dol- beault dgas” which works for formal neighborhoods of arbitrary analyt- ical embeddings. An algebraic proof of a theorem by M. Kapranov will be addressed at the end regarding the structure of such dga in the case of diagonal embedding.

CONTENTS 1. C -jet bundles 1 2. Holomorphic∞ jet bundles 5 3. Kapranov’s Theorem 11 References 13

1. C -JETBUNDLES The notion of C -jet bundle∞ provides an appropriate place where one can talk about Taylor∞ expansions (or jets) of smooth functions on a smooth manifold. Let X be a manifold and p ∈ X a point, for each nonnegative r integer r, we deﬁne the algebra Jp to be the quotient of C (X) by the ideal (r) ∞ Ip = {functions whose derivatives up to order r all vanish at p}.

In fact, if mp denotes the maximal ideal of the commutative algebra C (X) consisting of functions vanishing at p, we have ∞ (r) r+1 Ip = mp . The Taylor expansion of order r (or r-jet) of a function f at p is deﬁned to be the equivalence class r r r r+1 jpf := [f]p ∈ Jp = C (X)/mp ∞ Key words and phrases. ... 1 2 SHILIN YU

r Note that Jp is determined by local data around p, so in fact we should use the algebra C (X)p of germs of smooth functions at p instead of C (X). But this algebra itself∞ is a quotient of C (X), so everything is ﬁne. Indeed,∞ let np be the ideal of all global smooth functions∞ vanishing at some neighborhood of p, then there is a natural isomorphism ∼ C (X)/np = C (X)p. This is not true in the holomorphic∞ world.∞ If we choose a local coordinate chart (U, xi) containing p = (p1, . . . , pn), then there is an isomorphism r ∼ r+1 Jp = C[x1, . . . , xn]/(x1 − p1, . . . , xn − pn)

r i1 in [f]p 7 ai1,...,in (x1 − p1) ··· (xn − pn) X|I|≤r where → 1 ∂i1+···+in f ai ,...,i = (p) 1 n i ! . . . i ! i1 in 1 n ∂x1 ··· ∂xn Consider the above construction for any r, we have an inverse system 0 1 1 C = Jp Jp Jp ··· with connecting maps being the natural quotient maps. So we can define the space of -jets ← ← ← J := Jr p lim− p r ∞ ∞ r which is of infinite dimension. Moreover, the above isomorphisms for Jp are compatible with the inverse system,so← under the local chart (U, xi) we have an isomorphism J ∼ [x , . . . , x ]/(x − p , . . . , x − p )r+1 = x − p , . . . , x − p . p = lim− C 1 n 1 1 n n C 1 1 n n r J K ∞ Remark 1.1. By definition of inverse limit, we have the natural ’Taylor ex- ← pansion’ map

jp : C (X) Jp . A result by E. Borel ([1]) says that∞ this map∞ is surjective. In other words, there always exists (locally) a C -function→ with the given Taylor expansion. Again this fails when it comes to∞ holomorphic functions.The kernel of jp is

( ) \ (r) \ r+1 Ip = Ip = mp . ∞ ∞ ∞ r=0 r=0 NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES 3

r r Next I will define the jet bundles J (resp. J ) whose fibres are Jp (resp. Jp ). For this purpose, I only need to tell you the∞ sheaf of smooth sections of∞ the bundles. Let us assume that dimX = 1 for simplicity. For any open subset V ⊂ X, among the giant garbage of all ’discrete’ sections G r r s : V Jp, s(p) ∈ Jp, ∀ p ∈ V, p∈V I only look at those with the→ following property: for any y ∈ V, there exists r a local chart (U, x) containing y, such that, under the isomorphisms Jp = C[x]/(x − p)r+1, r r s(p) = [a0(p) + a1(p)(x − p) + ··· + ar(p)(x − p) ] ∈ Jp r where a0, . . . , ar ∈ C (U). We denote the set of all such sections by C (V, J ). It has a C (U)-module∞ structure defined by pointwise addition and∞ multi- plication,∞ and moreover is a commutative algebra. One can show that this really defines a sheaf over X and provides a realization of Jr as a smooth vector bundle. The only thing need to be checked is that the condition above is independent of the choice of local charts, which is merely an exercise using the chain rule. In fact, from the definition we get a trivialization of Jr on any local chart (U, x) with the frame [1], [x − p], [(x − p)2],..., [(x − p)r]. J J = Jr In a similar way one can define , or just by lim− . A section of the jet bundles does∞ not necessarily∞ come from an actual smooth function. In fact, the general sections are much more than those induced by functions. Nevertheless, we have canonical← maps of algebras jr : C (X) C (X, Jr). and ∞ ∞ j : C (X) → C (X, J ) r under which the images of∞ a function∞ are∞ called∞ prolongations. However, j is not a C (X)-linear map. It is a differential→ operator of order r. From the∞ description above of local sections of the jet bundles, one can see there are actually two variables x and p involved, though x is a formal variable. This inspires us to think of sections of jet bundles as ’formal functions’ on X × X near the diagonal. This is the first step towards the general concept of formal neighborhood. Let’s see how it works. We write X × X as X0 × X00 to distinguish the two factors and denote the corresponding projections by pr0 : X0 × X00 X0 and pr00 : X0 × X00 X00

→ → 4 SHILIN YU respectively. There is a natural splitting of the tangent bundle T(X0 × X00) = pr0∗TX0 ⊕ pr00∗TX00 or in shorthanded notation T(X0 × X00) = TX0 ⊕ TX00 The diagonal map ∆ : X , X0 × X00 embeds X as a closed submanifold, whose image we also write as ∆ by abuse of notation. A local chart (U, x) 0 00 0 00 0 00 0 of X gives rise to a chart (U→ × U , x , x ) of X × X , with x = x ⊗ 1 and x00 = 1 ⊗ x as coordinate functions. Then one can ’realize’ a local section of Jr on U as a function of the form 0 00 0 0 00 0 0 00 0 r f(x , x ) = a0(x ) + a1(x )(x − x ) + ··· + ar(x )(x − x ) on U0 × U00. More precisely, consider the closed ideal of the algebra C (X0 × 00 X ): ∞ 00 (r) 0 00 V1V2 ··· Vlf|∆ = 0, ∀ Vj ∈ C (TX ), (1) I∆ := f ∈ C (X × X ) . ∀ 1 ≤ j ≤ l, 0 ≤ l ≤ r.∞ ∞ In other words, it consists of functions on X0 × X00 whose derivatives in X00- directions restricted on the diagonal vanish up to r-th order. The algebra of smooth functions on the r-th order formal neighborhood of the diagonal then can be defined as (r) 0 00 (r) C (X∆ ) := C (X × X )/I∆ . Note that any function defined∞ only∞ on an open neighborhood of the diag- (r) onal also gives an equivalence class in C (X∆ ), but by multiplying with a bump function near the diagonal one can∞ still take a function globally define on X0 × X00 as a representative of the same class. There is an obvious isomorphism

(r) =∼ r (2) τ1 : C (X∆ ) − C (X, J ), [f] 7 sf, ∞ ∞ where sf is defined by → r → sf(p) = jp(f|{p}×X 00 ), that is, we restrict f on the fibre {p} × X00, identified naturally with X, and take its r-jet at (p, p), which is the intersection of {p} × X00 with the diagonal. 0 (r) Moreover, it is an C (X )-algebra isomorphism, if we endow C (X∆ ) with 0 ∗ 0 0 00 the C (X )-module∞ structure via the map pr1 : C (X ) C (X∞× X ). Unfortunately∞ the inverse of τ1 cannot be written∞ down∞ in a clean way. For the construction of the inverse we need partition of→ unity. Take a locally finite over of X by local charts, so that one can write a section s of Jr as a sum NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES 5 of sections si, each supported in some local chart (Ui, xi) of the cover. Then on Ui, si is of the form r si(p) = [a0(p) + a1(p)(xi − p) + ··· + ar(p)(xi − p) ], ∀ p ∈ Ui. We have correspondingly a function 0 00 0 0 00 0 0 00 0 r fi(xi, xi ) = a0(xi) + a1(xi)(xi − xi) + ··· + ar(xi)(xi − xi) 0 00 0 00 0 00 on Ui × Ui ⊂ X × X . The open subsets Ui × Ui also form a locally finite cover of an open neighborhood of the diagonal, so we can form f = i fi, which is a function defined near the diagonal and thus gives an element in (r) −1 P C (X∆ ). Easy to check it is equal to τ (s). (r) (r) ∞The only unsatisfying thing is that, in our definition of C (X∆ ) or I∆ , knowledge about the special splitting of the ambient manifold∞ is assumed. But we observe that the definition doesn’t change if we take vector fields of arbitrary direction in (1)! (Exercise) So we come up with a more intrinsic definition (3) 0 00 (r) 0 00 V1V2 ··· Vlf|∆ = 0, ∀ Vj ∈ C (T(X × X )), I∆ := f ∈ C (X × X ) . ∀ 1 ≤ j ≤ l, 0 ≤ l∞≤ r. ∞ In this way one can immediately generalize the definition to the case of general embeddings X , Y. Of course, the reader might already notice that the ideals we’re talking about here are nothing but again powers of the ideal of functions vanishing along→ the submanifold. We don’t want to emphasize this, however, since it’s no longer true in the following discussion about holomorphic formal neighborhoods.

2. HOLOMORPHICJETBUNDLES Now let’s consider holomorphic jet bundles. Let X be a complex manifold with structure sheaf OX of germs of holomorphic functions. Mimicking what we did in the C -case, for a given point p ∈ X, there is the space of r-jets of holomorphic functions∞ at p, r r+1 Jp := Op/mp , where Op is the stalk of OX at p and mp ⊂ Op is the maximal ideal of germs of holomorphic functions vanishing at p. Under a local (holomorphic) chart (U, zi) containing p, we have an isomorphism r ∼ r+1 Jp = C[z1, . . . , zn]/(z1 − p1, . . . , zn − pn) . 6 SHILIN YU

r Let me be lazy again and assume dimCX = 1. J as a smooth vector bundle has smooth sections which are locally of form r r s(p) = [a0(p) + a1(p)(z − p) + ··· + ar(p)(z − p) ] ∈ Jp , p ∈ U r for some local chart (U, z), where a0, . . . , ar ∈ C (U). To give Jp its holomorphic structure, I just declare that s is holomorphic∞ if and only if those ai’s are holomorphic for some (any) chart. It’s an easy exercise to check the definition works. So locally the sections [1], [z − p],..., [(z − p)r] give a local holomorphic frame of J r. Thus the ∂-derivation on the Dol- beault complex Ω0,•(J r) can be written locally as r ∂s = ∂a0 ⊗ [1] + ∂a1 ⊗ [z − p] + ··· + ∂ar ⊗ [(z − p) ]. Can we fit J r again into the diagonal picture? A first try is to consider all functions on X0 × X00 which are holomorphic along X00-direction but only smooth in X0-direction. Well, this works but again we appeal to the special feature of the product X0 × X00. To overcome this, we consider all smooth functions on X0 × X00, but then take the quotient by an appropriate equivalence relation so that we only memorize holomorphic derivatives of our functions. To begin with, one notice that there is an alternative definition of r the fiber Jp : r (r) Jp = C (X)/ap (r) ∞ where ap is the closed ideal consisting of all smooth functions whose holomorphic derivatives vanish at p up to order r. Next, one just modify (1) to define ∗ 1,0 00 0 00 ∆ (V1V2 ··· Vlf) = 0, ∀ Vj ∈ C (T X ), (4) ar := f ∈ C (X × X ) . ∀ 1 ≤ j ≤ l, 0 ≤ l ≤ r.∞ ∞ In other words, Vi’s in the definition are (1, 0)-tangent vector fields in the direction of X00. As before we can drop the restriction on directions as in (3), so we finally come up with (5) ∗ 1,0 0 00 0 00 ∆ (V1V2 ··· Vlf) = 0, ∀ Vj ∈ C (T (X × X )), ar := f ∈ C (X × X ) . ∀ 1 ≤ j ≤ l, 0 ≤ l∞≤ r. ∞ Then we have a new model for J r (more precisely, C (X, J r)): r 0 00 ∞ A(X∆) := C (X × X )/ar. ∞ Note that ar is just the ideal of smooth functions vanishing along the diago- r+1 nal, but ar 6= a0 . NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES 7

As in the C -case, we have a C (X0)-algebra isomorphism

∞ (r) =∼ ∞ r (6) σ1 : A(X∆ ) − C (X, J ), [f] 7 s[f], such that ∞ r s[f](→p) = jp(f|{p}×X 00 ), → which is however independent of the choice of representative function f. −1 The inverse σ1 can be constructed using partition of unity in the same as −1 τ1 in the previous section. 0 (r) (r) Next, I’ll extend the algebra A (X∆ ) = A(X∆ ) to a differential graded • (r) 0,• r algebra A (X∆ ), which is isomorphic to the Dolbeault complex Ω (J ), yet in such a way that it works for arbitrary embedding. Let me just throw • the definition and explain in a moment. We define a closed dg-ideal ar of 0,• 0 the Dolbeault dga A (X × X) with ar = ar as the zero-th component: ∗ 1,0 k 0,k ∆ (LV LV ··· LV ω) = 0, ∀ Vj ∈ C (T (X × X)), a := ω ∈ A (X × X) 1 2 l r ∀ 1 ≤ j ≤ l, 0 ≤ l ≤ r. ∞ ∗ • where ∆ is the pullback map of differential forms. If we can check that ar is invariant under the ∂-derivation of A0,•(X × X), then the quotient algebra • (r) 0,• • A (X∆ ) := A (X × X)/ar is also a dga. Before we do that, let me point out that all this gadget works for arbitrary closed embedding i : X , Y, once we substitute the ambient ∗ ∗ • manifold X × X by Y and ∆ by i in the definition of ar : (7) ∗ → 1,0 k k 0,k i (LV LV ··· LV ω) = 0, ∀ Vj ∈ C (T Y), a = a := ω ∈ A (Y) 1 2 l r X/Y,r ∀ 1 ≤ j ≤ l, 0 ≤ l ≤ r. ∞ Thus for any closed embedding i : X , Y and r ∈ N we can associate a “Dolbeault dga” • (r) • • A (XY ) := A→(Y)/ar which can be thought of as the Dolbeault complex on the rth-order formal neighborhood of X in Y. • Let’s verify that ar is indeed a dg-ideal. First notice that, if V is a (1, 0)- vector field and ω is a (0, k)-form, then the Lie derivative LV ω is still a (0, k)- form on Y. Moreover, this operation is linear with respect to V, i.e., if g is a smooth function, then LgV ω = g · LV ω. Indeed, by Cartan’s formula,

LV ω = ιV dω + dιV ω. 8 SHILIN YU

But ιV ω = 0 since we’re contracting a (1, 0)-vector ﬁeld with a (0, k)-form! So we have LV ω = ιV dω = ιV (∂ω + ∂ω) = ιV ∂ω, which is obviously linear in V. We also use this equality to compute the commutator of ∂ with LV :

∂(LV ω) − LV (∂ω) = ∂(ιV ∂ω) − ιV ∂∂ω

= ∂(ιV ∂ω) + ιV ∂∂ω

(8) = (∂ ◦ ιV + ιV ◦ ∂)∂ω

= ι∂V ∂ω

= L∂V ω Let me explain the last two equalities. Here we extend the contraction ι to an operation 0,k 1,0 1,l 0,k+l ι(·)(·): A (T Y) × AY AY such that 0,0 →1,• 1,0 ιη⊗V ξ = η ∧ (ιV ξ), ∀ η ∈ AY , ξ ∈ AY , ∀ V ∈ C (T Y). Similarly, we can also deﬁne an extension of the Lie derivative∞ 0,k 1,0 0,l 0,k+l L(·)(·): A (T Y) × AY AY by 0,• 1,0 Lη⊗V ζ := ιη⊗V ∂ζ = η ∧ LV ζ, ∀ η, ζ ∈ A→Y , ∀ V ∈ C (T Y). By these notations, one can show that ∞

(9) ι∂V = ∂ ◦ ιV + ιV ◦ ∂ = [∂, ιV ] •,• (e.g., using local coordinate system). Note that ιV is an operator on AY of degree −1, thus according the Koszul sign convention, the commutator of ∂ and ιV is indeed ∂ ◦ ιV + ιV ◦ ∂ instead of the one with minus sign. Remark 2.1. There is nothing fantastic about our extension of the Cartan’s 1,• 0,• 1,0∗ formula. In fact, one can identify the complex AY with AY (T Y) via 0,• 1,0∗ =∼ 1,• γ : AY (T Y) − AY , ω ⊗ µ 7 ω ∧ µ 0,• 1,0∗ 0,• 1,0 where ω ∈ AY and µ ∈ C (T Y). There is a natural pairing AY (T Y) 0,• 1,0∗ with AY (T Y), denoted by∞ → → 0,k 1,0 0,l 1,0∗ 0,k+l h·, ·i : AY (T Y) × AY (T Y) AY Then under the isomorphism γ, we can relate h·, ·i with the contraction ι via |η| → hω ⊗ V, η ⊗ µi = (−1) ιω⊗V (η ∧ µ). NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES 9

One can use this relation to translate the Lebniz rule of the Dolbeault differ- 1,• ential ∂ on h·, ·i into the equality (9) about ι on AY .

What we’ve just done can be packaged into one single natural homomorphism of dg-Lie algebras θ : 0,•(T 1,0Y) er• ( 0,•, 0,•). A D C AY AY

In local holomorphic coordinates z1, . . . , zn, it is given by ∂ → ∂ θ φ (fdzI) = Lφ ∂ (fdzI) = φ ∧ dzI. ∂zi ∂zi ∂zi 0,• 1,0 The Dolbeault differential in AY (T Y) corresponds, via the homomorphism θ, to the adjoint operator [∂, −] : Der• (A0,•, A0,•) Der•+1(A0,•, A0,•) C Y Y C Y Y • Observe that if we substitute, in the deﬁnition (7) of ar , those LVi ’s by Lie 0,• 1,0 derivatives LVi with respect to Vi ∈ A→Y (T Y), nothing will be changed. • 0,• 1,0 Indeed, if ω ∈ ar and Vi ∈ AY (T Y), 1 ≤ i ≤ l, l ≤ r, then we also have ∗ i LV1 ··· LVl ω = 0. Moreover, by (8), the commutator l [∂, L ··· L ] = L ··· L ··· L V1 Vl V1 ∂Vi Vl i=1 X 0,• • is still a differential operator on AY of order ≤ l. Thus if ω ∈ ar , then l i∗L ··· L ∂ω = ∂i∗L ··· L ω − i∗L ··· L ··· L ω = 0, V1 Vl V1 Vl V1 ∂Vi Vl i=1 X • for any 0 ≤ l ≤ r, which means that ∂ω also lies in ar . Hence our Dolbeault dga is well-deﬁned.

• (r) Let’s get a taste of this abstractly defined dga A (XY ) by studying an easy example. Let X = C and Y = C × C = {(z, w) | z, w ∈ C} and the embedding i : X Y, i(z) = (z, 0) identifies X with the submanifold of Y defined by the equation w = 0.A 0 smooth function f on Y belongs→ to ar if and only if, by means of Taylor expansion, it can be written as f(z, w) = wr+1 · g(z, w) + w · h(z, w) 10 SHILIN YU

0 0,0 for some g, h ∈ C (Y). Hence ar is the ideal of AY = C (Y) generated by r+1 functions w and∞w and there is an isomorphism ∞ r ∼ 1 ∂kf A0(X(r)) = A0,0/a0 −= A0,0 ⊗ [w]/(w)r+1, [f] 7 (z, 0) · wk. Y Y r X C C r k! ∂wk k=0 X For a smooth (0, 1)-form→ → ω = f(z, w)dz + g(z, w)dw,

1 0 ∗ it lies in ar if and only if f ∈ ar . No condition to put on g since i dw = 0. Thus 1 0 0,0 ar = ar · dz + AY · dw and there is an isomorphism

1 (r) 0,1 1 =∼ 0,1 r+1 A (X )Y = AY /ar − AX ⊗C C[w]/(w) deﬁned by r → 1 ∂kf [fdz + gdw] 7 (z, 0)dz ⊗ wk. r k! ∂wk k=0 X Put these two isomorphisms together,→ we obtain an isomorphism between dgas • (r) 0,• r+1 A (XY ) = AX ⊗C C[w]/(w) .

Now let’s go back to the case of the diagonal embedding and see how to • (r) r identify our new-defined dga A (X∆ ) with the Dolbeault complex of J . It’s of the same spirit as definition (6) of τ1 in the previous section, but one needs a little bit more than that to deal with (0, k)-forms with k ≥ 1. Let me illustrate the case when k = 1. General cases follow easily from a similar 1 argument. Given [ω] ∈ ar , the goal is to construct a corresponding a section 0,1 r 0,1 sω of the vector bundle Hom(T X, J ). For any W ∈ Tp X at some point 0,1 0 00 p ∈ X, we form ∆∗W ∈ T(p,p)(X × X ), a (0, 1)-tangent vector along the diagonal at (p, p) ∈ X0 × X00 by pushforward. We can always extend this vector to a local ’anti-holomorphic’ (0, 1)-vector field Wf on an open neighborhood of (p, p), such that for any (1, 0)-vector field V on X0 × X00, the Lie bracket [V, Wf] is again of (1, 0)-type (Exercise!). Finally contract Wf with ω, restrict the resulted function along {p} × X00 and take the jet at (p, p). In short, we have r s (W) = j ((ι ω)| 00 ). [ω] p We {p}×X NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES 11

One can check this deﬁnition is independent of the choice of Wf and representative form ω. To see this, one only needs to notice that, because of the way we choose Wf, there is V V ··· V (ι ω) = ι (L L ··· L ω) 1 2 l We We V1 V2 Vl 1,0 k for any Vi ∈ C (T (X × X)). The case of A for arbitrary k is similar. Thus we obtain a homomorphism∞ of graded algebra • (r) 0,• r σ1 : A (X∆ ) Ω (J ), [ω] 7 s[ω] More explicitly, under some local chart (U, z) containing p, hence (U0 × U00, z0, z00) containing (p, p), suppose→ → ω = f(z0, z00)dz0 + g(z0, z00)dz00 ∈ Ω0,1(U0 × U00),

1 (r) which defines an equivalence class [ω] ∈ A (U∆ ), then " r # 1 ∂if ∂ig s (p) = dz ⊗ (p, p) + (p, p) (z − p)i , ∀ p ∈ U. [ω] i! ∂z00i ∂z00i i=0 X Using this local expression, one can check that σ1 is also a homomorphism of dgas. Also from this we see that there is an isomorphism of dgas • (r) ∼ 0,• r+1 A (U∆ ) = Ω ⊗ C[dz]/(dz) , or written in more general form for arbitrary dimension, • (r) ∼ 0,• r i ∗ (10) A (U∆ ) = ΩU (⊕i=0S T X), where we identify T ∗X with the (1, 0)-cotangent bundle Ω1,0X. Moreover, • r • 0 0∗ • 0 • r A (X∆) is an A (X )-algebra via pr : A (X ) A (X∆) and σ1 respects this −1 structure. Hence the inverse of σ1 can be constructed by extending σ1 on the zero-th component in the following way: → −1 0∗ −1 0,• 0 r σ1 (η ⊗ s) = [pr (η)] · σ1 (s), ∀ η ∈ Ω (X ), s ∈ C (J ). ∞ 3. KAPRANOV’S THEOREM The rest of the notes will contribute to the understanding the holomorphic structure of the formal neighborhood of the diagonal embedding, i.e., the • ( ) dga (A (X∆ ), ∂). As we’ve seen before, at least in some local chart, we have some∞ isomorphism • ( ) ∼ 0,• ^ ∗ A (U∆ ) = ΩU (S(T X)) ∞ 12 SHILIN YU where S^(T ∗X) = SiT ∗X ∞ i=0 Y is the bundle of complete symmetric algebras generated by T ∗X and the dif- 0,• ^ ∗ ∗ ferential on ΩU (S(T X)) is the usual ∂-derivation induced from that of T X. This isomorphism is just obtained by taking the inverse limit of isomorphisms (10) for all formal neighborhoods of finite orders. In the global case, however, we don’t have such an isomorphism if X is not affine. By this, I mean that one can always find an isomorphism between the two as graded algebras (actually there are plenty of such isomorphisms), but it might not be compatible with the usual ∂ on S^(T ∗X). What we can do is to correct the holomorphic structure on S^(T ∗X) to make it compatible. It • ( ) may sound trivial because one can just transfer the differential on A (U∆ ) 0,• ^ ∗ to one on ΩU (S(T X)) via the chosen isomorphism. It requires some work,∞ however, to write down explicitly the transferred differential. For this purpose we need to pick an isomorphism with transparent geometric meaning so that the new differential could be expressed in terms of the geometry of X. Now suppose that X is equipped with a Kahler¨ metric h. Let ∇ be the canonical (1, 0)-connection in TX associated with h, so that (11) [∇, ∇] = 0 in Ω2,0(End(TX)). and it is torsion-free, which is equivalent to the condition for h to be Kahler.¨ For most of the time here, however, I will use ∇ as a connection on the cotangent bundle T ∗X. In other words, I think of ∇ as a differential operator ∇ : T ∗X T ∗X ⊗ T ∗X such that → h∇(α),V ⊗ Wi = h∇V α, Wi where α is any section of T ∗X and V and W are any sections of TX. I can also apply ∇ iteratively to get sections of higher order tensors of T ∗X. Moreover, I will put a constant family of ∇’s on X0×X00 along the X00-fibers, so that we get a differential operator only in X00-directions with respect to the decomposition T(X0 × X00) = TX0 ⊕ TX00: ∇ : T ∗X00 T ∗X00 ⊗ T ∗X00 where I still use the same notation ∇. Then we can define → ∼ ∗ • ( ) = 0,• ^ ∗ exp : A (XX×X) − Ω (S(T X)) ∞ → NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES 13 by ∗ ∗ ∗ ∗ 2 ∗ n ^ ∗ expσ([f] ) = (∆ f, ∆ ∇f, ∆ (∇) f, ··· , ∆ (∇) f, ··· ) ∈ S(T X) where ∞ ∇f := ∂00f is the (1, 0)-differential of f along X00-fibers, so it lies in T ∗X00. And ∇if = ∇i−1∂00f, i ≥ 2 To show that ∇if indeed lies in symmetric tensors, one has to resort to the torsionfreeness and flatness of ∇. The curvature of ∇˜ = ∇ + ∂ is just R = [∂, ∇] ∈ Ω1,1(End(TX)) = Ω0,1(Hom(TX ⊗ TX, TX)) which is a Dolbeault representative of the Atiyah class αTX of the tangent bundle. In particular one has the Bianchi identity: ∂R = 0 in Ω0,2(Hom(TX ⊗ TX, TX)) Actually, by the torsion-freeness we have R ∈ Ω0,1(Hom(S2TX, TX))

We then deﬁne tensor ﬁelds Rn, n ≥ 2, as higher covariant derivatives of the curvature: 0,1 2 ⊗(n−2) (12) Rn ∈ Ω (Hom(S TX ⊗ TX , TX)),R2 := R, Ri+1 = ∇Ri

In fact Rn is totally symmetric, i.e., 0,1 n 0,1 ∗ n ∗ Rn ∈ Ω (Hom(S TX, TX)) = Ω (Hom(T X, S T X)) by the ﬂatness of ∇ (??).

REFERENCES [1] Borel, Sur quelques points de la th´eoriedes fonctions, Ann. Ecole Norm. sup.,. Ser. 3, 12, 1895

DEPARTMENT OF MATHEMATICS,PENNSYLVANIA STATE UNIVERSITY,UNIVERSITY PARK, PA 16802, USA