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Notes on Formal Neighborhoods and Jet Bundles

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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. Holomorphic1 jet bundles 5 3. Kapranov’s Theorem 11 References 13 1. C -JET BUNDLES The notion of C -jet bundle1 provides an appropriate place where one can talk about Taylor1 expansions (or jets) of smooth functions on a smooth manifold. Let X be a manifold and p 2 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) 1 Ip = ffunctions whose derivatives up to order r all vanish at pg: In fact, if mp denotes the maximal ideal of the commutative algebra C (X) consisting of functions vanishing at p, we have 1 (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 2 Jp = C (X)=mp 1 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 itself1 is a quotient of C (X), so everything is fine. Indeed,1 let np be the ideal of all global smooth functions1 vanishing at some neighborhood of p, then there is a natural isomorphism ∼ C (X)=np = C (X)p: This is not true in the holomorphic1 world.1 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) XjIj≤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 1 1 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 1 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 that1 this map1 is surjective. In other words, there always exists (locally) a C -function! with the given Taylor expansion. Again this fails when it comes to1 holomorphic functions.The kernel of jp is ( ) \ (r) \ r+1 Ip = Ip = mp : 1 1 1 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 the1 sheaf of smooth sections of1 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) 2 Jp; 8 p 2 V; p2V I only look at those with the! following property: for any y 2 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) ] 2 Jp r where a0; : : : ; ar 2 C (U). We denote the set of all such sections by C (V; J ). It has a C (U)-module1 structure defined by pointwise addition and1 multi- plication,1 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 vec- tor 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 does1 not necessarily1 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 1 1 j : C (X) ! C (X; J ) r under which the images of1 a function1 are1 called1 prolongations. However, j is not a C (X)-linear map. It is a differential! operator of order r. From the1 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 func- tions’ 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 ): 1 00 (r) 0 00 V1V2 ··· Vlfj∆ = 0; 8 Vj 2 C (TX ); (1) I∆ := f 2 C (X × X ) : 8 1 ≤ j ≤ l; 0 ≤ l ≤ r:1 1 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 defined1 only1 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 can1 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; 1 1 where sf is defined by ! r ! sf(p) = jp(fjfpg×X 00 ); that is, we restrict f on the fibre fpg × X00, identified naturally with X, and take its r-jet at (p; p), which is the intersection of fpg × 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 )-module1 structure via the map pr1 : C (X ) C (X1× X ). Unfortunately1 the inverse of τ1 cannot be written1 down1 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) ]; 8 p 2 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) 1The only unsatisfying thing is that, in our definition of C (X∆ ) or I∆ , knowledge about the special splitting of the ambient manifold1 is assumed.
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