The Ext algebra of a quantized cycle

Given a quantized analytic cycle $(X, \sigma)$ in $Y$, we give a categorical Lie-theoretic interpretation of a geometric condition, discovered by Shilin Yu, that involves the second formal neighbourhood of $X$ in $Y$. If this condition (that we call tameness) is satisfied, we prove that the derived Ext algebra $\mathcal{RH}om_{\mathcal{O}_Y}(\mathcal{O}_X, \mathcal{O}_X)$ is isomorphic to the universal enveloping algebra of the shifted normal bundle $\mathrm{N}_{X/Y}[-1]$ endowed with a specific Lie structure, strengthening an earlier result of C\u{a}ld\u{a}raru, Tu, and the first author This approach allows to get some conceptual proofs of many important results in the theory: in the case of the diagonal embedding, we recover former results of Kapranov, Markarian, and Ramadoss about (a) the Lie structure on the shifted tangent bundle $\mathrm{T}_X[-1]$ (b) the corresponding universal enveloping algebra (c) the calculation of Kapranov's big Chern classes. We also give a new Lie-theoretic proof of Yu's result for the explicit calculation of the quantized cycle class in the tame case: it is the Duflo element of the Lie algebra object $\mathrm{N}_{X/Y}[-1]$.

The diagonal embedding case.Let be X a complex manifold or a smooth algebraic variety over a field of characteristic zero.Thanks to the celebrated result of Hochschild, Kostant and Rosenberg (cf.[10]), the Hochschild homology and cohomology groups of the sheaf O X are given by HH i (O X ) = Ω i X and HH i (O X ) = Λ i T X .These isomorphisms can be upgraded at the level of derived categories, and are called (geometric) HKR isomorphisms.For more history on this topic, we refer the reader to the paper [8], as well as references therein.In the present paper, we will be especially interested in the geometric HKR isomorphism involving Hochschild cohomology: this isomorphism is an additive sheaf isomorphism between the sheaf of polyvector fields S(T X [−1]) = ⊕ i≥0 Λ i T X [i] and the derived Hom sheaf of the diagonal p 1 * RHom O X×X (O X , O X ).Here we view X as the diagonal inside X × X (which is the geometric counterpart of looking at an algebra as a bimodule over itself) and p 1 is the first projection.
Both members of this isomorphism have multiplicative structure: the wedge product on polyvector fields, and the Yoneda product on the derived Hom complex.Taking the cohomology on both sides, for any non-negative integer p, the corresponding isomorphism between the algebras ⊕ p i=0 H p−i (X, Λ i T X ) and Ext i X×X (O X , O X ) is not multiplicative in general (contrarily to the homological geometric HKR).It has been conjectured by Kontsevich [13] and later proved by Van den Bergh and the first author [2] that one can use the square root of the Todd class Td (X) on ⊕ i H i (X, Ω i X ) of the tangent sheaf T X to "twist" the global cohomological HKR isomorphism in order to get an isomorphism of algebras.
The Todd class of X is also intimately related to the geometry of the diagonal of X in another way: it is the correction term appearing on the Grothendieck-Riemann-Roch theorem, so thanks to the Lefschetz formalism it can be interpreted as a restriction of the Grothendieck cycle class to the diagonal itself.This has been hinted in the pioneering work of Toledo-Tong (see [17, §6]), and formalized using HKR isomorphisms in an unpublished manuscript of Kashiwara around 1992.Kashiwara's account can be found in [7], where the second author proves in fact that the Todd class is the Euler class of O X , proving a conjecture of Kaschiwara-Schapira [12].
The above results can be re-interpreted very naturally in Lie-theoretic terms after the works of Kapranov [11] and Markarian [14]: -The shifted tangent sheaf T X [−1] is a Lie algebra object in the derived category D b (X).
-Thanks to results of Markarian [14] and Ramadoss [15], the universal enveloping algebra of this Lie algebra object is indeed the derived Hom sheaf, and that the geometric HKR isomorphism can be re-interpreted as the Poincaré-Birkhoff-Witt (PBW) isomorphism.-Every element F in D b (X) is naturally a representation of T X [−1], and via the PBW isomorphism the character of this representation (which is a central function on U(T X [−1])) can be identified with the Chern character of F .-The Todd class becomes the derivative of the multipication map in the universal enveloping algebra, and is therefore the Duflo element of T X [−1].-The isomorphism HKR • ι √ Td (X) from [13,2] can be seen as a Duflo isomorphism (see [6]) for the Lie algebra object T X [−1].We refer to [4], [3] for further analogies between Lie theory and algebraic geometry.
1.2.More general embeddings: tame quantized cycles.In the present paper, we are interested in the more general situation where we replace the diagonal embedding ∆ X → X × X by an arbitrary closed immersion X → Y, where X is a smooth closed subscheme of an ambiant smooth scheme Y.
In [1], Arinkin and Cȃldȃraru gave a necessary and sufficient condition for an additive generalized geometric HKR isomorphism to exist between RHom O Y (O X , O X ) and S(N X/Y [−1]): the condition is that N X/Y extends to a locally free sheaf on the first infinitesimal neighborhood of X in Y.The Lie theoretic interpretation of the first order neighborhood and of the above geometric condition has been given in [3] by Cȃldȃraru, Tu, and the first author.
Earlier on, in Kashiwara's 1992 unpublished manuscript, a more restrictive condition is introduced: Kashiwara deals with subschemes with split conormal sequence, which means that the map from X to its first infinitesimal neiborhood in Y admits a global retraction (in this case, any locally free sheaf on X extends at order one in Y).On the Lie side, this corresponds to pairs h ⊂ g that split as h-modules, these are usually called reductive pairs.In [8], the second author developed Kashiwara's construction in this framework.The data of a subscheme X of Y together with such a retraction σ is called a quantized cycle, and to such a cycle it is possible to associate geometric HKR isomorphisms, as well as a quantized cycle class q σ (X) living in ⊕ i≥0 H i (X, Λ i N * X/Y ) that generalizes the Todd class in the diagonal case.Recently, answering a question raised by the second author in the article [8], Yu has shown in [18] the following result: given a quantized cycle (X, σ) in Y such that σ * N X/Y extends to a locally free sheaf on the second infinitesimal neighborhood of X in Y, -The quantized cycle q σ (X) class is completely determined by the geometry of the second infinitesimal neighborhood of X in Y. -It can be expressed by an explicit formula similar to that of the usual Todd class.Yu's proof is based on direct calculation using the dg Dolbeault complex as well as homological perturbation theory.In this paper we provide a Lie theoretic explanation of Yu's results.We introduce the notion of tame quantized cycle corresponding to Yu's condition: a quantized cycle (X, σ) is tame if σ * N X/Y extends to a locally free sheaf at the second order.We can list all the conditions that can be investigated on the cycle X (each condition being more restrictive than the previous one), and the corresponding conditions for Lie algebra pairs: If one of the two last conditions is satisfied, the object N X/Y [−1] is naturally a Lie object in D b (X), but this is no longer the case if we drop the tameness assumption.In full generality (that is without any specific quantization conditions), the algebraic structure of N X/Y [−1] has been investigated in [4]: it is a derived Lie algebroid, whose anchor map is given by the extension class of the normal exact sequence of the pair (X, Y).Hence, our setting can be understood as the weaker universal hypotheses for which this derived algebroid is a true Lie object in the symmetric monoidal category D b (X).
where all horizontal arrows are algebra morphisms, and all vertical arrows are isomorphisms.
1.4.Plan of the paper.The paper is organized as follows: - §2 is devoted to some recollection about Lie algebra objects in a categorical setting.We claim no originality for this material, but we were not able to find the desired results in the form we needed in the literature.For instance, most references are written for abelian categories while we work in the slightly more general Karoubian framework.- §3 deals with three different topics.§3.1 gives universal formulas for the multiplication map U(g) ⊗ g → U(g) via the PBW isomorphism.Up to our knowledge, this result has never been proved in this level of generality.In §3.2, we define an algebraic condition that characterizes uniquely the Duflo element of a Lie object in a symmetric monoidal category.This will be the key ingredient to our Lie-theoretic proof of Yu's result.In §3.3, we introduce the "tame condition" for pairs of Lie algebras, which is a Lie theoretic analog of Yu's condition.- §4 recollects previous results on (first and second order) infinitesimal neighborhoods, HKR isomorphisms and quantized cycles.This is were we state the geometric tameness condition, a-k-a Yu's condition.-in §5 we explain that the geometric tameness condition coincides with the Lie theoretic tameness condition for the pair of Lie algebra objects in D b (X).We get in particular that in the tame case, N X/Y [−1] is a Lie algebra object in D b (X) and we describe its universal enveloping algebra in geometric terms.Using the Lie-theoretic results of §2-3, we are able to give enlightening and simple proofs of results of Ramadoss about Kapranov big Chern classes (diagonal case), and Yu's formula for the quantized cycle class.
-In §6 we use the above to get a description of the Ext algebra of a tame quantized cycle.We show in particular that it is completely determined by the second order infinitesimal neighborhood of X in Y. -We conclude the paper with a few perspectives in §7.
2. Universal enveloping algebras and the categorical PBW theorem 2.1.Preliminary results of linear algebra.
2.1.1.Partially antisymmetric tensors.Let k be a field of characteristic zero and let C be a k-linear symmetric monoidal category that is Karoubian (i.e.every idempotent splits 1 ) and such that countable direct sums exist.We can assume without loss of generality that C is a strict monoidal category (i.e. it is harmless to drop the parenthesizations of iterated tensor products from the notation).For any nonnegative integer n, the symmetric group S n acts naturally on V ⊗n , where V is an object of C. Let π n be the element (n!) −1 ∑ g∈S n g, considered as an idempotent element of the group algebra k[S n ].It induces a natural idempotent2 on V ⊗n , whose kernel is denoted by Λ n V and whose image is denoted by S n V. We therefore have a decomposition Assume that n is at least two and let Proof.For i in ⟦1, n − 1⟧, let τ i be the transposition in the group S n that switches i and i + 1.We first observe that (1 Hence the kernel of 1 + τ i acting on V ⊗n is a (split) sub-object of the kernel of π n acting on V ⊗n : In other words, the map Ψ n factors through Λ n V.In order to conclude, it is then sufficient to prove that ) → Λ n V admits a section.We claim that the right ideal in k[S n ] generated by 1 − τ i (1 ≤ i ≤ n − 1) contain all elements 1 − τ for arbitrary τ in S n .Indeed, for any elements g 1 , . . ., g d in the group algebra k[S n ], we have The claim follows again from the fact that the τ i generate S n .As a corollary, 1 − π sits in this ideal, so that we can choose elements (a i ) 1≤i≤n−1 in the group algebra such that As a consequence we get that the map The simplest nontrivial case is n = 3.In this case we can take for instance Proposition 2.3.The pair (V, α) is a Lie algebra object if and only if there exists The Jacobi identity is equivalent to the identity with the right inverse of Ψ 3 given by Lemma 2.1.This gives a morphism β : 2 and a short calculation provide the following explicit formula: This morphism β is the only possible candidate to fulfil the desired condition Hence This condition is clearly implied by the Jacobi identity, but it is in fact equivalent to it.Indeed, 2u Algebras satisfying the PBW isomorphism.

2.2.1.
The morphisms c k p .Let V be an object in C and let A be a unital augmented algebra object in C together with an algebra morphism ∆ : T(V) → A. From now on, we require Here T(V) := ⊕ n≥0 V ⊗n is equipped with the concatenation product, S(V) := ⊕ n≥0 S n V and the map S(V) → T(V) is the direct sum of direct factor embeddings S n V ⊂ V ⊗n .Note that A carries a split increasing filtration: . Moreover, ∆ is a filtered morphism for the obvious degree filtration on T(V).We now consider the restriction ∆ p of the filtered morphism ∆ −1 + • ∆ on each homogeneous component V ⊗p ; it decomposes as follows: where c k p ∈ Hom C (V ⊗p , S k V).Since A is augmented, c 0 p vanishes.We further make the following 3 Only as objects of C, not as algebras.

Assumption (A2)
For any non-negative integer p the morphism V ⊗p → Gr p A ≃ S p V is the canonical projection π p .
Note that this morphism is nothing but ∆ p followed by the projection onto S p V. This second assumption on A ensures that the restriction of c k p to S p V is zero if k ≤ p − 1, and c p p is the canonical projection π p from V ⊗p to S p V. In particular, if 1 ≤ k ≤ p − 1, then c k p lives naturally in Hom C ( Λ p V, S k V).
2.2.2.The Lie bracket.We define α = c 1 2 : Λ 2 V → V. Since A is augmented, we can identify F 1 A with 1 ⊕ V (via ∆ + ), where 1 is the monoidal unit of C. Lemma 2.4.Let m : A ⊗2 → A be the associative product.Then the morphism V ⊗2 → A defined by Proof.As ∆ is an algebra morphism, we have that The morphism α defines a Lie structure on V.

Induction formulae for c k p .
We can now provide explicit induction formulae for the morphisms c k p .Recall that for 0 ≤ k ≤ p − 1, we consider c k p as an element of Hom C ( Λ p V, S k V).Proposition 2.6.Given a pair (V, A) as above, the coefficients c k p are determined as follows: where m (2) := m • (m ⊗ id).Applying ∆ −1 + followed by the projection on the direct factor S k V we get Hence both members of the induction relation agree on Universal algebras in the categorical setting.

2.3.1.
Reverse PBW theorem.We can now prove our first main result: assuming that the algebra A satisfies the PBW theorem (i.e.Assumptions (A1) and (A2)), we prove that it is the universal enveloping algebra of V endowed with the Lie bracket 2c 1 2 .Proposition 2.7 (Reverse PBW).If A satisfies Assumptions (A1-A2) of §2.2.1, then A is a universal enveloping algebra of the Lie algebra (V, 2α).
Proof.For any associative algebra object B in C, with product m B : B ⊗2 → B, we denote by B Lie = (B, µ B ) the Lie algebra object which is B endowed with the Lie bracket µ B = m B • (1 − τ 1 ).Lemma 2.4 tells us that the direct factor inclusion (∆ + ) |V : V → A is a morphism of Lie algebra objects from (V, 2α) to A Lie in C.
Assume now to be given a morphism f : V → B. By the universal property of the tensor algebra, it defines an algebra morphism f : T(V) → B. Lemma 2.8.If f is a Lie algebra morphism (from (V, 2α) to B Lie ) then f factors through a unique morphism g : A → B.
Proof of the Lemma.It is sufficient to prove that f = f • s • ∆, where s is a section of ∆.Indeed, the only possible choice is g = f • s.Here we use the section s given by ∆ −1 + : A → S(V) ⊂ T(V).We will prove by induction on p that f Note that this identity is obviously satisfied when restricted to S p V ⊂ V ⊗p .The only thing left to prove is thus that -For p ∈ {0, 1} the result is obvious.
-For p = 2, we have -Let us now assume that the required equality holds for a given p ≥ 2. We compute Finally one can prove that in the same way as for the case when p = 2. □ End of the proof of Proposition 2.7.In order to conclude one has to prove that g is indeed a morphism of algebras: Categorical construction of the product on S (V).In this section, we prove the following delicate result: Theorem 2.9.There exists an algebra A that satisfies Assumptions (A1-A2).
Remark 2.10.In [5, Lemma 1.3.7.5], the authors provide an explicit multiplication law m ⋆ on S(V) in the case when C is the category of graded vector spaces (or more generally any abelian category), and prove that it is associative.Then the algebra A = (S(V), m ⋆ ) satisfies all required properties.Their proof should carry on as well for a general C, but the formulas defining m ⋆ are daunting to write down in the categorical setting.This is why we provide another approach, which is close in spirit of [5], but more adapted to the categorical framework.
The proof is long and technical.Before starting, let us introduce some notation that will be used constantly in the sequel: -On S(V) we have an associative and commutative product m 0 defined as the composition -We write τ p,q for the transposition (p, q) in the symmetric group, as well as for the corresponding action on V ⊗n .
-We consider the morphism ω (V, µ) : where µ := 2α.We leave it as an exercise to check that the image of ω (V, µ) indeed factors through S n−1 V ⊗ V ⊂ V ⊗n .We often simply write ω as the choice of (V, µ) is clear from the context.
Proof of Theorem 2.9.Let (V, α) be a Lie algebra object in C. For any integer n ≥ 2, we are going to construct a pair (m ⋆ , π) that satisfies the seven following properties: And its unit is given by the canonical direct factor embedding 1 ⊂ S ≤n (V).
C The action of the Lie algebra V on S ≤n (V) respects the product . D The action of the Lie algebra V on S <n (V) is the commutator for m ⋆ with degree one elements: F The restriction of π to S ≤n (V) is the identity map.G On S n−1 (V) ⊗ V, the morphism m ⋆ is given by the explicit formula To see where condition (G) comes from, we refer the reader to Lemma 3.2.We argue by induction on n.
Initial step.The first interesting case is n = 2. Then m ⋆ is defined by m 0 + α and π is given by m ⋆ itself on V ⊗ V.It is easy to check that all properties are satisfied.
Inductive step: definitions.Assume that the induction hypotheses are satisfied at level n.We define the product "m ⋆ " : inductively for 1 ≤ i ≤ n − 1 as follows: - -Lastly we define "m ⋆ " = m ⋆ for lower degrees.
-Symbols in between quotes are the ones that we've just defined for the induction step.For these ones we are not yet allowed to use properties (B-G), which we in fact have to prove.Note that property (G) holds on the nose, as this is exactly how we defined "m ⋆ " .-In the symmetric group S ℓ , we write τ i,..., j for the permutation τ j−1 . . .τ i+1 τ i (i < j).
Property (F) is satisfied.
First of all, if i + j ≤ n then "m ⋆ " = m ⋆ and this is true because π preserves the product and is the identity on S ≤n (V) according to (F).Now, if i + j = n + 1, then it follows from the definition of "m ⋆ " and from what we've just observed that the following diagram commutes4 : It is therefore sufficient to prove that "m ⋆ " coincides with m 0 on S n+1 (V) ⊂ S n (V) ⊗ V.This follows from the fact that the restriction of ω to S n+1 (V) ⊂ S n (V) ⊗ V vanishes.
A useful identity.
We will transform the second term.The first observation is that 5 .This gives Hence we get We claim that for 1 ≤ k ≤ n − 1, the two terms corresponding to the index k in the two sums cancel.This follows from the identity on S n (V) ⊗ V that we prove as follows: , so it acts trivially on S n−1 (V).This gives (3).This being done, we obtain that This finishes the proof of the identity.
Theorem 2.11.Let V be an object of C and α be an element of Hom C (V, Λ 2 V).
-The system of equations has a solution (c k p ) 1≤k≤p if and only if α is a Lie bracket on V.If it exists, this solution is unique.
-Given a Lie algebra object (V, α) in C, let (c k p ) 1≤k≤p be coefficients satisfying (4).If we define a product m ⋆ on S(V) by the formula , then m ⋆ is associative and A = (S(V), m ⋆ ) is a universal enveloping algebra of (V, 2α).Besides, the PBW theorem holds.
Proof.Assume that α is a Lie bracket.Theorem 2.9 gives an algebra A satisfying Assumptions (A1-A2) of §2.2.1.Hence Proposition 2.6 provides the existence of the c k p .Uniqueness is clear.Conversely, if the equations ( 4) are satisfied, then we have Thanks to Proposition 2.3, α satisfies the Jacobi identity.Now assume that α is a Lie bracket, and let A be an algebra satisfying Assumptions (A1-A2) of §2.2.1.Then we can transport the algebra structure on S(V) via the isomorphism ∆ + : we have This gives the result.□

Distinguished elements in the universal enveloping algebra
3.1.The derivative of the multiplication map.We borrow the notation from the previous Section: (V, α) is a Lie algebra object in C and m ⋆ is the associative product on S(V) from Proposition 2.11.Our aim is to give a closed formula for the restriction φ of m ⋆ to S(V) ⊗ V ⊂ S(V) ⊗ S(V).We use the notation introduced in the beginning of §2.3.2, especially the morphism ω defined in (1).
We consider the morphism ϖ We often simply write ϖ as the choice of V is clear from the context.
with the multiplication morphism m 0 of the symmetric algebra S(V).
We first need an intermediate result of categorical linear algebra.
Lemma 3.2.The restriction of τ p+1, n+1 to S n V ⊗ V followed by the projection V ⊗n+1 → Λ n+1 V equals: Note that τ i,i = 1 by convention.Also remark that, if p ∈ {0, n}, then one of the two sums in the above expression is empty and thus vanishes.
Proof.First recall that the projection V ⊗n+1 → Λ n+1 V is given by 1 − π n+1 .Remark that the subgroup S n of S n+1 corresponds to elements in S n+1 fixing p + 1 act trivially on τ p+1, n+1 (S n V ⊗ V).Besides, S n+1 /S n consists of the (n + 1) left classes (τ i,p+1 S n ) 1≤i≤n+1 .Hence, restricted to τ p+1, n+1 (S n V ⊗ V), we have the identity Then observe 6 that on S n V ⊗ V ⊂ V ⊗n+1 , we have the following identities: and similarly . Using this, we compute: which is indeed (n + 1) times the desired map.□ Corollary 3.3.Let n, ℓ be non-negative integers with 0 ≤ ℓ ≤ n − 1.For any p ∈ {1, . . ., n}, on where λ(n, ℓ, p) are rational numbers satisfying the induction relations Proof.We proceed by induction.The case ℓ = 0 is easy: this amounts to observe that c n n • τ p,n = π n = m 0 on S n−1 V ⊗ V.For the general case, on S n V ⊗ V we have where the first equality follows from Lemma 3.2, the second one from Corollary ??, and the last one from the induction hypothesis.□ Example 3.4.We have In order to prove Theorem 3.1, it remains to show the following: We start by simplifying a little bit the induction relation given in Corollary 3.3.To do so, we claim that for ℓ ≥ 1, ∑ n k=1 λ(n, ℓ, k) vanishes.This follows from a direct calculation: Hence for ℓ ≥ 2, we see that Remark that this relation is also valid for ℓ = 1.
Let us now show by induction on ℓ that λ(n, ℓ, n + 1) doesn't depend on n, and equals to B ℓ ℓ!
• We have already seen that it is true for ℓ = 1.We now assume that the induction hypothesis is true at rank ℓ.For any i, we put • We define modified power sums T k (n) as follows: The proof of the claim is by induction on s.For s = 1, we have The induction step follows from the fact that ∑ We take s = ℓ − 1.We have Now we have the formula which is x.□ 3.1.1.Linear algebra computations.All along this Subsection, we assume that V is a dualizable object in C and we denote by V * its dual7 : in particular, we have a coevaluation map ϵ : 1 → V * ⊗ V and an evaluation map δ : V ⊗ V * → 1 that satisfy the "snake" identity -One shows easily that the restriction of id As a consequence the direct factor, inclusion S n V → S n−1 V ⊗ V can be re-written as the restriction of (id which also equals -We have an adjunction between the functor V ⊗ − and the functor V * ⊗ −: for any two objects where the bijection is given by sending For instance, the element µ * in Hom C (V, V * ⊗ V) is understood as the adjoint action, and we have -We also have an adjunction in the reverse way between the functor − ⊗ V and the functor − ⊗ V * : where the bijection is given by sending -Taking into account that S p (V * ) is canonically isomorphic to S p (V) * , we get a canonical "contraction map" element c p ∈ Hom C (S n V ⊗ S p V * , S n−p V) given as n! (n − p)! times the adjoint to the . One the other hand, using (5), we obtain that ω equals (c 1 ⊗ id V ) • (id V ⊗n ⊗ µ * ) and thus We now introduce a convenient notation.Let X, Y, Z be three objects in C and let (A, m A ) be an associative algebra object in C. We then have a k-linear associative composition product defined as follows: for every ϕ : Y → A ⊗ Z and every ψ : In particular, if X = Y = Z then we get that • turns Hom C (Y, A ⊗ Y) into a k-linear associative algebra.We are interested in the case (A, m A ) = (S(V * ), m 0 ).
as morphisms from S n V ⊗ X to S n−p−q V ⊗ Z. 8 The morphism ω has been defined at the beginning of §3.1.
Proof.First of all, in view of the expression for c p in terms of c 1 's, it is sufficient to prove the Lemma for p = 1.Then, in view of the definition of the product •, we have that the r.h.s.
Hence it is sufficient to show the following identity Then observe that, from the very definition c 1+q , we have that As a consequence, it is sufficient to have that which is obvious.
Proof.Setting Y = V, we get a family of maps (µ * ) •p ∈ Hom C (V, S p V * ⊗ V).Lemma 3.6 gives the result.
Proof.First observe that it is sufficient to prove it for p = 1 (using the associativity of •).Then note that since µ : Therefore µ * • ϵ = 0, and we are done.□ 3.1.2.The trace identity.For every two objects Y, Z in C one has a linear map defined as follows: where ε is given by ϵ followed by the symmetry morphism Proposition 3.9.For every p, we have that Proof.Using the fact that c 1 defines an action of V * on S(V) by derivations, we get that the r.h.s. is Then observe that we have Finally, We are done.□

The Duflo element.
We borrow the notation from the previous paragraphs and introduce the Duflo ) .
This has to be understood as a formal expression in terms of the "invariant polynomials" More formally, for any N ≥ 0, we write formally where x i n , and P i is a polynomial independant from N, and of total degree i if each variable y k Then for any p ≥ 0, we have d p = P p (ν 1 , . . ., ν p ) .

Torsion morphisms.
Let ℓ ∈ N and let a be a morphism from an arbitrary object X to S ≤ℓ V and (a n ) 0≤n≤ℓ be the graded components of a.
Definition 3.10.We say that such a morphism a is an ℓ-torsion morphism if m ⋆ •(a⊗id V ) factors through S ℓ+1 V ⊂ S ≤ℓ+1 V.
Our main result is: Remark that this theorem tells nothing about the existence of ℓ-torsion morphisms.
Proof.Thanks to Theorem 3.1, a is an ℓ-torsion morphism if and only if the system of equations holds for 1 ≤ k ≤ ℓ.Each condition corresponds to the vanishing of the ℓ − k + 1 th graded piece of the element m ⋆ • (a ⊗ id V ).Thanks to Proposition 3.9, we get Let us explain how it works on the first terms.
-For k = 1, the first equation is ) .
-For k = 2, the second equation is and we get -For k = 3, the third equation is To conclude, it suffices to prove the induction relation involving the polynomials P i .For this we fix the variables x 1 , . . ., x k , and put The left-hand side is the homogeneous term of degree k (in the variables x i ) in the product Taking the sum in p, we get ∑ Tameness for pairs of Lie algebras.

3.3.1.
Setting.Assume to be given a triplet (g, h, n), where g is a Lie algebra, h is a Lie subalgebra of g, and g = h ⊕ n as h-module.This makes perfect sense in any abstract k-linear symmetric monoidal category C. We denote by µ g and µ h the Lie brackets on g and h respectively; and by π h and π n the two projections from g to h and n respectively.We also define α and β in Hom C (n ⊗2 , n) and Hom C (n ⊗2 , h) respectively by the formulas α = π n • µ g |n ⊗2 and β = π h • µ g |n ⊗2 .
Definition 3.12.The triplet (g, h, n) is called tame if the morphism Lemma 3.13.Given a tame triplet (g, h, n), the morphism α defines a Lie structure on n.Besides, n becomes a Lie object in the symmetric monoidal category of h-modules.
Proof.We claim that we have Using this, the Jacobi identity for (g, µ g ) restricted to n ⊗3 at the source, and projected to n at the target, gives the Jacobi identity for (n, α).□

The envelopping algebra U(n).
Since n is a Lie algebra object in the category of h-modules in C, the algebra object U(n) is naturally endowed with an action by derivation of h.This action is simply induced by the adjoint action of h on the tensor algebra T(n).We define a morphism g ⊗ U(n) → U(n) componentwise as follows: -The morphism p : n ⊗ U(n) → U(n) is the multiplication in U(n).
-The morphism q : h ⊗ U(n) → U(n) is the action of h on U(n).
Lemma 3.14.Given a tame triplet (g, h, n), the above morphism endows U(n) with a g-module structure.
Proof.For any elements x and y in C, be denote by τ the symmetry isomorphism x ⊗ y ∼ − → y ⊗ x.We check componentwise (that is on n ⊗ n, n ⊗ h and on h ⊗ h) that the map (p, q) defines a g-action.
-We have on n 0 by tameness = 0 .
-Since h acts by derivation on U(n), we have on n ⊗ h ⊗ U(n) the equality The induced representation.The aim of this section is to prove the following theorem: Theorem 3.15.Given a tame triplet (g, h, n), the induced g-module Ind g h 1 C of the trivial h-module 1 C exists, and is naturally isomorphic to the g-module U(n) 9 .
Proof.The proof goes in several steps.We want to prove that U(n) satisfies the universal property of the induced representation, that is that for any g-module V, Hom h (1 C , V) ≃ Hom g (U(g), V).
First, we claim that the induced representation Ind g 0 1 C exists and is isomorphic to U(g).This means that Hom C (1 C , V) ≃ Hom g (U(g), V).The morphism is obtained by attaching to each φ in Hom C (1 C , V) the map Its inverse if simply the precomposition with the map 1 C → U(g).Let us consider the following diagram: V The vertical arrow is simply given by the left action of U(g) on the unit element element 1 of U(n).Here the dashed arrow is a morphism in C, the plain arrows are morphism of h-modules and the double arrows are morphisms of g-modules.We have a diagram of morphism spaces We now claim the following: -The map U(g) → U(n) admits a section, in particular it has a kernel N in the category C.
-The map U(g) ⊗ h → U(g) factors through the kernel N, and the induced map U(g) ⊗ h → N is an isomorphism.To prove the two claims, we make heavy use of the categorical PBW theorem (Proposition 2.11).If we endow U(n) and U(g) with their natural filtrations, then the action of g on U(n) is of degree 1 with respect to this filtration.The induced map g ⊗ Gr p U(n) → Gr p+1 U(n) is simply given by the projection g → n followed by the multiplication map n ⊗ S p n → S p+1 n.Therefore, we see that the natural map U(g) → U(n) is a filtered map of degree zero, and that the induced graded map S(g) → S(n) is induced by the projection from g to n.We can now consider the map an isomorphism, since the associated graded map is the identity, and the filtration splits (as objects of C).
Let us prove the second claim.First we remark that the composite map vanishes.This gives the factorization of the map U(g) ⊗ h → U(g) by N. We can endow N with the filtration induced by U(g).Since for each non-negative integer p the morphism F p U(g) → F p U(n) 9 The g-module structure on U(n) has been introduced in §3.3.2.admits a section, the natural map from Gr p N to the kernel of Gr p U(g) → Gr p U(n) is an isomorphism.Thus, using the PBW isomorphism, we have a commutative diagram Hence the map U(g) ⊗ h → N is a degree one map whose associated graded map is an isomorphism.Hence it is an isomorphism.
We now come back to the main proof.The only thing that remains to prove is that the composite morphism factors through Hom g (U(n), V).This is equivalent to prove that the composition vanishes.The image of a map φ is the morphism which can also be written as the composition which allows to finish the proof.□ Remark now that we have a priori two algebra structures on the algebra U(n) h : the first one is the natural one induced by the algebra structure on U(n), and the second one is obtained using Frobenius duality, and Theorem 3.15, and the natural composition on Hom g (U(n), U(n)): . These two structure are in fact compatible, this is the content of the following:

Theta morphisms.
In this section, we study some properties related to sheaves on a trivial first order thickening of a smooth scheme.Let us fix the setting: X is a smooth k-scheme, V is a locally free sheaf on X, and S is the (split) first order thickening of X by V; that is O S = V ⊕ O X and V is a square zero ideal in O S .We denote by j : X → S and σ : S → X the natural morphisms.Let F be an element of D − (X).Then the exact sequence Proposition 4.1.The following properties are valid: (i) For any φ : F → G in D − (X), we have (ii) For any F in D − (X), we have (iii) For any F in D − (X), the composition (iv) The morphism σ * Θ F vanishes.
Proof.(i), (ii) and (iv) are straightforward.For (iii), Θ F is a special occurrence of a residual Atiyah morphism, and we apply [9,Prop. 4.9].□ 4.1.2.Infinitesimal HKR isomorphism.In this section, we describe the infinitesimal cohomological HKR isomorphism attached to a split square-zero extension of a smooth scheme.
For any non-negative integer p, we define a morphism ∆ p : O X → T p (V [1]) in the derived category D b (S ) as follows: ∆ 0 = id and Proposition 4.2.For any vector bundles E 1 and E 2 on X, there is a canonical isomorphism obtained by precomposing by ∆ p ⊗ id σ * E 1 .Besides, this isomorphism is compatible with the Yoneda product for the pair (E 1 , E 2 ).
Proof.The first part of the proof is well known and follows from the existence of a canonical locally O S -free resolution of V on S (see [1]).The compatibility with the Yoneda product follows from routine calculations using Proposition 4.1.□ It is also possible to derive the internal Hom with respect to the second variable instead of the first one: this gives the infinitesimal counterpart of Kashiwara's dual HKR isomorphism (see [8]).To do so, we replace the morphism ∆ 1 by the dual Atiyah morphism Then for any integer p we construct the morphism ) → O X as we did before.Then the dual infinitesimal HKR isomorphism takes the following form: Proposition 4.3.For any vector bundles E 1 and E 2 on X, there is a canonical isomorphism obtained by postcomposing by ∆ ′ p ⊗ id σ * E 2 .Besides, this isomorphism is compatible with the Yoneda product for the pair (E 1 , E 2 ).
In particular, the composition where we implicitly use the isomorphism in the derived category D b (X), we call it the torsion of E. It is easy to see that the vector bundle E is entirely determined by the couple (E, τ E ).In particular, τ E vanishes if and only if E is isomorphic to σ * E Proposition 4.6.For any locally free sheaf E on S , the components of the morphism Proof.There is a natural morphism j * P 1 S (E) → P 1 X (E) making the diagram commutative.On the other hand, which is the identity of the first factor and the natural inclusion on the second one.Hence it is isomorphic to σ * E (by taking the difference of the two factors).This gives a commutative diagram and the result follows.□ 4.2.Sheaves on a second order thickening.

Setting and cohomological invariants.
As before, let us fix a pair (X, V) where V is a locally free sheaf on X, and let S be the split first order thickening of X by V. We are interested by ring spaces W which underlying topological space X satisfying the following conditions: There exists a map S → W which is locally the quotient by S 2 V.
Let k : X → W be the composite map, and let us denote by ⟨V⟩ the ideal sheaf of X in W, which is a sheaf of O S -modules.We can attach to W two cohomology classes: -The class α in Ext 1 (V, S 2 V) is the extension class of the exact sequence is the obstruction of lifting σ to W. To see how this class is defined, it suffices to remark that the sheaf of retractions of k inducing σ on S is an affine bundle directed by the vector bundle Der(O X , S 2 V), which is Hom(Ω 1 X , S 2 V).Lemma 4.7.The map W → (α, β) is a bijection between isomorphism classes of ring spaces W satisfying (9) and Ext 1 (V, S 2 V) ⊕ Ext 1 (Ω 1 X , S 2 V).Proof.The subsheaf of the sheaf of automorphisms of the ringed space {X, O X ⊕ V ⊕ S 2 V} that induce the indentity morphism after taking the quotient by the square zero ideal S 2 V is isomorphic to Der(O X , S 2 V) ⊕ Hom(V, S 2 V), a couple (D, φ) corresponding to the automorphism given by the 3 This gives the required result.□ 4.2.2.The second order HKR class.Let E be a locally free sheaf on X.The functor that associates to any open set of U the set of locally O W -free extensions of σ * E to W is an abelian gerbe, whose automorphism sheaf is Hom(E, S 2 V ⊗ E).Hence this gerbe is classified by a cohomology class in Ext 2 (E, S 2 V ⊗ E).
Definition 4.8.For any locally free sheaf E on X, the class of the gerbe of locally free extensions of σ * E on W is called the second order HKR class of E, and is denoted by γ σ (E).
Remark 4.9.The terminology is justified as follows: in [1], the authors introduce the HKR class of a vector bundle on X in the case S is not globally split, it mesures the obstruction to lift the bundle from X to S .Here we are defining the same kind of obstruction classes when S is trivial, but W is not.
As in [1], the class γ σ (E) can be computed explicitly: Proposition 4.10.For any locally free sheaf E on X, the class γ σ (E) is obtained (up to a nonzero scalar) as the composition Proof.For the first HKR class, this is [1, Prop.2.11].We present an alternative and somehow more down to earth proof.Let us assume that the map X → W admits two retractions β 1 and β 2 , and let D = β 1 − β 2 be the corresponding element in Hom(Ω 1 X , V). Assume also that E admits a regular connexion ∇ on X.We claim that the map Ξ : = β * 1 (g) .Ξ(s).Let us now fix a covering (U i ) i∈I of X, and assume that on each U i , there is a retraction β i of the map X → W and E admits a regular connexion ∇ i .We put D i j = β i − β j .Then we have isomorphisms . This element is a Chech representative of the class of the gerbe of locally free extensions of σ * E on W.
, then (c i j ) defines a 1-cochain with values in Hom(E, S 2 V ⊗ E) and we have which can be split us to some nonzero constant factors as the sum of the boundary of the cochain (c i j ) i, j and the Yoneda product of the 1-cocycles (∇ i − ∇ j ) i, j and (D i j ⊗ id E ) i, j that represent at X (E) and β ⊗ id E respectively.This gives the required formula.□ 4.3.Quantized cycles.

Setting, and tameness condition.
Let Y be a smooth k-scheme, and let X be a smooth and closed subscheme of Y.We denote by i : X → Y the injection of X in Y.
Let S denote the first formal neighbourhood of X in Y. Let us assume that the closed immersion j : X → S admits a retraction σ : S → X (that is S is a globally trivial square zero extension of X by N * X/Y ); this is equivalent to say that the conormal sequence of the pair (X, Y) splits.In this case, we say that (X, σ) is a quantized cycle in Y (see [8]).
Assume that (X, σ) is a quantized cycle, and let W be the second formal neighborhood of X in Y, and let k : X → W be the corresponding inclusion.According to Lemma 4.7, the ringed space W is entirely encoded by two classes α and β introduced in the previous section; α is the extension class of the exact sequence ) that measures the obstruction to the existence of a retraction of k that extends σ.Definition 4.11.A (X, σ) quantized cycle (X, σ) is tame if the locally free sheaf σ * N * X/Y on S extends to a locally free sheaf on W. Remark 4.12.If k admits a retraction q : W → X such that q |S = σ, that is if β vanishes, then (X, σ) is automatically tame: the locally free sheaf q * N * X/Y on W extends σ * N * X/Y .In this case, we say that (X, σ) is 2-split.
given by the 2 × 2 matrix Proof.The locally free sheaf (Ω 1 Y ) |S depends only on the second formal neighbourhood W of X in Y. Let us consider an automorphism of the trivial ringed space O X ⊕ N * X/Y ⊕ S 2 N * X/Y given by a couple (d, φ), where d is a derivation from O X to S 2 N * X/Y and φ is a linear morphism from N * X/Y to S 2 N * X/Y .Assume that we are in the local situation, so that we can take coordinates: X = U ⊂ k n and Y = U × V where V ⊂ k r .Then we can represent the morphisms d and φ by sequences of regular maps (Z k i, j (x)) 1≤k≤n,1≤i, j≤r and (Λ ℓ i, j (x)) 1≤i, j,ℓ≤r that are symmetric in the indices (i, j).The automorphism of W can be lifted to an automorphism of Y given by the formula The result follows by computing the pullback of the forms dx k , t i dx k , dt j , t i dt j restricted to S by the above automorphism.□ Corollary 4.14.The composition is given by the matrix Proof.This is obtained by putting together Proposition 4.6 and Proposition 4.13, together with the functoriality of the Atiyah class.□ 4.3.3.Quantized HKR isomorphism.As noticed in [1], the composition is an isomorphism in D + (Y), where the first map is the antisymetrization map.For quantized cycles, it is also possible to produce a left resolution of the sheaf O X (the Atiyah-Kashiwara resolution) that computes directly RHom O Y (O X , O X ), this construction is done in [8].Both constructions are in fact compatible (see [8,Thm. 4.13]).Let us now give the corresponding HKR isomorphism.For any non-negative integer p, we decompose the morphism ∆ p as the sum ∆ − p + ∆ + p according to the decomposition [1]).For any sheaf F on X, the functor For any locally free sheaves E 1 , E 2 on X, there is a canonical isomorphism in D b (X) obtained by precomposing with ∆ + p ⊗ id σ * E 1 .We also give Kashiwara's dual version: Proposition 4.16.For any locally free sheaves E 1 , E 2 on X, there is a canonical isomorphism in D b (X) obtained by postcomposing by 5. Tame quantized cycles 5.1.Structure constants.
5.1.1.Definition.We fix a quantized analytic cycle (X, σ) in Y.By Proposition 4.15, there exist unique coefficients (c Similarly, using Proposition 4.16, we can define dual coefficients: there exist unique coefficients (c Proof.We take the relation p , take the tensor product by id T p N X/Y [−p] , and precompose with the co-evaluation map [1]) component of c k p vanishes, so c k p factors through Λ p (N * X/Y [1]).(iii) The coefficient c p p is the canonical inclusion of S p (N * X/Y [1]) in T p (N * X/Y [1]).
Proof.For (i), we apply σ * to the relation defining the c k p and use Proposition 4.1 (iv).To prove (ii), let π p be the projection from T p (N * X/Y [1]) to S p (N * X/Y [1]).Then For (iii) this is a purely local question, and we can use the Koszul complex as in [8].
It defines a morphism f from N * X/Y to S 2 N * X/Y [1] in D b (S ).Applying Proposition 4.2, f can be written as u The morphism u is the image under σ * of ( 11), so it is α.The morphism v can be computed locally, it is the the natural symetrization map from T 2 N * X/Y to S 2 N * X/Y .Let us now consider the diagram 0 0 O S [1] / / S 2 N * X/Y [2] in D b (W).Since the composite arrow from O X to O S [1] vanishes, the morphism f where all horizontal arrows are algebra morphisms.
Strategy of proof.Let us first discuss the statement, as well as the main points involved in the proof.
The object RHom O Y (O X , O X ) is always an algebra object in the category D b (Y), but this structure doesn't always come in full generality from an algebra object structure on RHom ) is an isomorphism in D + (X), which is not always the case.
Assuming that we have an algebra structure on RHom r O Y (O X , O X ) and on RHom ℓ O Y (O X , O X ) making the top row of each diagram if Theorem 5.8 multiplicative, the statement follows directly from Proposition 5.3 the reverse PBW Theorem (that is Theorem 2.7), without using the tameness condition at all: indeed, condition (A1) is (10), and condition (A2) is Lemma 5.2.Therefore, the main difficulty lies in the construction of this algebra structure on RHom r O Y (O X , O X ) and on RHom ℓ O Y (O X , O X ).We will provide three different proofs corresponding to different geometric contexts: Case A: ∞-split Assume that X admits a global retract in Y that lifts σ.This is the easiest case, but it covers the case of the diagonal injection and is therefore sufficient to prove the results of Kapranov, Markarian and Ramadoss that are presented in the next section.The reader only interested in this can skip the two other cases.
Case B: 2-split Assume that X admits a retract at order two that lifts σ (that is β = 0).In this case, we can adapt the former proof to the second formal neighborhood, using the second Corollary of the decomposition Lemma (Corollary 5.7).
Case C: tame The general case: (X, σ) is tame.This requires the first Corollary of the decomposition Lemma (Corollary 5.6) as well as the full strength of the categorical PBW Theorem (Theorem 2.11).□ Proof of Theorem 5.8.We follow the aforementioned plan of proof, and discuss successively the three cases A, B and C. We recall the following notation: There is a natural morphism of algebra objects , which gives a morphism of algebra objects since f * is monoidal.Now it suffices to remark that ) inherits naturally from an algebra structure and the morphism (12) becomes an algebra morphism The same trick works for the functor RHom r .This settles Case A.

Case B Let us consider the map
It admits a section, given by the composition Hence p admits a kernel K, which can be explicitly described as follows: where the last map is obtained componentwise by precomposing with [1]).Assume now that there exists a retraction q : W → X that extends the first order retraction σ.Then the composition But this map is identically zero, since according to proposition 5.7, ∆ p − ∑ p k=1 c k p ∆ + k vanishes in the derived category D b (W).We can now conclude: the map is a morphism of algebra objects and K is a split sub-object that maps to zero, so that the composition is zero.Therefore, in the decomposition the object K is an ideal object, so that RHom ℓ O Y (O X , O X ) inherits of a natural algebra structure, for which p is a multiplicative morphism.This finishes the proof.Again, the whole proof works in the same way for the functor RHom r O Y .Case C If we consider α as a morphism from Λ 2 (N * X/Y [1]) to N * X/Y [1] in the opposite category of D b (X), we notice that the induction relations provided by Corollary 5.6 are exactly the same as the ones proved in Proposition 2.6 (this is why we took the same notation c k p ). Hence (N * X/Y [1], α) is a Lie algebra object in the opposite derived category of X.Now according to the second part of the categorical PBW Theorem (Theorem 2.11), we can define an algebra structure on S(N * X/Y [1]) using the coefficients c k p and there is a natural multiplicative morphism from T(N * X/Y [1]) to S(N * X/Y [1]) endowed with this structure.To conclude, it suffices to notice that the following diagram is commutative, which is nothing but the fact that the "geometric" coefficients c k p are the same as the "algebraic" coefficients c k p , that is Corollary 5.6.□ 5.2.The results of Kapranov, Markarian, Ramadoss and Yu.In this section, we provide a new light on the foundational result in this theory: the construction of the Lie algebra structure on T X [−1], due to Kapranov [11] and Markarian [14], and the computation of its universal enveloping algebra, due to Markarian [14], and Ramadoss [16].Then we prove Ramadoss formula [15] that computes the big Chern classes of a vector bundle introduced by Kapranov in [11].

5.2.1.
The Lie algebra T X [−1].Given a smooth scheme/manifold X, we consider the special case of the quantized cycle (∆ X , pr 1 ) in the product X × X.
-The object (Ω 1 Proof.The only thing we must prove is that α identifies with the Atiyah class of Ω 1 X , which is wellknown: this follows from looking at the diagram 0 / / pr 1 * I 2 where the vertical maps are given by differentation with respect to the second variable (so that they are all linear).The right vertical map is the isomorphism given by the quantization pr 1 , and the left vertical map is the symetrization morphism.□  Proof.We switch from Lie algebras objects to Lie coalgebras objects, so that we see Ω 1 Y|X [1] as a Lie coalgebra in D b (X).This Lie coalgebra is described by Corollary 4.14: first the diagram commutes, so the morphism Ω 1 Y|X → Ω 1 X is a morphism of Lie coalgebras objects.Next the tameness of the pair (in the sens of Definition 3.12) can be explicited as follows: if consider the morphism [1], tameness means the vanishing of the composite morphism [1].The first morphism is the class β, and the second one is (β ⊗ id N * X/Y [1] ) • at N * X/Y [1] .Thanks to Proposition 4.10, this is γ σ (N * X/Y ), and we are done.For the last point, we remark that the composition N * X/Y [1] → and then taking the trace on E (without antisymmetrizing on the factor T p Ω 1 X ).Theorem 5.12.For any vector bundle E on X, we have ĉp (E) = ) Hence the morphism we want to look at is (modulo isomorphism on the target) the composition [1]).We can factor the first arrow through RHom r (O X , O S ), and the composition 7.2.Beyond the tame case.At first sight, the diagonal cycle X → X × X admits two distinguished quantizations (i.e.first order retractions): the two projections pr 1 and pr 2 .However, since the space of quantizations of a cycle is an affine space, this gives a whole line of quantizations, namely tpr 1 +(1−t)pr 2 for t in the base field k.
In general these quantizations are never tame except for t = 0 or t = 1, that is for the two projections.
However, the value t = 1 2 is special since the corresponding quantized cycle fits in a very interesting family of quantized cycles: every fixed-point locus X = Y ι of an involution ι of Y defines naturally a quantized cycle.In our former example, Y = X × X and ι(x, x ′ ) = (x ′ , x).In Lie algebraic terms this corresponds to symmetric pairs.We will study further on this case in future work.

Theorem 3 . 11 .
If a is an ℓ-torsion morphism, then a = c(d ⊗ a ℓ ), where c(d ⊗ −) means ∑ p c p (d p ⊗ −), the sum being in fact automatically finite.

Remark 4 . 4 .
The object σ * RHom O S (O X , O X ) is a ring object in D + (X).Propositions 4.2 and 4.3 give two a priori different isomorphisms between this ring object and TV * [−1].In the next Lemma, we will provide an identity relating ∆ p and ∆ ′ p that shows that these two isomorphisms are in fact the same.Lemma 4.5.Let W be any element in D b (X), let W * = RHom O X (W, O X ) be the (naïve) derived dual of W, and let φ be in Hom D b (X) (W, T k V * [−k]).Then the following diagram

Proof. 4 . 1 . 3 .
The first point follows directly from Proposition 4.1 (i).The second point follows from the fact that for any A, B in D b (X), the isomorphism Hom D b (X) (A, B) ≃ Hom D b (X) (B * , A * ). are left to the reader.□ Torsion and Atiyah class.For any locally free sheaf E on S , let E = j * E. The exact sequence 0

4. 3 . 2 .
Restriction of the Atiyah class.The aim of this section is to describe another intrinsic description of the classes α and β.Proposition 4.13.The torsion of the locally free sheaf

Theorem 5 .Theorem 5 . 11 .
9 allows to give a Lie-theoretic interpretation of the tameness condition (Definition 4.11) for quantized cycles.To a quantized cycle (X, σ) in Y, we have an exact sequence 0 → TX[−1] → TY[−1] |X → N X/Y [−1] → 0 and σ provides a splitting of this sequence.The triplet (TX[−1], TY[−1] |X , N X/Y [−1]) is a reductive pair of Lie objects, as defined in §3.3.Besides this pair is tame in the sense of Definition 3.12 if and only if (X, σ) is tame in the sense of Definition 4.11.In this case, the dual of α defines a Lie structure on N X/Y [−1].

Remark 5 . 13 .
p (X) • c k (E) where c k p are the universal elements in Ext p−k (Λ k X, T p X) associated to the Lie algebra T X [−1].Proof.On X × X, we have∆ p = ∑ p k=1 c k p • ∆ + k .This gives pr 2 * (∆ p ⊗ id pr * 1 E ) = p ∑ k=1 (c k p ⊗ id E ) • pr 2 * (∆ + k ⊗ id pr * 1 E ).The result follows by taking the trace on E. □ As explained in[15], the total Chern class of a vector bundle E can be interpreted in terms of the representation of the Lie algebra TX [−1].Indeed E defines a representation of T X [−1], whith is (the dual of) the Atiyah class of E, hence a map from U(T X [−1]) to End (E).Its trace defines a map from U(T X [−1]) to O X , which is exactly ∑ p c p (E) via the isomorphism Hom D b (X) (U(T X [−1]), O X ) ≃ ⊕ p H p (X, Ω p X ).5.2.3.The quantized cycle class.Let us recall the definition of the quantized cycle class introduced in[8].For any quantized analytic cycle (X, σ) in Y, we consider the compositionω X/Y ≃ RHom r (O X , O Y ) → RHom r O Y (O X , O X ) ≃ S(N X/Y [−1])(13)where the last isomorphism is the dual HKR isomorphism.Let d be the codimension of X in Y. Proposition 5.14.Assume that (X, σ) is tame.Then the morphism (13) is a d-torsion morphism for the Lie algebra N X/Y [−1].Proof.We must prove that the compositionN X/Y [−1] ⊗ ω X/Y → N X/Y [−1] ⊗ S(N X/Y [−1]) ≃ N X/Y [−1] ⊗ U(N X/Y [−1]) → U(N X/Y [−1])vanishes, where the last map is given the multiplication in the algebra U(N X/Y [−1]).We can rewrite this map (using duality) as a morphism from U(N X/Y [−1]) to N * X/Y [1] ⊗ U(N X/Y [−1]), and the question reduces to the vanishing of the map ω X/Y → S(N X/Y [−1]) ≃ U(N X/Y [−1]) → N * X/Y [1] ⊗ U(N X/Y [−1]) Using Theorem 5.8, we have a commutative diagram RHom r O Y (O X , O X )

7 . Conclusion and perspectives 7 . 1 .
RHom r (O X , O S ) → RHom r (O X , O X ) → RHom r (O X , N * X/Y[1]) vanishes, since it is obtained by composing two successive arrows of a distinguished triangle.□ Theorem 5.15.[18]Let (X, σ) be a tame quantized cycle in Y. Via the isomorphismRHom D b (Y) (ω X/Y , S(N X/Y [1])) ≃ H 0 (X, S(N * X/Y [1])) ≃ S(N * X/Y [1]) N X/Y [−1] , the image of the morphism (13) is the Duflo element of N X/Y [−1].canbe written as φ • ∆ − p + ψ where ψ belongs to F p−1 R X/Y , and we conclude by induction since elements in F p−1 R X/Y map to zero in A(2)  X/Y .□ State of the art in the tame case.In this paper we introduced a tameness condition for a quantized cycle (X, σ) in Y.Under this assumption we proved that the shifted normal sheaf N X/Y [−1] is endowed with a Lie bracket and that the sheaf of derived O Y -linear endomorphisms of O X is isomorphic to the universal enveloping algebra of N X/Y [−1].Using this, we were able to:-identify the quantized cycle class with the Duflo element of the Lie algebra object N X/Y [−1], recovering and reinterpreting in Lie algebraic terms a result Yu. -describe explicitly the Ext algebra Ext • Y (O X , O X ).
1.3.The results.The two main geometric result we prove in this paper deals with tame quantized analytic cycle.Both rely heavily on abstract result on Lie algebras in symmetric monoidal categories, that we will discuss after the statements.The first result is the explicit computation of the enveloping algebra of the Lie algebra object NX/Y [−1].Besides, the objects RHom ℓ O Y (O X , O X ) and RHom r O Y (O X , O X )are naturally algebra objects in the derived category D b (X), and there are commutative diagrams Theorem A. Let (X, σ) be a tame quantized cycle in Y.The class α defines a Lie coalgebra structure on N * X/Y [1], hence a Lie algebra structure on N X/Y [−1].
This gives the result.□ As a corollary, we get our key technical result: Proposition 5.4 (Decomposition lemma).Assume that the quantized cycle (X, σ) is tame.For any nonnegative integer i, we can write Θ T i+1 N * We argue by induction on p.For p = 2, this is Proposition 5.3.Assume that the property holds at the order p − 1.The morphism ∆ p − Then we use the induction hypothesis.□ 5.1.3.Main result.We can come to our main result.Theorem 5.8.Let (X, σ) be a tame quantized cycle in Y.The class α defines a Lie coalgebra structure on N * X/Y [1], hence a Lie algebra structure on N X/Y [−1].Besides, the objects RHom ℓ O Y (O X , O X ) and RHom r O Y (O X , O X ) are naturally algebra objects in the derived category D b (X), and there are commutative diagrams 1] is exactly α. □ 5.2.2.Big Chern classes.Let us recall that for any vector bundle E on X, the big Chern classes ĉp (E) of E live in H p (X, T p Ω 1 X ), they are obtained by composing the morphisms at E , id Ω 1 X ⊗ at E , . . ., id Ω p−1 X⊗ at E