Formality of derived intersections

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Julien Grivaux

To cite this version:

Julien Grivaux. Formality of derived intersections. Documenta Mathematica, Universität Bielefeld, 2014, 19, pp.1003–1016. �hal-01301444�

JULIEN GRIVAUX

Abstract. We study derived intersections of smooth analytic cycles, and provide in some cases necessary and sufficient conditions for this intersection be formal. In particular, ifXis a complex submanifold of a complex manifoldY, we prove thatXcan be quantized if and only if the derived intersection ofX²and∆Yis formal in D^b X².

Contents

1. Introduction 1

2. Notations and conventions 3

3. Quantized analytic cycles and AK complexes 3

4. Linear intersections and excess contributions 4

5. Restriction of the AK complex 5

6. Formality in the split case 6

7. The reverse implication 7

8. An alternative proof 9

9. The formality criterion for quantization 10

10. Perspectives 11

References 12

1. Introduction

The theory of non-transverse intersections is now a classical topic in algebraic geometry, which is exposed in the classical book [4]. The main idea goes as follows: if two (smooth) subschemes X and Y of a smooth complex algebraic scheme Z intersect along an irreducible scheme T, then the refined intersection of X and Y consists of the two following data: the set-theoretic intersectionT = X∩Y, and an elementξof the Chow group CH^r(T) whereris the difference between the expected codimension of X∩Y (that is the sum of the codimensions ofX andY) and the true codimension of X∩Y. If bN is the restriction of N^∗_X/Z to T, the conormal excess bundleEis an algebraic vector bundle of rankrdefined by the exact sequence

0−→ E −→bN−→N^∗_T/Y −→0.

Then the excess intersection classξis given byξ= cr(E^∗). This formula is a particular case of the excess intersection formula for refined Gysin morphisms, we refer the reader to [4,§6.2 and

§6.3] for more details.

Recall now that

ch : K(Z)⊗_ZQ−→CH(Z)⊗_ZQ

is an isomorphism, so that dealing with Chow rings is the same as dealing withK-theory (mod- ulo torsion). Intersection theory in K-theory is much easier and goes back to [3]: the product

1

ofO_X andO_Y is just the alternate sum of the classes of the sheaves Tor_Oⁱ

Z(OX,O_Y), which is the element

Λ−1E:= X

i≥0

(−1)ⁱ h ΛⁱEi

.

Then chr(Λ−1E) is exactly the excess classξintroduced above. For a geometric interpretation of intersection in K-theory, we refer the reader to the classical Tor formula [7,§5.C.1].

For applications, it is essential to lift K-theoretic results at the level of derived categories. The intersection product of two algebraic cycles X and Y in this framework is simply the derived tensor product ϑ := O_X⊗^L_O

ZO_Y in D^b(Z), which is much more complicated to study. First of all, the element ϑ admits canonical lifts in D^b(X) or D^b(Y), but does not always comes from an object in D^b(T). Thusϑdoesn’t live on the set-theoretic intersection of X andY. Next, for any nonnegative integer p, there is a canonical isomorphism betweenH^−p(ϑ) andΛ^pE. There is therefore a natural candidate for ϑ, namely the formal object s(E) := Ld

i=0ΛⁱE[i]. It is a rather delicate question to decide if the object ϑ is formal in D^b(X), D^b(Y) or in D^b(Z). The first important result in this direction was given by Arinkin and C˘ald˘araru for self-intersections.

Their result is a far-reaching generalisation of the Hochschild-Kostant-Rosenberg isomorphism corresponding to the self-intersection of the diagonal in X × X. Let us state it in the analytic category¹:

Theorem ([1]). Let X, Y be smooth complex manifolds such that X is a closed complex sub- manifold of Y. Then j^∗_X/Y j_X/Y∗O_X is formal in D^b(X) if and only if N^∗_X/Y can be lifted to a holomorphic vector bundle on the first formal neighbourhood of X in Y.

In particular, if the injection of X into its first formal neighbourhood in Y admits a retrac- tion σ, which means in the terminology of [5] that (X, σ) is a quantized analytic cycle, then j^∗_X/Y jX/Y∗O_X is formal. This fact was discovered by Kashiwara in the beginning of the nineties [6], and his proof has been written down in [5]. In this paper, we are interested in more gen- eral intersections than self-intersections. We will deal with cycles intersecting linearly, which means that the intersection is locally biholomorphic to an intersection of linear subspaces. We will also make the additional assumption that one of the cycles, say X, can be quantized. Our main result runs as follows²:

Theorem 1.1. Let Z be a complex manifold, let (X, σ)be a quantized analytic cycle in X, and Y be a smooth complex submanifold of Z such that X and Y intersect linearly. Put T = X∩Y, and letNbbe the restriction ofNX/Zto T , and letEbe the associated conormal excess bundle on T . Then the object j^∗_Y/Z jX/Z∗O_X is formal in the derived categoryD^b(Y)if and only if and only if the excess conormal exact sequence

0−→ E −→bN−→N^∗_T/Y −→0 is holomorphically split.

Two extreme cases of this theorem are already known: the case of self-intersections and the case of transverse intersections. For self-intersections, the formality follows directly from Arinkin- Cald˘ar˘aru’s result, since X is assumed to be quantized. For transverse intersections, there are no higher Tor sheaves so that j^∗_Y/Z j_X/Z∗O_X ' j_T/Y∗O_T. An important point must be noticed:

1Since we are concerned with applications in complex geometry, we will state throughout the paper the results in the analytic category; they are also valid in the algebraic category as well.

2The author was informed that a similar result has been announced independently by Arinkin, C˘ald˘araru and Hablicsek [2].

although j^∗_Y/Z jX/Z∗O_X is locally formal, and hence comes locally from D^b(T), this property is no longer true globally in general if the excess exact sequence is not split; we provide counter- examples.

One of the principal applications of Theorem 1.1 is the existence of a formality criterion in order that an analytic cycle be quantized:

Theorem 1.2. Let X be a smooth complex submanifold of a complex manifold Y. Then X can be quantized if and only if the object j^∗_X2/Y²O_∆Y is formal inD^b(X²).

This result follows directly from Theorem 1.1 by applying the trick of the reduction to the diagonal. Thus, the existence of an extension to the conormal bundle at the first order or the existence of an infinitesimal retraction for a complex analytic cycle are both equivalent to the formality of a derived intersection: the first one involves the pair (X,X) inYand the second one involves the pair ∆Y,X²

inY². However, it would be very interesting to know ifO_X⊗^L_O

YO_X is always formal in D^b(Y), since no obstruction to the formality of this complex is known up the the author’s knowledge. We conjecture that it is indeed the case, and provide some evidence towards this conjecture.

Acknowledgments.This paper grew up from conversations with Daniel Huybrechts and Richard Thomas, I would like to thank them both for many interesting discussions on this topic.

2. Notations and conventions

All complex manifolds considered are smooth and connected. By submanifold, we always mean closedsubmanifold. IfXandY are complex analytic spaces such thatX is a complex subspace ofY, we denote by jX/Y the corresponding inclusion morphism, and we denote by XY the first formal neighbourhood ofX inY. For any holomorphic vector bundleEon a complex manifold X, we denote bys(E) the formal complexL

k≥0Λ^kE[k].

For any ringed complex space (X,O_X), we denote by D^b(X) the bounded derived category of the abelian category of coherent sheaves of O_X-modules on X. For any holomorphic map f: X −→ Y between complex manifolds, we denote by f^∗: D^b(Y) −→ D^b(X) the associated derived pullback. The non-derived pullback functor is denoted by H⁰(f^∗).

3. Quantized analytic cycles andAKcomplexes

We recall terminology and basics constructions from [5]. LetZbe complex manifold. Aquan- tized analytic cycleinZis a pair (X, σ) such thatXis a closed connected complex submanifold of Z and σ is a retraction of the injection of X into its first formal neighbourhood in Z. If we denote the latter by XZ, thenσ is a morphism of sheaves ofCX-algebras from O_X toO_X

Z. Equivalently,σis a splitting of the Atiyah sequence

(1) 0−→N^∗_X/Z −→ O_X

Z −→ O_X −→0

in the category of sheaves of CX-algebras, that is an isomorphism of sheaves of CX-algebras betweenO_X

Z and the trivial extension of O_X by N^∗_X/Z. The set of possible quantizationsσof X is either empty or an affine space over Hom

Ω¹X,N^∗_X/Z

. For any quantized analytic cycle (X, σ) inZ, there is a canonical complex P_σ ofO_X

Z-modules, called theAtiyah-Kashiwara complex, defined as follows: for any nonnegative integerk,

(P_σ)−k = Λ^kO⁺_X¹O_X

Z = Λ^k+1N^∗_X/Z⊕Λ^kN^∗_X/Z.

Besides, (Pσ)−kis acted on byO_X

Zviaσas follows:O_Xacts in the natural way by multiplication, and N^∗_X/Zacts onΛ^k+1N^∗_X/Z by 0 and onΛ^kN^∗_X/Zby the formula

i.(i₁∧. . .∧ik)=i∧i₁∧. . .∧ik. The differential ¹_kd−kis given by the composition

Λ^k+1N^∗_X/Z⊕Λ^kN^∗_X/Z −−−−^(0,id)→ Λ^kN^∗_X/Z −−−−^(id,→⁰⁾ Λ^kN^∗_X/Z⊕Λ^k−1N^∗_X/Z. The complexP_σ is a left resolution ofO_X overO_X

Z which neither projective nor flat. However, the morphism

j^∗_X/Z jX/Z∗O_X ' j^∗_X/ZP_σ −→ P_σ ⊗_O

XZ O_X =s N^∗_X/Z

is an isomorphism, called the quantized Hochschild-Kostant-Rosenberg isomorphism (see [5, Proposition 4.3]). If σis fixed, so that the isomorphism betweenO_X

Z and N^∗_X/Z ⊕ O_X is fixed, and ifϕis in Hom

Ω¹X,N^∗_X/Z

, then P_σ+ϕ = Λ^k+1N^∗_X/Z⊕Λ^kN^∗_X/Z whereO_X acts onΛ^k+1N^∗_X/Z by multiplication, onΛ^kN^∗_X/Zby

f.(i₁∧. . .∧ik)=(ϕ(d f)∧i1∧. . .∧ik, f i1∧. . .∧ik),

and N^∗_X/Z acts by wedge product. If N^∗_X/Z admits a holomorphic connection ∇: N^∗_X/Z −→ Ω¹X ⊗ N^∗_X/Z, we denote by (Λ^k∇)_k≥1 the associated connections of the exterior powers of N^∗_X/Z. Then there is an associated natural isomorphism betweenP_σ andP_σ+ϕgiven in any degree−kby

(2) (i, j)→(i+{ϕ∧Λ^k∇}(j), j).

4. Linear intersections and excess contributions

Let us consider three complex manifolds X, Y and Z such that X and Y are closed complex submanifolds ofZ. LetT = X∩Y. We will say thatXandY intersectlinearlyif for any pointt inT, there exist local holomorphic coordinates ofZneartsuch thatXandY are given by linear equations in these coordinates. If this is the case, T is smooth, and there exists a holomorphic vector bundleEonT fitting in the exact sequence

(3) 0−→ E −→^α bN−→^π N^∗_T/Y −→0

wherebN= j^∗_T/XN^∗_X/Z. The bundleEis called theexcess intersection bundle. If two cyclesXand YinZintersect linearly then for any nonnegative integerk, Tor^k_O

Z(OX,O_Y)' jT/Y∗Λ^kE. The two extreme cases of linear intersections are transverse intersections (E={0}) and self-intersections (X= YandE=N^∗_X/Z).

We give computations in local coordinates which will be useful in the sequel. By definition of linear intersections, we can find local holomorphic coordinates

n(xi)1≤i≤p,(yj)1≤j≤q,(tk)1≤k≤r,(zl)1≤l≤s

o

onZin a neighbourhoodUof any point inT such that locally

X={x₁= 0} ∩. . .∩ {xp =0} ∩ {t₁ = 0} ∩. . .∩ {tl = 0}

Y ={y1= 0} ∩. . .∩ {yq =0} ∩ {t1 = 0} ∩. . .∩ {tl = 0}

The excess bundleEis the bundle spanned bydt₁, . . . ,dtr. LetK be the Koszul complex onU associated with the regular sequence

x₁, . . . ,xp,t₁, . . . ,tr

defining X, and let τbe the local infinitesimal retraction of XZ inX induced by the projection (x,y,t,z)→(0,y,0,z), and there is a quasi-isomorphism

(4) ν: s(E)|T∩U −→ K|Y∩U

We have also a natural morphism of complexesγ fromK to jX/Z∗(Pτ)_|U given as follows (see the proof of [5, Proposition 3.2]): ifI ⊂ {1. . . ,p}andJ ⊂ {1. . . ,r}, then

γ

f(x,y,t,z) dxI ∧dtJ=









p

X

a=1

∂f

∂xa

(0,y,0,z)dxa+

r

X

b=1

∂f

∂tb

(0,y,0,z)dtb









∧dxI ∧dtJ

⊕ f(0,y,0,z) ∧dxI∧dtJ. (5)

5. Restriction of theAKcomplex We keep the notation of§4. The cartesian diagram of analytic spaces

TY

//

XZ

Y ^//Z

yields a canonical morphism from O_X

Z

|Y toO_T

Y which is in fact an isomorphism (this can be checked using local holomorphic coordinates as in the end of§3). If we restrict the sequence 1 toY, we get the exact sequence

Tor¹_O

Z(OX,O_Y)−→bN−→

O_X

Z

|Y −→ O_T −→0 which is exactly the sequence

0−→ E −→bN−→ O_T

Y −→ O_T −→0

obtained by merging the sequences (1) and (3) corresponding to the pair (T,Y).

We turn now to the quantized situation, where we can describe explicitly the restriction to Y functor from Mod

O_X

Z

to Mod O_T

Y

. Assume that (X, σ) is a quantized analytic cycle inZ.

Let us define a sheaf of ringsOb_T onT by the formulaOb_T = σ∗O_X

Z

|T. Then we have an exact sequence

0−→ E −→Ob_T −→ O_T

Y −→ 0.

The inclusion morphism fromTY toXZ can be described as follows: O_X

Z is anO_X-module via σ. Since N^∗_X/Z acts on O_X

Z, bN acts on Ob_T and O_T

Y = Ob_T/E.Ob_T so that for any sheaf F of O_X

Z-modules,

(6) F_|Y = (σ∗F)|T

E.(σ∗F)|T

·

Let us now consider the Leray filtration onΛ^kN. The p-th piece Fb pof this filtration is the image ofΛ^pE ⊗Λ^k−pbN inΛ^kbN. Since Gr₀

Λ^kbN

= Λ^kN^∗_T/Y, it follows from (6) that the restriction of the AK complexP_σ toY is given for any nonnegative integerkby the formula

(7) (P_σ)_−k|Y = Λ^k+1N^∗_T/Y⊕Λ^kbN.

If δk: Λ^kbN → Λ^kN^∗_T/Y is the k^th wedge product ofπ, the differential of the complex (P_σ)_|Y is given via (7) by the composition

Λ^k+1N^∗_T/Y⊕Λ^kbN−−−−^{(0, δ}→^k) Λ^kN^∗_T/Y −−−−^(id,→⁰⁾ Λ^kN^∗_T/Y⊕Λ^k−1N.b

so that the local cohomology sheaves of (Pσ)|Yare

(8) H^−kn

(Pσ)|Y

o' kerδk ' jT/Y∗

nF₁ Λ^kbNo

. Remark that for any k the natural inclusion F1

Λ^kNb

,→ (Pσ)−k|Y is O_T

Y-linear, hence also O_Y-linear. Thanks to (8), we have a quasi-isomorphism in D^b(Y):

(9) (P_σ)_|Y 'M

k≥0

jT/Y∗

nF₁ Λ^kNbo

[k].

6. Formality in the split case Let us define a morphism

Ψσ : j^∗_Y/Z jX/Z∗O_X ' j^∗_Y/Z jX/Z∗P_σ −→(Pσ)|Y ' M

k≥0

jT/Y∗

nF₁ Λ^kbNo

[k]

in D^b(Y) using (9). If (3) splits, then for any positive integerk, any splitting of this sequence yields an isomorphism between F1

Λ^kbN

andLk

`=1Λ<sup>`</sup>E ⊗Λ^k−`N^∗_T/Y.

Proposition 6.1. Let(X, σ), Y, Z be as above; and assume that the excess exact sequence(3) splits. Then the composite morphism

Θσ : j^∗_Y/Z jX/Z∗O_X −→^Ψσ M

k≥0

jT/Y∗

nF₁ Λ^kbNo

[k]−→ jT/Y∗s(E) is an isomorphism inD^b(Y).

Proof. Let us describeΨσ locally. For this we use the computations in local coordinates per- formed at the end of§4. Taking the same notations, the Koszul complexK is a free resolution onU of jX_U/U∗O_XU. Let us denote X∩U,Y∩U, andT∩UbyXU,YU, andTU respectively. If τis the local quantization ofXadapted to our choice of local holomorphic coordinates andγis defined by (5), we claim that the diagram

K_|YU

γ|YU

jT_U/Y_U∗s(E)|T_U ν|YU

oo ∼

 _

(Pτ)|YU

L

k≥0 jT_U/Y_U∗

nF₁ Λ^kbNo

∼ [k]

oo

commutes. Indeed, ifJ ⊂ {1. . . ,r}, then a local section f(z)dtJofΛ^rEis mapped byγ◦νto the section (0, f(z)dtJ) ofP_τ(the derivatives in formula (5) don’t appear since f doesn’t depend on the variablesxandt). Hence we get a commutative diagram

j^∗_Y/ZjX/Z∗O_X

|U j^∗_Y

U/U j_XU/U∗K

∼ //

∼

oo ∼ _K|Y

U

γ|YU

j_TU/Y_U∗s(E)|TU ν|YU

oo ∼

 _

j_TU/Y_U∗s(E)|TU

j^∗_Y/ZjX/Z∗O_X

|U

Θτ

44j^∗_Y

U/UjX_U/U∗P_τ

oo ∼ //_(Pτ)|Y

U

L

k≥0 jT_U/Y_U∗

nF1

Λ^kbNo [k]

oo ∼ // _jT

U/Y_U∗s(E)|T_U

in D^b(YU). This proves that Θτ is an isomorphism. Then we can prove easily that the same results holds if we replace σby τ. Indeed, if U is small enough, we can pick a holomorphic

connection on (N^∗_X/Z)|U; and we have a commutative diagram L

k≥0 jT_U/Y_U∗

nF₁ Λ^kbNo

[k] ^{∼ //}

id

(Pσ)_|YU

∼

L

k≥0 jT_U/YU∗

nF₁ Λ^kbNo

[k] ^{∼ //}(P_τ)_|YU

where the left vertical isomorphism is given by (2). This yields the result.

Remark that even if the excess sequence (3) if not split, we always have an Atiyah morphism at_σ: j^∗_Y/Z j_X/Z∗O_X −→ j_T/Y∗E[1]

which is obtained by the composition (10) j^∗_Y/Z jX/Z∗O_X −^Ψ−→^σ M

k≥0

jT/Y∗

nF₁ Λ^kNbo

[k]−→ jT/Y∗

nF₁ bNo

[1]= jT/Y∗E[1]

where the last arrow is the projection on the factor corresponding tok = −1. This morphism induces an isomorphism on the local cohomology sheaves in degree−1 of both sides.

7. The reverse implication

Let us now assume that j^∗_Y/Z jX/Z∗O_X is formal in D^b(Y). Remark that we have a canonical morphism

(11) j^∗_Y/Z jX/Z∗O_X −→ jT/Y∗ O_T ⊕ E[1]

which is an isomorphism on the local cohomology sheaves of degrees 0 and −1. The first component is given by the composition

p: j^∗_Y/Z jX/Z∗O_X −→H⁰ j^∗_Y/Z

jX/Z∗(OX) ' jT/Y∗O_T

and the second one is the Atiyah morphismat_σ defined by (10). Therefore, ifβ = L

k≥0βk is an isomorphism between j^∗_Y/Z jX/Z∗O_X and jT/Y∗ L

k≥0Λ^kE[k]

, we can replaceβ₀bypandβ₁ byat_σ. At usual, we denote byP_σthe AK complex associated with (X, σ). Let us consider the morphism

m: jT/Y∗s(E) ⊗^L

O_Y jT/Y∗s(E)

β^−1L⊗β⁻¹

−→∼

j^∗_Y/Z jX/Z∗O_XL

⊗O_Y

j^∗_Y/Z jX/Z∗O_X

−→∼ j^∗_Y/Z

jX/Z∗O_X⊗^L

O_ZjX/Z∗O_X

' j^∗_Y/Z

jX/Z∗O_X⊗^L

O_ZP_σ ∼

−→ j^∗_Y/Z jX/Z∗

ns N^∗_X/Zo

−→ j^∗_Y/Z jX/Z∗

nOX ⊕N^∗_X/Zo

−→ jT/Y∗

OT ⊕ E[1]⊕N [1]b ,

where the last morphism is built using (11) on the first factor. Note that all arrows occurring in the expression ofmare isomorphisms, except the two last ones which induce isomorphisms on local cohomology sheaves of degrees 0 and −1. Hence H⁻¹(m) is an isomorphism. Now we can write

τ^≥−1

jT/Y∗s(E)⊗^L

O_Y jT/Y∗s(E)

' jT/Y∗

E[1] ⊗_OY O_T

⊕ jT/Y∗

O_T ⊗_OY E[1]

⊕τ^≥−1

jT/Y∗O_T⊗^L

O_YjT/Y∗O_T ' jT/Y∗E[1] ⊕ jT/Y∗E[1] ⊕ jT/Y∗

O_T ⊕N^∗_T/Y[1]

.

so thatH⁻¹(m)' j_T/Y∗

E ⊕ E ⊕N^∗_T/Y

. Hence we have an isomorphism H⁻¹(m) : E ⊕ E ⊕N^∗_T/Y −→ E ⊕^∼ bN

in Mod (Y), hence in Mod (T). We denote itn. Using the natural morphism O_Z −→ j_X/Z∗O_X, we get a commutative diagram

j^∗_Y/ZO_Z⊗^L

O_Y j^∗_Y/Z jX/Z∗O_X ^{∼ //}

j^∗_Y/Z O_Z⊗^L

O_Z jX/Z∗O_X

j^∗_Y/Z j_X/Z∗O_X⊗^L_O

Y j^∗_Y/Z j_X/Z∗O_X ^{∼ //} j^∗_Y/Z

j_X/Z∗O_X⊗^L_O

Z j_X/Z∗O_X Applying the functorH⁻¹we have another commutative diagram, namely

E ^{id //}

(0,id,0)

E

(id,α)

E ⊕ E ⊕N^∗_T/Y ^{n //}E ⊕bN

The proof that the second vertical arrows is (id, α) is purely local, let us briefly explain how to do it. The morphism O_Z −→ jX/Z∗O_X can be lifted locally on any small open set U to a morphism of complexesO_ZU −→ K whereK is the Koszul complex introduced in §4. Hence we must compute the action of the morphism

(OU ⊗O_U K)|Y_U −→(K ⊗O_U K)|Y_U

on the local cohomology sheaves of degree−1, which is an elementary calculation.

We can do the same argument after switching the two factors of the tensor product, and consider the diagram

j^∗_Y/ZO_X⊗^L_O

Y j^∗_Y/ZO_Z ^{∼ //}

j^∗_Y/Z

jX/Z∗O_X⊗^L_O

ZO_Z

j^∗_Y/Z jX/Z∗O_X⊗^L

O_Y j^∗_Y/Z jX/Z∗O_X ^{∼ //} j^∗_Y/Z

jX/Z∗O_X⊗^L

O_Z jX/Z∗O_X which gives

E ^{id //}

(id,0,0)

E

(−id,α)

E ⊕ E ⊕N^∗_T/Y ^{n //} E ⊕bN Therefore wee see that the composite map

E ⊕N^∗_T/Y ' {0} ⊕ E ⊕N^∗_T/Y ,→ E ⊕ E ⊕N^∗_T/Y −→ E ⊕ⁿ bN−−−→^α+id bN

is an isomorphism whose first component is 2α. This proves that N^∗_T/Y is a direct factor ofEin bN, so that (3) is holomorphically split.

8. An alternative proof

In this section, we sketch an alternative proof of the splitting of the excess exact sequence (3) when j^∗_Y/Z j_X/Z∗O_X is formal in D^b(Y). For the formalism of Grothendieck’s six operations in our context, we refer the reader to [5,§5.2].

Recall that there arecanonicalisomorphismsH⁻ⁱ

j^∗_Y/Z j_X/Z∗O_X

' j_T/Y∗

ΛⁱE

. Therefore, if the derived intersection j^∗_Y/Z jX/Z∗O_X is formal, we can pick an isomorphismϕ between jT/Y∗s(E) and j^∗_Y/Z j_X/Z∗O_X and we can assume without loss of generality that the induced isomorphisms in local cohomology are the canonical ones. Ifddenotes the codimension ofT inY, we have

Hom_D^b_(Y)

j_T/Y∗s(E), j^∗_Y/Z j_X/Z∗O_X ' Hom_D^b_(T)

s(E), j^!_T/Y{j^∗_Y/Z j_X/Z∗O_X} ' Hom_D^b_(T)

s(E), j^∗_T/Y{j^∗_Y/Z jX/Z∗O_X}⊗^L

O_T j^!_T/YO_Y ' Hom_D^b_(T)

s(E), j^∗_T/X{j^∗_X/Z jX/Z∗O_X}⊗^L

O_T j^!_T/YO_Y ' Hom_D^b_(T)

s(E), j^∗_T/X n s

N^∗_X/Zo L

⊗O_T Λ^dNT/Y[−d]

' Hom_D^b_(T)

s(E),s bN

⊗_OT Λ^dNT/Y[−d]

' M

q−p≥d

Ext^q−p−d_O

T

Λ^pE,Λ^qbN⊗O_T Λ^dNT/Y

' M

q−p≥d

Ext^q−p−d_O

T

Λ^pE ⊗_OT Λ^dN^∗_T/Y,Λ^qbN . Using this decomposition, we writeϕ =P

p,qϕp,q. Then for any nonnegative integer pthe mor- phismH^−p(ϕ) is determined byϕp,p+d. Besides, it is possible to check using local holomorphic coordinates that for any p, the mapϕp,p+dis a retraction of the natural projection

Λ^p+dbN'Fp

Λ^p+dbN

→Grp

Λ^p+dbN

'Λ^pE ⊗Λ^dN^∗_T/Y.

The reason for this is that we have normalised the action of the isomorphism ϕ on the local cohomology sheaves. Ifrdenotes the rank of the excess bundleE, then transpose of the map

E^∗ 'Λ^r−1E ⊗Λ^rE^∗ ' Λ^r−1E ⊗Λ^dN^∗_T/Y⊗Λ^r+dbN^∗−−−−−−−−−→^{ϕr−1,r−1+d⊗id} Λ^r+d−1bN⊗Λ^r+dNb^∗ 'bN^∗ is a retraction ofαqed.

This method allows to provide pathological examples:

Proposition 8.1. Assume that the natural surjection Λ^dNb⊗Λ^dNT/Y −→ O_T does not split.

Then j^∗_Y/Z jX/Z∗O_X does not live in the image ofD^b(T)inD^b(Y).

Proof. Assume that there exist an elementθin D^b(T) such that j^∗_Y/Z jX/Z∗O_X = jT/Y∗θ. There is a canonical morphismO_Y ' j^∗_Y/ZO_Z −→ j^∗_Y/Z jX/Z∗O_X inducing the natural morphismO_Y −→ O_T on the local cohomology sheaves of degree 0. Then using the chain

Hom_D^b_(Y)

O_Y, jT/Y∗θ

' Hom_D^b_(T)

j^∗_T/YO_Y, θ

−→Hom_D^b_(Y)

jT/Y∗O_T, j^∗_Y/Z jX/Z∗O_X

we get a morphism fromO_T to j^∗_Y/Z jX/Z∗O_X inducing the canonical isomorphism in local coho- mology of degree 0. Thanks to the previous calculation:

Hom_D^b_(Y)

j_T/Y∗O_T, j^∗_Y/Z j_X/Z∗O_X

'M

q≥d

H^q−d Λ^qbN⊗Λ^dN_T/Y. The component in H⁰ Λ^dbN⊗Λ^dNT/Y

must map to 1 via the mapΛ^dbN⊗Λ^dNT/Y −→ O_T, which

contradicts our assumption.

9. The formality criterion for quantization

In this section, we prove Theorem 1.2. This is in fact a straightforward consequence of Theorem 1.1. Indeed, ifX is a closed submanifold ofY, the cycles∆^Y andX² are smooth submanifolds ofY² intersecting linearly along∆^X. Observe that∆^Y can be obviously quantized (for instance by one of the two projections), and that the excess exact sequence (3) on ∆^X is the conormal exact sequence

0−→N^∗_X/Y −→Ω¹Y|X −→ Ω¹X −→0.

Thanks to Theorem 1.1, j^∗_X2/Y²O_∆Y is formal in D^b X²

if and only if the conormal exact se- quence of the pair (X,Y) is holomorphically split, that is if and only ifX can be quantized.

To give a clearer picture of this result, we can draw the following diagram:

No necessary and sufficient condition is known for the formality

ofO_X⊗^L

O_YO_X. j^∗_X/YO_X is formal if and only if the conormal bundle

N^∗_X/Z extends toXZ

j^∗_X2/Y²O_∆Y is formal if and only if the conormal exact

sequence of (X,Y) splits.

O_X²⊗^L_O

Y2 O_∆Y D^b(X)

D^b(Y)

D^b X²

D^b Y² jX/Y∗

j_Y/Y²∗

jX²/Y²∗

10. Perspectives

In view of the preceding discussion, we propose a very natural conjecture, which predicts the formality ofallderived self-intersections in the bounded derived category of the ambient com- plex manifold.

Conjecture 10.1. For any pair(X,Y), the objetO_X⊗^L_O

Y O_X is always formal inD^b(Y).

As an evidence for this conjecture, we provide the following formality result for truncated objects of length two (which corresponds in classical terms to the degeneracy at the page E₂ of a spectral sequence):

Proposition 10.2. For any nonnegative integer p,τ^≥−(p+1)τ^≤−p O_X⊗^L_O

YO_X

is formal inD^b(Y).

Proof. Let j = j_X/Y andϑ = j^∗ j∗O_X. The Atiyah morphism at: j∗O_X −→ j∗N^∗_X/Y[1] in the derived category D^b(Y) yields by adjunction a morphism ϑ −→ N^∗_X/Y[1], which induces an isomorphism in local cohomology of degree−1. Thereforeτ^≥−1ϑ' O_X⊕N^∗_X/Y[1] in D^b(X). We can go a step further in D^b(Y): let us consider the morphism

j∗ϑ' O_X⊗^L_O

YO_X −^at−−−^⊗→^at N^∗_X/Y[1]⊗^L_O

YN^∗_X/Y[1]−−−→^∧ Λ²N^∗_X/Y[2].

It induces an isomorphism in local cohomology of degree−2. Therefore τ^≥−2j∗ϑ' O_X ⊕N^∗_X/Y[1]⊕Λ²N^∗_X/Y[2]

in D^b(Y). Recall now that j∗ϑis a ring object in D^b(Y), the ring structure being given by the composition of morphisms

j∗ϑ⊗^L

O_Y j∗ϑ−→ j∗(ϑ⊗^L

O_X j^∗ j∗ϑ)−→ j∗(ϑ⊗^L

O_Xϑ)' j∗ j^∗ O_X⊗^L

O_YO_X

−→ j∗ϑ.

Let us consider the morphism τ^≤−1j∗ϑ^⊗p −→ j∗ϑ^⊗p −→ j∗ϑwhere the last arrow is the ring multiplication. Since the left-hand side is concentrated in degrees at most−p, this morphism can be factored throughτ^≤−pj∗ϑ. Applying the truncation functorτ^≥−p+1, we get a morphism

τ^≥−p+1 τ^≤−1j∗ϑ^⊗p −→τ^≥−p+1τ^≤−pj∗ϑ.

Remark now that the natural morphismτ^≥−p+1 τ^≤−1j∗ϑ^⊗p −→τ^≥−p+1 τ^≥−2τ^≤−1j∗ϑ^⊗p

is an iso- morphism, so that we obtain a morphism

θ: τ^≥−p+1 τ^≥−2τ^≤−1 j∗ϑ^⊗p −→τ^≥−p+1τ^≤−pj∗ϑ.

We have seen thatτ^≥−2τ^≤−1 j∗ϑis formal; this yields τ^≥−p+1 τ^≥−2τ^≤−1 j∗ϑ^⊗p '

N^∗_X/Y[1]^⊗p

⊕

N^∗_X/Y[1]⊗p−1

⊗Λ²N^∗_X/Y[2]

.

Besides, in local cohomology, θ is simply the wedge-product. Using the antisymmetrization morphisms (see [5,§2.1]), we obtain a quasi-isomorphism

Λ^pN^∗_X/Y[p]⊕Λ^p+1N^∗_X/Y[p+1]−→ τ^≥−p+1 τ^≥−2τ^≤−1j∗ϑ^⊗p −−→^θ τ^≥−p+1τ^≤−pj∗ϑ

Remark 10.3. The crucial point in this proof is thatτ^≥−2τ^≤−1 j∗ϑis formal in D^b(Y). However, τ^≥−2τ^≤−1ϑis not always formal in D^b(X). In fact, this object is formal if and only if N^∗_X/Yextends at the first order (see [1]). This reflects the fact that j∗is not faithful.

References

[1] Dima Arinkin and Andrei C˘ald˘araru. When is the self-intersection of a subvariety a fibration? Adv. Math., 231(2):815–842, 2012.

[2] Dima Arinkin, Andrei C˘ald˘araru, and Marton Hablicsek. On the formality of derived intersections.Preprint, 2013.

[3] Armand Borel and Jean-Pierre Serre. Le th´eor`eme de Riemann-Roch.Bull. Soc. Math. France, 86:97–136, 1958.

[4] William Fulton.Intersection theory. Springer-Verlag, Berlin, 1998.

[5] J. Grivaux. The Hochschild-Kostant-Rosenberg isomorphism for quantized analytic cycles.Int. Mat. Res. No- tices, 2012.

[6] M. Kashiwara. Unpublished letter to Pierre Schapira, 1991.

[7] Jean-Pierre Serre.Local algebra. Springer-Verlag, Berlin, 2000.

CNRS, I2M, UMR 7373, CMI, Universit´e deProvence, 39,rueFr´ederic´ Joliot-Curie, 13453 MarseilleCedex 13, France.

E-mail address:jgrivaux@math.cnrs.fr