tovena/transpisa.pdfIndex theorems for subvarieties transversal to a holomorphic foliation Marco Abate, Filippo Bracci, Francesca Tovena April 2004 0. Introduction In 1982, C. Camacho

Index theorems for subvarieties transversal to a holomorphic foliation

Marco Abate, Filippo Bracci, Francesca Tovena

April 2004

0. IntroductionIn 1982, C. Camacho and P. Sad [CS] proved the existence of separatrices for singular holomorphic foliationsin dimension 2. One of the main tools in their proof was the following index theorem:

Theorem 0.1: (Camacho-Sad) Let S be a compact Riemann surface embedded in a smooth complexsurface M . Let F be a one-dimensional singular holomorphic foliation defined in a neighbourhood of Sand such that S is a leaf of F . Then it is possible to associate to any singularity q ∈ S of F a complexnumber ιq(F , S) ∈ C, the index of the foliation along S at q, depending only on the local behavior of Fnear q, such that ∑

q∈Sing(F)

ιq(F , S) =∫

S

c1(NS),

where c1(NS) is the first Chern class of the normal bundle NS of S in M .

The index can be defined as follows. Let A = (Uα, zα) be an atlas of M adapted to S, that is suchthat Uα ∩ S = z1

α = 0 for all indices α such that Uα ∩ S = ∅. Locally, the foliation F is generated by alocal vector field

Xα = X1α

∂

∂z1α

+ X2α

∂

∂z2α

,

with X1α|S ≡ 0 because S is a leaf of F ; in particular, q ∈ S is a singularity of F if and only if X2

α(q) = 0.Then the index is defined by

ιq(F , S) = Resq

(∂(X1

α/X2α)

∂z1α

∣∣∣∣S

dz2α

); (0.1)

it is not difficult to check that it is independent of the adapted chart chosen.Thus Theorem 0.1 gives a quantitative connection between the way S sits in M (the first Chern class

of NS is also equal to the self-intersection number S ·S of S) and the behavior of singular foliations tangentto S.

This theorem has been subsequently generalized in several ways, first allowing the existence of singu-larities of the Riemann surface S (Lins-Neto [Li], Suwa [S1]), and then to singular holomorphic foliationsdefined in a neighbourhood of a possibly singular subvariety S of a complex manifold M , without restrictionson the dimension of M or the codimension either of F or of S (see Lehmann [L], Lehmann-Suwa [LS1, 2]and references therein). In particular, using Cech-de Rham cohomology Lehmann and Suwa (see [S2] for asystematic exposition) developed a very complete theory of indices for holomorphic foliations along a pos-sibly singular leaf at singularities, either of the foliation or of the leaf. One of their results is the followinggeneralization of Theorem 0.1:

Theorem 0.2: (Lehmann-Suwa) Let S be a compact connected subvariety of codimension m in a complexmanifold M of dimension n. Assume that S is a locally complete intersection, and that there is a smoothvector bundle N over M extending the normal bundle NS of (the regular part of) S in M (such a vectorbundle N always exists if S is an hypersurface). Let F be a singular holomorphic foliation of dimension pdefined in a neighbourhood of S and leaving S invariant. Let Σλ be the decomposition in connectedcomponents of the singular set Sing(S)∪

(Sing(F)∩S

). Finally, let ϕ be a homogeneous symmetric polynomial

of degree d > n − p − m. Then we can associate to each connected component Σλ a residue

Resϕ(F , NS ; Σλ) ∈ H2n−2(m+d)(Σλ; C),

depending only on the local behavior of F near Σλ, such that∑λ

(iλ)∗Resϕ(F , NS ; Σλ) = [S] ϕ(N) in H2n−2(m+d)(S; C), (0.2)

2 Marco Abate, Filippo Bracci, Francesca Tovena

where iλ: Σλ → S is the inclusion, and ϕ(N) denotes the class obtained evaluating ϕ in the Chern classesof N .

In particular, when n = 2 and m = p = d = 1, we recover Theorem 0.1. Furthermore, if the sin-gular component Σλ is an isolated point (and d = n − m), we can often compute Resϕ(F , NS ; Σλ) usingGrothendieck residues; see [S2] for details.

In these theorems, the subvariety S is always supposed to be invariant by the foliation. Recent results byCamacho [C], Camacho and Lehmann [CL], and by Camacho-Movasati-Sad [CMS] have opened a different,and somewhat unexpected, venue of research: it is possible to get index theorems also when S is transversalto the foliation F , if one is willing to assume that S sits into the ambient manifold M in a particularly niceway. For instance, one might assume that M is the total space of a line bundle over S (identifying S withthe zero section of M); see [C, CL]. For more general subvariety S, Camacho, Movasati and Sad proved thefollowing

Theorem 0.3: (Camacho-Movasati-Sad) Let S be a compact Riemann surface embedded in a smoothcomplex surface M . Assume that S is fibered embedded and 2-linearizable in M , that is that there existsan atlas A = (Uα, zα) adapted to S such that the changes of coordinates satisfy

∂z2β

∂z1α

≡ 0, (0.3)

and∂2z1

β

∂(z1α)2

∣∣∣∣∣S

≡ 0. (0.4)

Let F be a singular holomorphic foliation of dimension one, and let SingS(F) denote the set of points of Swhere F is (singular or) tangent to the normal direction spanned by ∂/∂z1

α, which is well-defined becauseof (0.3), and assume that SingS(F) = S. Then

∑q∈SingS(F)

ιq(F , S) =∫

S

c1(NS),

where ιq(F , S) is again defined by (0.1).

Thus if S is fibered embedded and 2-linearizable into M we recover the Camacho-Sad index formulawithout assuming that the foliation is tangent to S; the information on the way S sits into M replace theinformation on the way S is placed with respect to F .

The aim of this paper is to give versions of this theorem in higher dimensions, or when S is singular.To describe our main results, let us first describe what happens in the discrete case, that is for holomorphicself-maps of a complex manifold.

In 2001, studying the local dynamics of holomorphic maps f tangent to the identity (that is defined ina neighbourhood of the origin in C

n and such that f(O) = O and dfO = id), the first author discovered anunexpected analogy between the problem of the existence of separatrices for singular holomorphic foliationsand the problem of existence of invariant curves for maps tangent to the identity. One component of thisanalogy, instrumental in his proof of the Leau-Fatou flower theorem in dimension 2 [A], is an index theoremvery similar to Theorem 0.1:

Theorem 0.4: ([A]) Let S be a compact Riemann surface embedded in a smooth complex surface M , andlet f be a germ about S of a holomorphic self-map of M such that f |S = idS . Assume that df acts as theidentity on the normal bundle NS of S in M , and that f is tangential to S (see below). Then it is possibleto associate to any point q ∈ S a complex number ιq(f, S) ∈ C, the index of f along S at q, depending onlyon the local behavior of f near q, and vanishing at all but a finite number of points, such that

∑q∈Sing(F)

ιq(F , S) =∫

S

c1(NS).

Index theorems for subvarieties transversal to a holomorphic foliation 3

This time the index is defined as follows. Let A = (Uα, zα) be an atlas adapted to S. Then if wewrite f = (f1

α, f2α) in local coordinates, we can consider the local meromorphic function

kα(z2α) = lim

z1α→0

f1α(zα) − z1

α

z1α

(f2

α(zα) − z2α

) ;

we shall say that f is tangential to S if kα ≡ ∞ (and it is not difficult to check that this condition isindependent of α). Then the index ιq(f, S) is defined by

ιq(f, S) = Resq

(kα dz2

α

), (0.5)

and again it does not depend on the adapted chart chosen.It should be remarked that, as first avatar of things to come, the same proof yielded a version of

Theorem 0.4 for any self-map f , not necessarily tangential, if M was the total space of a line bundle over S.Soon after [A] was completed, the second and third authors [BT] found a version of Theorem 0.4 for

possibly singular compact Riemann surfaces; thus it become natural to try and generalize Theorem 0.4 in away similar to what Lehmann and Suwa did starting from Theorem 0.1.

Let M be an n-dimensional complex manifold, S a (reduced, irreducible, possibly singular) subvarietyof M of codimension m, and f a germ about S of holomorphic self-map of M fixing S pointwise. Thenin [ABT] we defined the following two main objects:

– the order of contact νf ∈ N of f with S;– the canonical section Xf , a section over S of the coherent sheaf TM,S⊗(N ∗

S)⊗νf , where N ∗S is the conormal

sheaf of S and TM,S is the sheaf of germs of holomorphic sections over S of the restriction TM |S to Sof the tangent bundle of M .

Roughly speaking, the order of contact measures how close to the identity in a neighbourhood of S themap f is; and the canonical section indicates how f would move S if it did not keep it fixed.

The canonical section can be thought of as a section of the sheaf Hom((NS)⊗νf , TM,S), and thus itdetermines a canonical distribution Ξf ⊂ TM |S , the image of (NS)⊗νf through Xf . It turns out that,from a dynamical point of view, the most interesting case is when f is tangential, that is when the canonicaldistribution is tangent to (the regular part of) S (when S is a compact Riemann surface in a smooth complexsurface M this definition of tangential reduces to the one introduced above). In this case, it turns out thatthe dynamics of f is concentrated around the singular points of S and Xf : indeed, it is possible to provethat if p ∈ S is a smooth point of S not singular for Xf , then there is no infinite orbit of f arbitrarily closeto p.

For the sake of simplicity, let us now assume that S is a hypersurface of M ; in [ABT] are described resultswith S of codimension greater than 1, but in a slightly different spirit and of a more technical character.When S is a hypersurface and f is tangential to S, we can use the canonical section to define a holomorphicaction (sometimes called a partial (1, 0)-connection; see Section 3 for a precise definition) of N⊗νf

S on NS .Then following the ideas introduced by Lehmann and Suwa it is possible to prove the following generalizationof Theorem 0.4:

Theorem 0.5: ([ABT]) Let S be a compact, complex, reduced, irreducible, possibly singular, hypersurfacein an n-dimensional complex manifold M , and let f be a germ about S of a holomorphic self-map of Mfixing S pointwise. Assume that f is tangential to S, and let Σλ be the decomposition in connectedcomponents of the singular set Sing(Xf ) ∪ Sing(S). Then there exist complex numbers Res(Xf , S,Σλ) ∈ C,depending only on the local behavior of f near Σλ, such that

∑λ

Res(Xf , S,Σλ) =∫

S

cn−11 (NS),

where c1(NS) is the first Chern class of the normal bundle NS of S in M . Furthermore, when Σλ is anisolated point there is an explicit formula for computing Res(Xf , S,Σλ) using a Grothendieck residue.

Now, while in Theorem 0.2 the hypothesis that F is tangential to S looks completely natural, thecorresponding hypothesis here that f is tangential to S might look somewhat artificial. Actually, this is not


the case: in [ABT] we showed that from a dynamical point of view this is the most interesting situation (if fis transversal to S the dynamics of f nearby S is much easier to determine), and furthermore f is naturallytangential to S in important applications (for instance when blowing-up a non-dicritical map tangent to theidentity).

However, already in [ABT] we noticed that we do not need f to be tangential to S if we ask somethingmore on the way the regular part of S sits into M . To describe the exact hypotheses we need, which are (ofcourse) much weaker than asking M to be the total space of a line bundle over S, let us recall a couple ofdefinitions.

Let S be a (smooth) complex submanifold of a complex manifold M . We shall say that S splits into Mif the exact sequence

O −→ TS −→ TM |S −→ NS −→ O

splits as sequence of vector bundles over S, that is there is a projection σ:TM |S → TS which is the identityon TS. It turns out (see Section 1) that S splits into M if and only if the exact sequence

O −→ IS/I2S −→ OM/I2

S −→ OS = OM/IS −→ O

splits as sequence of OS-modules, where OM is the sheaf of germs of holomorphic functions on M , and IS

is the ideal sheaf of S. Furthermore, if S splits into M it is possible to introduce a structure of OS-moduleon IS/I3

S such thatO −→ I2

S/I3S −→ IS/I3

S −→ IS/I2S −→ O

becomes an exact sequence of OS-modules (see Section 2 for details). Then we shall say that S is comfortablyembedded in M if the latter sequence splits too.

In terms of adapted atlases, it turns out that S splits into M if there is an atlas A = (Uα, zα) adaptedto S (that is, such that Uα ∩ S = z1

α = · · · = zmα = 0, where m is the codimension of S) such that

∂zpβ

∂zrα

∣∣∣∣∣S

≡ 0, (0.6)

for all r = 1, . . . , m and p = m+1, . . . , n, where n = dimM . Furthermore, S is comfortably embedded in Mif there is an adapted atlas satisfying (0.6) and

∂2zrβ

∂zsα∂zt

α

∣∣∣∣∣S

≡ 0, (0.7)

for all r, s, t = 1, . . . , m. Clearly, the zero section of a vector bundle is comfortably embedded in the totalspace of the bundle. Furthermore, both splitting and being comfortably embedded can be characterized bythe vanishing of a suitable sheaf cohomology class; in particular, Stein submanifolds are always split andcomfortably embedded.

Then the proof of Theorem 0.5 can be adapted to yield the following index theorem:

Theorem 0.6: ([ABT]) Let S be a compact, complex, reduced, irreducible, possibly singular, hypersurfacein an n-dimensional complex manifold M , such that the regular part of S is comfortably embedded in M , andlet f be a germ about S of holomorphic self-map of M fixing S pointwise. If σ:TM |S → TS is the splittingprojection, set X = (σ ⊗ id) Xf ; roughly speaking, the image of X is the projection of the canonicaldistribution on TS. [Actually, if df does not act as the identity on the normal bundle of NS we shouldslightly change the definition of X to take in account the action of df ; but after this technical adjustement,everything works.] Assume that X ≡ O, and let Σλ be the decomposition in connected components of thesingular set Sing(X) ∪ Sing(S). Then there exist complex numbers Res(X, S, Σλ) ∈ C, depending only onthe local behavior of f near Σλ, such that

∑λ

Res(X, S, Σλ) =∫

S

cn−11 (NS).


Moreover, when Σλ is an isolated smooth point of S there is an explicit formula for computing Res(X, S, Σλ)using a Grothendieck residue.

Thus our investigations in the discrete setting yielded index theorems for subvarieties transversal toholomorphic self-maps. This suggests that comfortably embedded might be the right condition to get indextheorems for subvarieties transversal to holomorphic foliations. And indeed the main result of this paper isthe following

Theorem 0.7: Let S be a compact, complex, reduced, irreducible, possibly singular, subvariety of codimen-sion m of an n-dimensional complex manifold M , such that the regular part S′ of S is comfortably embeddedin M . Assume that there exists a coherent sheaf of OM -modules N on M such that N ⊗ OS′ = NS′ (thiscan always be arranged if S is smooth, or it is an hypersurface). Let F be a singular holomorphic foliationof dimension p on M , and let Fσ = σ(F ⊗OS), where σ:TM |S′ → TS′ is the splitting projection. Assumethat Sing(Fσ) = S, and that Fσ is involutive (which is automatic if p = 1 or p = n − m). Let Σλbe the decomposition in connected components of the singular set Sing(Fσ) ∪ Sing(S). Finally, let ϕ be ahomogeneous symmetric polynomial of degree d > n − p − m. Then we can associate to each connectedcomponent Σλ a residue

Resϕ(Fσ,N ; Σλ) ∈ H2n−2(m+d)(Σλ; C),

depending only on the local behavior of F near Σλ, such that∑λ

(iλ)∗Resϕ(Fσ,N ; Σλ) = [S] ϕ(N ) in H2n−2(m+d)(S; C), (0.8)


In particular, if we take n = 2, m = p = d = 1 and S smooth we get Theorem 0.3 because a fiberedembedded 2-linearizable Riemann surface is comfortably embedded.

This paper is structured as follows. In Section 1 we shall present several different characterizationsof splitting submanifolds. In Section 2 we shall introduce the notions of k-splitting and k-comfortablyembedded submanifolds, and discuss relations between these notions and infinitesimal neighbourhoods ofthe submanifold. Section 3 is devoted to the notion of holomorphic action. Finally, in Section 4 we shallprove Theorem 0.7 together with two other index theorems for subvarieties transversal to a foliation.

1. Holomorphic splittingIn this and the next section we shall study in detail a few conditions on the embedding of a subvariety intoa complex manifold. To do so, let us recall a few general facts on sequences of sheaves.

Definition 1.1: We say that an exact sequence of sheaves (of abelian groups, rings, modules. . . )

O −→ R ι−→S p−→T −→ O (1.1)

on a variety S splits if there is a morphism τ : T → S of sheaves (of abelian groups, rings, modules. . . ) suchthat p τ = id. Any such morphism is called a splitting morphism.

Let us recall a few standard facts on splitting sequences of sheaves.

Lemma 1.1: LetO −→ R ι−→S p−→T −→ O

be an exact sequence of sheaves of abelian groups over a variety S. Then the sequence splits iff there is amorphism of sheaves of abelian groups σ:S → R such that σ ι = id. Furthermore, the morphism σ andthe splitting morphism τ : T → S are such that σ τ = O and

ι σ + τ p = id . (1.2)

Proof : Assume that (1.1) splits as sequence of sheaves of abelian groups. Then p (id−τ p) = O; thereforefor every s ∈ S there is a unique σ(s) ∈ R such that ι

(σ(s)

)= s − τ

(p(s)

). It is clear that ι is a morphism,


that (1.2) holds, and that σ τ = O. Furthermore, for every r ∈ R we have ι(r) = ι(r) − τ(p(ι(r)

)), and

hence σ ι = id, as required.Conversely, assume that σ is given. For any t ∈ T choose s ∈ S such that t = p(s), and set

τ(t) = s − ι(σ(s)

). If s′ ∈ S is also such that t = p(s′), we have s′ − s = ι(r) for some r ∈ R, and

hences′ − ι

(σ(s′)

)= s + ι(r) − ι

(σ(s) + σ

(ι(r)

))= s − ι

(σ(s)

),

that is τ is well-defined. Then it is easy to check that τ is a splitting morphism satisfying (1.2) and στ = O.

Definition 1.2: If the sequence (1.1) splits, the morphism σ:S → R whose existence is asserted by theprevious lemma is called left splitting morphism.

Remark 1.1: If (1.1) splits as sequence of sheaves of O-modules, for a suitable sheaf O of rings, thenit is easy to check that the left splitting morphism is still a morphism of sheaves of O-modules. However, if(1.1) splits as sequence of sheaves of rings, the left splitting morphism is not a morphism of sheaves of rings;we shall later discuss its algebraic properties in our setting.

As soon as one splitting morphism exists, it is easy to see how many there are:


be a splitting exact sequence of sheaves of abelian groups over a variety S. Then there is a 1-to-1 correspon-dance between splitting morphisms and elements of H0

(S, Hom(T ,R)

). More precisely, if τ0: T → S is a

splitting morphism, with corresponding left splitting morphism σ0:S → R, all the splitting morphisms are ofthe form τ0−ιϕ, with corresponding left splitting morphism σ0+ϕp, for a unique ϕ ∈ H0

(S, Hom(T ,R)

).

Proof : Since p (τ0 − ι ϕ) = p τ0 = id, it is clear that τ0 − ι ϕ is a splitting morphism for anyϕ ∈ H0

(S, Hom(T ,R)

). Conversely, if τ is another splitting morphism, then p(τ0−τ) = O, and thus τ0−τ

must be of the form ι ϕ for a unique ϕ ∈ H0(S, Hom(T ,R)

). Finally, the assertion on the left splitting

morphisms follows from (1.2).

Remark 1.2: If τ0 and τ = τ0 − ι ϕ are two splitting morphisms of an exact sequence of O-modules,then ϕ is a morphism of O-modules too. However, if τ0 and τ = τ0 − ι ϕ are two splitting morphisms of anexact sequence of rings, then ϕ is not a morphism of rings.

Actually, it turns out that if (1.1) splits as sequence of sheaves of abelian groups but R and T also aresheaves of O-modules, then (1.1) splits as sequence of sheaves of O-modules:


be a splitting exact sequence of sheaves of abelian groups over a variety S. Assume moreover that R and Tare sheaves of O-modules for a suitable sheaf of rings O over S. Then there is a unique structure of sheaf ofO-modules on S such that ι, p and the splitting morphisms are morphisms of sheaves of O-modules.

Proof : Let us first prove the uniqueness. If S has a structure of sheaf of O-modules so that (1.1) splits as asequence of sheaves of O-modules, then for every x ∈ S, f ∈ Ox and s ∈ Sx we must have

f · s = f ·[ι(σ(s)

)+ τ

(p(s)

)]= ι

(f · σ(s)

)+ τ

(f · p(s)

), (1.3)

where σ and τ are the splitting morphisms and we are using (1.2). Thus the O-module structure of S isuniquely determined by the morphisms and the structures of R and T . Conversely, it is easy to check that(1.3) defines a structure of sheaf of O-modules on S such that ι, p, σ and τ become morphisms of sheaves ofO-modules.

Remark 1.3: If we choose a different splitting morphism, the O-module structure of S changes. Indeed,if we denote by ·0 the structure induced by a splitting morphism τ0, and by · the structure induced by thesplitting morphism τ = τ0 − ι ϕ, where ϕ ∈ H0

(S, Hom(T ,R)

)is considered as a morphism of sheaves of

abelian groups only, thenf · s = f ·0 s + ι

[f · ϕ

(p(s)

)− ϕ

(f · p(s)

)].


In particular, τ0 and τ induce the same O-module structure iff ϕ is a morphism of sheaves of O-modules.

Let us now introduce the sheaves we are interested in. Let M be a complex manifold of dimension n,and let S be a reduced, globally irreducible subvariety of M of codimension m ≥ 1. We denote by OM thesheaf of germs of holomorphic functions on M ; by IS the subsheaf of OM of germs vanishing on S; and by OS

the quotient sheaf OM/IS of germs of holomorphic functions on S. Furthermore, let TM denote the sheafof germs of holomorphic sections of the holomorphic tangent bundle TM of M , and ΩM the sheaf of germsof holomorphic 1-forms on M . Finally, we shall denote by TM,S the sheaf of germs of holomorphic sectionsalong S of the restriction TM |S of TM to S, and by ΩM,S the sheaf of germs of holomorphic sections along Sof T ∗M |S . It is easy to check that TM,S = TM ⊗OM

OS and ΩM,S = ΩM ⊗OMOS . We stress that TM,S

(respectively, ΩM,S) is not the sheaf on S induced from TM (respectively, ΩM ) by the injection S → M , butonly a quotient of this sheaf.

For k ≥ 1 let πk:OM → OM/IkS be the canonical projection; when f ∈ OM we shall often write [f ]k

for πk(f). It is easy to check that the map d2:OM/I2S → ΩM,S given by

d2[f ]2 = df ⊗ [1]1

is well-defined; furthermore, its restriction to IS/I2S is a morphism of sheaves of OS-modules. Then if we

define the sheaf ΩS of S by settingΩS = ΩM,S

/d2(IS/I2

S)

we get the conormal sequence of sheaves of OS-modules associated to S:

IS/I2S

d2−→ΩM,Sp−→ΩS −→ O. (1.4)

If we apply the functor HomOS(·,OS) to the conormal sequence we get the normal sequence of sheaves of

OS-modules associated to S:O −→ TS

ι−→TM,Sp2−→NS , (1.5)

where TS = HomOS(ΩS ,OS) is the tangent sheaf of S, NS = HomOS

(IS/I2S ,OS) is the normal sheaf of S,

and p2 is the morphism dual to d2.The first condition we shall consider on the way the variety S sits inside M is:

Definition 1.3: Let S be a reduced, globally irreducible subvariety of a complex manifold M . We saythat S splits into M if there exists a morphism of sheaves of OS-modules σ: ΩS → ΩM,S such that pσ = id.

In all the papers we consulted, splitting subvarieties are supposed non-singular to start with. This isactually a consequence of our definition:

Theorem 1.4: Let S be a reduced, globally irreducible subvariety of a complex manifold M . Assume thatS splits in M . Then S is non-singular, and the morphism d2: IS/I2

S → ΩM,S is injective.

Proof : Let us consider the sequence

O −→ K −→ ΩM,Sp−→ΩS −→ O,

where K = d2(IS/I2S) = Ker(p). This is an exact sequence of OS-modules, which splits; let σ: ΩS → ΩM,S

be the splitting morphism. Fix x ∈ S. Then σ((ΩS)x

)is a direct addend of the projective module (ΩM,S)x;

therefore σ((ΩS)x

)is itself projective and thus, being OS,x Noetherian, it is OS,x-free. But σ is an injective

OS-morphism; therefore (ΩS)x itself is OS,x-free for any x ∈ S.Now, both ΩM,S and ΩS are coherent sheaves of OS-modules, and S is globally irreducible; therefore

ΩS is locally OS-free of constant rank r. Checking the rank at a point in the regular part of S we see that rmust be equal to the dimension of S, and hence [Ha, Theorem II.8.15] S is non-singular, and [Ha, TheoremII.8.17] d2 is injective.


As a consequence, when S splits into M the sequence

O −→ IS/I2S

d2−→ΩM,Sp−→ΩS −→ O (1.6)

is a splitting exact sequence of OS-modules, and we also have a left-splitting morphism τ : ΩM,S → IS/I2S .

Conversely, if we had a OS-morphism τ : ΩM,S → IS/I2S such that τ d2 = id, then d2 would have to be

injective, and (1.6) would have been a splitting sequence of OS-modules, and thus S would split into M .In other words, even without assuming d2 injective, the existence of such a morphism τ is equivalent to Ssplitting into M .

Remark 1.4: If S splits into M then the normal sequence

O −→ TSι−→TM,S

p2−→NS −→ O (1.7)

is a splitting exact sequence of OS-modules too. In case S is (reduced) locally complete intersection and(1.5) is exact then S is non-singular. Indeed, since S is locally complete intersection then (1.4) is exact andIS/I2

S is locally free. Thus NS is locally free. Therefore, since TM,S is locally free as well, it follows that TS

must be locally free and (1.5) locally splits. Consider a local splitting σ : NS → TM,S . Since IS/I2S is locally

free, then the dual of NS is (naturally isomorphic to) IS/I2S . Therefore σ defines a local left splitting for

(1.4). This means that (1.4) is locally splitting and therefore ΩS is locally free, which implies S non-singular.

The aim of this section is to describe several equivalent characterizations of splitting subvarieties. Todo so we need a few more definitions.

Definition 1.4: Let S be a reduced, globally irreducible subvariety of a complex manifold M . Forany k ≥ h ≥ 0 let θk,h:OM/Ik+1

S → OM/Ih+1S be the canonical projection given by θk,h[f ]k+1 = [f ]h+1;

when h = 0 we shall write θk instead of θk,0. Then a k-th order lifting is a morphism of sheaves of ringsρ:OS = OM/IS → OM/Ik+1

S such that θk ρ = id. In other words, a k-th order lifting is a splittingmorphism for the exact sequence of rings

O −→ IS/Ik+1S −→ OM/Ik+1

S

θk−→OS −→ O. (1.8)

Remark 1.5: A k-th order lifting ρ yields, roughly speaking, a way to extend up to the k-th orderelements of OS to a neighbourhood in M . Indeed, given x ∈ S and f ∈ OS,x, if g1, g2 ∈ OM,x are such that[g1]k+1 = [g2]k+1 = ρ(f), then g1|S ≡ g2|S ≡ f and g1 − g2 ∈ Ik+1

S .

Remark 1.6: In the next section we shall need to know that ρ([1]1) = [1]k+1 for any k-th order liftingρ:OS → OM/Ik+1

S , a fact which is not an automatic consequence of being a morphism of rings but canbe proved as follows: from θk ρ = id we get [1]k+1 − ρ[1]1 ∈ IS/Ik+1

S , that is ρ[1]1 = [1 + h]k+1 fora suitable h ∈ IS . Now [1 + h]k+1 = ρ[1]1 = (ρ[1]1)(ρ[1]1) = [(1 + h)2]k+1 = [1 + 2h + h2]k+1, and so[h + h2]k+1 = O. But [1 + h]k+1 is a unit in OM/Ik+1

S ; therefore [h]k+1 = O, and ρ[1]1 = [1]k+1.

There is a more geometrical way of describing the existence of a k-th order lifting.

Definition 1.5: Let S be a reduced, globally irreducible subvariety of a complex manifold M . For k ≥ 0the k-th infinitesimal neighbourhood of S in M is the ringed space S(k) = (S,OM/Ik+1

S ) together with thecanonical inclusion of ringed spaces ιk:S = S(0) → S(k) given by ιk = (idS , θk). A k-th order infinitesimalretraction is a morphism of ringed spaces r:S(k) → S such that r ιk = id. More generally, for k ≥ h ≥ 0we shall say that S(k) retracts onto S(h) if there is a morphism of ringed spaces r:S(k) → S(h) such thatr ιh,k = id, where ιh,k = (idS , θk,h):S(h) → S(k) is the natural inclusion.

A k-th order infinitesimal retraction is given by a pair r = (η, ρ), with η:S → S and ρ:OS → OM/Ik+1S ,

such that (η, ρ) (idS , θk) = (idS , id); therefore η = idS and θk ρ = id. In other words, the existence of ak-th order infinitesimal retraction is exactly equivalent to the existence of a k-th order lifting.

As we shall momentarily see, there is a strong connection between splitting morphisms and first orderliftings. The connecting link is provided by the following definition:


Definition 1.6: Let O be a sheaf of rings, and M a sheaf of O-modules. A derivation of O in M is amorphism of sheafs of abelian groups D:O → M satisfying Leibniz rule:

D(f1f2) = f1 · D(f2) + f2 · D(f1).

More generally, if R is another sheaf of rings and θ:R → O is a morphism of sheaves of rings, a θ-derivationof R in M is a derivation of R in M, where the latter is considered as an R-module via restriction of thescalars. In other words, a θ-derivation is a morphism of sheaves of abelian groups D:R → M such that

D(r1r2) = θ(r1) · D(r2) + θ(r2) · D(r1)

for any r1, r2 in a stalk of R.

Now, the sheaf IS/I2S is naturally both an OS-module and an OM -module. Furthermore, the OM -

module structure is obtained by restriction of the scalars via the morphism π1:OM → OS , that is

f · [h]2 = [fh]2 = [f ]1 · [h]2

for all x ∈ S, f ∈ OM,x and h ∈ IS,x. This fact provides a key element in the proof of the following theorem,essentially due to Mustata and Popa (see [MP, Lemma 1.1] and [Ei, Proposition 16.12]):

Theorem 1.5: Let S be a reduced, globally irreducible subvariety of a complex manifold M . Then there isa 1-to-1 correspondance among the following classes of morphisms:

(a) morphisms σ: ΩS → ΩM,S of sheaves of OS-modules such that p σ = id;(b) morphisms τ : ΩM,S → IS/I2

S of sheaves of OS-modules such that τ d2 = id;(c) derivations D:OM → IS/I2

S such that D ι0 = π2|IS, where ι0: IS → OM is the canonical inclusion;

(d) θ1-derivations ρ:OM/I2S → IS/I2

S such that ρ ι1 = id;(e) morphisms ρ:OS → OM/I2

S of sheaves of rings such that θ1 ρ = id.

In particular, then, the following conditions are equivalent:

(i) S splits into M ;(ii) S has a first order lifting;(iii) S has a first order infinitesimal retraction.

Finally, if any (and hence all) of the classes (a)–(e) is not empty, then it is in 1-to-1 correspondance withthe following class of morphisms too:

(f) morphisms τ∗:NS → TM,S of sheaves of OS-modules such that p2 τ∗ = id.

Proof : We have already noticed that there is a 1-to-1 correspondance between classes (a) and (b), andbetween classes (f) and (b) if the latter is not empty. The gist of the rest of the proof is contained in thefollowing commutative diagram with exact rows and columns:

O −→ ISι0

−→ OMπ1−→ OS −→ O

D ↓ π2

∥∥∥O −→ IS/I2

S

ι1−→←−

ρ

OM/I2S

θ1−→←−

ρ

OM/IS −→ O

↑τ ↓ d2 d2

ΩM,S

↑σ ↓ p

ΩS

↓

O

.


Assume now that we have a morphism τ : ΩM,S → IS/I2S such that τ d2 = id; then putting

D = τ d2 π2

we get a derivation of OM in IS/I2S such that D ι0 = π2|IS

, as it is easily verified. Conversely, assumewe have a derivation D:OM → IS/I2

S such that D ι0 = π2|IS. The universal property of the module

of differentials yields a morphism of sheaves of OM -modules τ : ΩM → IS/I2S such that D = τ d, where

d:OM → ΩM is the usual differential. If we then define τ : ΩM ⊗OMOS = ΩM,S → IS/I2

S by

τ(ω ⊗ [f ]1) = [f ]1 · τ(ω),

we get a morphism of sheaves of OS-modules such that τ d2 = id, and it is easy to check that theseconstructions are inverses of each other.

Now assume we have a derivation D:OM → IS/I2S such that D ι0 = π2|IS

. Clearly, D kills I2S ;

therefore it defines a morphism of sheaves of abelian groups ρ:OM/I2S → IS/I2

S such that D = ρ π2. Since

π2|IS= D ι0 = ρ π2 ι0 = ρ ι1 π2|IS

and π2|ISis surjective, it follows that ρ ι1 = id. Finally, ρ is a θ1-derivation:

ρ([f ]2[g]2) = ρ([fg]2) = D(fg) = f · Dg + g · Df = [f ]1 · ρ[g]2 + [g]1 · ρ[f ]2 = θ1[f ]2 · ρ[g]2 + θ1[g]2 · ρ[f ]2.

Conversely, given a θ1-derivation ρ with ρ ι1 = id, we get a derivation D with D ι0 = π2|ISjust by

setting D = ρ π2, and these constructions are clearly inverses of each other.Finally, Lemma 1.1 shows that we have a 1-to-1 correspondance between morphisms ρ:OM/I2

S → IS/I2S

of sheaves of abelian groups such that ρ ι1 = id and morphisms ρ:OS → OM/I2S of sheaves of abelian

groups such that θ1 ρ = id; we are left to show that ρ is a θ1-derivation iff ρ is a morphism of sheaves ofrings.

First of all, if ρ is a θ1-derivation then ρ is a morphism of sheaves of rings: indeed,

ρ([f ]1[g]1) = ρ([fg]1) = [fg]2 − ι1ρ[fg]2 = [f ]2[g]2 − ι1([f ]1 · ρ[g]2 + [g]1 · ρ[f ]2

)= [f ]2[g]2 − [f ]2ι1ρ[g]2 − [g]2ι1ρ[f ]2= ([f ]2 − ι1ρ[f ]2)([g]2 − ι1ρ[g]2) = ρ([f ]1)ρ([g]1).

Conversely, if ρ is a morphism of sheaves of rings then ρ is a θ1-derivation: indeed,

ι1ρ([f ]2[g]2) = ι1ρ[fg]2 = [fg]2 − ρθ1[fg]2 = [f ]2[g]2 − ρ([f ]1[g]1)

= [f ]2[g]2 − (ρ[f ]1)(ρ[g]1) +([f ]2 − ρ[f ]1

)([g]2 − ρ[g]1

)= [f ]2

([g]2 − ρ[g]1

)+ [g]2

([f ]2 − ρ[f ]1

)= [f ]2ι1ρ[g]2 + [g]2ι1ρ[f ]2= ι1

(θ1[f ]2 · ρ[g]2 + θ1[g]2 · ρ[f ]2

),

and we are done because ι1 is injective.

Corollary 1.6: Let S be a reduced, globally irreducible subvariety of a complex manifold M . If S splitsinto M , then every splitting morphism induces a structure of OS-module on OM/I2

S . Furthermore, thisstructure is unique up to isomorphisms: given two splitting morphisms σ0, σ: ΩS → ΩM,S , with correspondingfirst order liftings ρ0, ρ:OS → OM/I2

S , there is an isomorphism χ:OM/I2S → OM/I2

S between the twoinduced structures such that the following diagram commutes:

O −→ IS/I2S

ι1−→ OM/I2

S

θ1−→←−

ρ

OS −→ O∥∥∥ ↓ χ

∥∥∥O −→ IS/I2

S

ι1−→ OM/I2

S

θ1−→←−ρ0

OS −→ O

.


Proof : The existence of the structure of OS-module follows from Lemma 1.3 and Theorem 1.5. For theuniqueness up to isomorphisms, arguing as in the previous proof and in Lemma 1.2 it is not difficult to checkthat ρ0−ρ = ι1 ϕ, where ϕ:OS → IS/I2

S is a derivation. Then Remark 1.3 shows that the relation betweenthe structure ·0 of OS-module induced by ρ0 and the structure · of OS-module induced by ρ is given by

[f ]1 · [g]2 = [f ]1 ·0 [g]2 − ι1(θ1[g]2 · ϕ[f ]1).

Then setting χ = id +ι1ϕθ1 we get an isomorphism between the two structures: indeed, χ−1 = id−ι1ϕθ1,and it is easy to check that χ([f ]1 · [g]2) = [f ]1 ·0 χ[g]2, as required. Finally, χ ι1 = ι1 and χ ρ = ρ0, andso χ is as claimed.

We have already noticed that a splitting subvariety is necessarily non-singular; therefore we can usedifferential geometric techniques, leading to another couple of characterizations of splitting submanifolds.

Definition 1.7: Let S be a complex submanifold of codimension m ≥ 1 in an n-dimensional complexmanifold M , and let (Uα, zα) a chart of M . We shall sistematically write zα = (z1

α, . . . , znα) = (z′α, z′′α), with

z′α = (z1α, . . . , zm

α ) and z′′α = (zm+1α , . . . , zn

α). We shall say that (Uα, zα) is adapted to S if either Uα ∩ S = ∅

or Uα ∩ S = Uα ∩ S = z1α = · · · = zm

α = 0. In particular, if (Uα, zα) is adapted to S then z1α, . . . , zm

α is a set of generators of IS,x for all x ∈ Uα ∩ S. An atlas U = (Uα, zα) of M is adapted to S if all itscharts are; then US = (Uα ∩ S, z′′α) | Uα ∩ S = ∅ is an atlas for S. The normal bundle NS of S in M isthe quotient bundle TM |S/TS; its dual is the conormal bundle N∗

S . If (Uα, zα) is a chart adapted to S, forr = 1, . . . , m we shall denote by ∂r,α the projection of ∂/∂zr

α|Uα∩S in NS , and by ωrα the local section of N∗

S

induced by dzrα|Uα∩S . Then ∂1,α, . . . , ∂m,α and ω1

α, . . . , ωmα are local frames over Uα ∩ S for NS and N∗

S

respectively, dual to each other.

Remark 1.7: From now on, every chart and atlas we consider on M will be adapted to S. Furthermore,we shall use Einstein convention on the sum over repeated indices. Indices like j, h, k will run from 1 to n;indices like r, s, t, u, v will run from 1 to m; and indices like p, q will run from m + 1 to n.

Remark 1.8: If (Uα, zα) and (Uβ , zβ) are two adapted charts with Uα ∩Uβ ∩ S = ∅, then it is easy tocheck that

∂zsα

∂zpβ

∣∣∣∣∣S

≡ O (1.9)

for all s = 1, . . . , m and p = m + 1, . . . , n.

When S is a non-singular submanifold of a complex manifold M , the sheaf ΩS is canonically isomorphicto the sheaf of germs of holomorphic sections of T ∗S, the sheaf TS is canonically isomorphic to the sheafof germs of holomorphic sections of TS, the sheaf NS is canonically isomorphic to the sheaf of germs ofholomorphic sections of NS , and the sheaf IS/I2

S is canonically isomorphic to the sheaf N ∗S of germs of

holomorphic sections of the conormal bundle. This isomorphism associates to [zrα]2 the local section ωr

α (andmore generally to any [h]2 ∈ IS/I2

S the local 1-form ∂h∂zs

α

∣∣∣S

ωsα), for any adapted chart (Uα, zα).

Definition 1.8: Let E ′ and E ′′ be two sheaves of locally free OS-modules over the same complexmanifold S. An extension of E ′′ by E ′ is an exact sequence of locally free OS-modules

O −→ E ′ −→ E −→ E ′′ −→ O.

If O → E ′ → E → E ′′ −→ O is another extension of E ′′ by E ′, we shall say that the two extensions areequivalent if there is an isomorphism χ: E → E of OS-modules such that

O −→ E ′ −→ E −→ E ′′ −→ O∥∥ ↓ χ∥∥

O −→ E ′ −→ E −→ E ′′ −→ O


commutes.

Remark 1.9: Almost by definition, an extension of E ′′ by E ′ splits iff it is equivalent to the trivialextension O → E ′ → E ′ ⊕ E ′′ → E ′′ → O.

Following Grothendieck and Atiyah (see [At]), we can associate to each equivalence classof extensions a cohomology class. Let

O −→ E ′ −→ E −→ E ′′ −→ O (1.10)

be an extension of sheaves of locally free OS-modules. Applying the functor Hom(E ′′, ·) to this sequence weget the exact sequence

O −→ Hom(E ′′, E ′) −→ Hom(E ′′, E) −→ Hom(E ′′, E ′′) −→ O. (1.11)

Let δ:H0(S, Hom(E ′′, E ′′)

)→ H1

(S, Hom(E ′′, E ′)

)be the connecting homomorphism in the long exact co-

homology sequence of (1.11).Then we can associate to the extension (1.10) the cohomology class

δ(idE′′) ∈ H1(S, Hom(E ′′, E ′)

),

and we have

Proposition 1.7: This procedure gives a 1-to-1 correspondance between equivalence classes of extensionsof E ′′ by E ′ and the cohomology group H1

(S, Hom(E ′′, E ′)

). In particular, an extension splits iff it corresponds

to the zero cohomology class.

Proof : [At, Proposition 1.2].

This suggests a cohomological characterization of splitting submanifolds:

Proposition 1.8: Let S be a complex submanifold of codimension m of a complex manifold M , and letU = (Uα, zα) be an adapted atlas. Then the cohomology class s ∈ H1

(S, Hom(NS , TS)

)associated to the

extensionO −→ TS −→ TM,S −→ NS −→ O

is represented by the 1-cocycle sβα ∈ H1(US ,Hom(NS , TS)

)given by

sβα =∂zs

β

∂zrα

∂zpα

∂zsβ

∣∣∣∣∣S

∂

∂zpα⊗ ωr

α ∈ H0(Uα ∩ Uβ ∩ S, TS ⊗N ∗S).

In particular, S splits into M iff s = O.

Proof : Over each Uα ∩ S we get a local splitting morphism τ∗α:NS |Uα∩S → TM |Uα∩S by setting

τ∗α(∂r,α) =

∂

∂zrα

.

Then τ∗α is a 0-cochain of Hom(NS , TM,S) lifting the 0-cocycle idNS

of Hom(NS ,NS); by definition ofconnecting homomorphism, then, sβα = τβ − τα represents the 1-cocycle δ(idNS

). Now,

(τ∗β − τ∗

α)(∂r,α) =∂zs

β

∂zrα

∣∣∣∣S

∂

∂zsβ

− ∂

∂zrα

=∂zs

β

∂zrα

∂zpα

∂zsβ

∣∣∣∣∣S

∂

∂zpα

for all r = 1, . . . , m, where we used (1.9), and we are done.

We can rewrite this characterization in a more useful form using the notion of splitting atlas.


Definition 1.9: Let U = (Uα, zα) be an adapted atlas for a complex submanifold S of codimen-sion m ≥ 1 of a complex n-dimensional manifold M . We say that U is a splitting atlas if

∂zpα

∂zrβ

∣∣∣∣∣S

≡ O

for all r = 1, . . . , m, p = m + 1, . . . , n and indices α, β such that Uα ∩ Uβ ∩ S = ∅. In other words, andrecalling (1.9), the jacobian matrices of the changes of coordinates become block-diagonal when restrictedto S.

Definition 1.10: Let τ∗:NS → TM,S be a splitting morphism for a complex submanifold S of codi-mension m ≥ 1 of a complex manifold M . We shall say that an adapted atlas U = (Uα, zα) is adaptedto τ∗ if τ∗(∂r,α) = ∂/∂zr

α for all r = 1, . . . , m and indices α such that Uα ∩ S = ∅.

Then

Proposition 1.9: Let S be a complex submanifold of codimension m ≥ 1 of a n-dimensional complexmanifold M . Then:

(i) S splits into M iff there exists a splitting atlas for S in M ;(ii) an atlas adapted to S is splitting iff it is adapted to a splitting morphism;(iii) if S splits into M , then for any splitting morphism there exists an atlas adapted to it.

Proof : (i) By Propositions 1.7 and 1.8, the existence of a splitting atlas clearly implies that S splits into M .Conversely, assume that S splits into M . Then by Propositions 1.7 and 1.8 we can find an adapted atlas U

and a 0-cochain c = cα ∈ H0(US , TS ⊗N ∗S) such that sβα = cβ − cα on Uα ∩ Uβ ∩ S, that is

∂zsβ

∂zrα

∂zpα

∂zsβ

∣∣∣∣∣S

= (cβ)qs

∂zsβ

∂zrα

∂zpα

∂zqβ

∣∣∣∣∣S

− (cα)pr (1.12)

on Uα ∩ Uβ ∩ S for all r = 1, . . . , m and p = m + 1, . . . , n, where we have written

cα = (cα)pr

∂

∂zpα⊗ ωr

α.

We then claim that the coordinates zrα = zr

α,zpα = zp

α + (cα)psz

sα,

(1.13)

restricted to suitable open subsets Uα ⊆ Uα, give a splitting atlas. Indeed, (1.12) yields∂zp

β

∂zrα

=∂zp

β

∂zsα

∂zsα

∂zrα

+∂zp

β

∂zqα

∂zqα

∂zrα

=∂zp

β

∂zrα

− (cα)qr

∂zpβ

∂zqα

+ R1

=∂zp

β

∂zrα

+ (cβ)ps

∂zsβ

∂zrα

− (cα)qr

∂zpβ

∂zqα

+ R1 = R1,

where each R1 denotes a (possibly different) term vanishing on S, and we are done.(ii) Let U = (Uα, zα) be an adapted atlas. Define local splitting morphisms τ∗

α:NS |Uα∩S → TM |Uα∩S

by setting τ∗α(∂r,α) = ∂/∂zr

α; we claim that in this way we get a global splitting morphism τ∗ iff U is asplitting atlas (necessarily adapted to τ∗). But indeed

τ∗α(∂r,α) − τ∗

β (∂r,α) =∂

∂zrα

−∂zs

β

∂zrα

∣∣∣∣S

∂

∂zsβ

=∂zp

β

∂zrα

∣∣∣∣∣S

∂

∂zpβ

,

and so τ∗α ≡ τ∗

β on Uα ∩ Uβ ∩ S for all α and β iff U is a splitting atlas.(iii) Let U = (Uα, zα) be a splitting atlas, adapted to the splitting morphism τ∗

0 , and choose anothersplitting morphism τ∗. Lemma 1.2 implies that τ∗ = τ∗

0 − ιϕ for a suitable ϕ ∈ H0(S, TS ⊗N ∗S). Therefore

there is a 0-cocycle c = cα ∈ H0(US , TS ⊗N ∗S) such that

τ∗(∂r,α) =∂

∂zrα

− (cα)pr

∂

∂zpα

.

Define new coordinates as in (1.13); the computations made in (i) show that we still get a splitting atlas. Fur-thermore, ∂/∂zr

α = τ∗(∂r,α) and the projection of ∂/∂zrα in NS is still ∂r,α; therefore the new atlas (Uα, zα)

is adapted to τ∗, as required.


Using splitting atlases it is very easy to describe the first order lifting associated to a given splittingmorphism:

Proposition 1.10: Let S be a complex submanifold of codimension m ≥ 1 of a n-dimensional complexmanifold M . Assume that S splits in M , let τ∗:NS → TM,S be a splitting morphism, and choose anatlas U = (Uα, zα) adapted to τ∗. For any g ∈ OS(Uα ∩ S) define gτ∗

α ∈ OM (Uα) by setting

gτ∗

α (z1α, . . . , zn

α) = g(zm+1α , . . . , zn

α).

Then the first order lifting ρ:OS → OM/I2S associated to τ∗ is given by ρ(g) = [gτ∗

α ]2.

Proof : This can be proved directly, but for the sake of completeness let us follows the path from τ∗ to ρdescribed in the proof of Theorem 1.5.

We know that τ∗ is locally given by τ∗(∂r,α) = ∂/∂zrα. Therefore the dual morphism τ : ΩM,S → N ∗

S isgiven by

τ(dzjα ⊗ [1]1) =

ωj

α for j = 1, . . . , m,0 for j = m + 1, . . . , n.

The canonical isomorphism between N ∗S and IS/I2

S sends ωrα to [zr

α]2. Therefore the θ1-derivation ρ associ-ated to τ is given by

ρ([f ]2) = τ(df ⊗ [1]1) =[

∂f

∂zrα

zrα

]2

for all f ∈ OM . Hence

ρ([f ]1) =[f − ∂f

∂zrα

zrα

]2

.

Now take g ∈ OS and any f ∈ OM such that g = [f ]1. Since in a suitable Uα we can write

f(z′α, z′′α) = f(O, z′′α) +∂f

∂zrα

(O, z′′α)zrα + R2,

where R2 ∈ I2S(Uα), and g(z′′α) = f(O, z′′α), it follows that

ρ(g) = [f(O, z′′α)]2 = [gτ∗

α ]2,

as claimed.

Our last characterization of splitting submanifolds compares the embedding in the ambient manifoldwith the embedding (as zero section) in the normal bundle. It turns out that a submanifold S splits in Miff its embedding in M coincides at the first order with its embedding in NS :

Proposition 1.11: Let S be a submanifold of a complex manifold M . Then S splits into M iff its firstinfinitesimal neighbourhood S(1) in M is isomorphic to its first infinitesimal neighbourhood SN (1) in NS ,where we are identifying S with the zero section of NS .

Proof : The main observation here is that if E is any vector bundle over S, then TE|S is canonically iso-morphic to TS ⊕ E. When E = NS this implies that the projection TNS |S → NS on the second directsummand induces an isomorphism NOS

→ NS , where NOSis the normal bundle of S (or, more precisely,

of the zero section of NS) in NS ; in particular, then, S always splits in NS (see also Example 1.1 later on).Furthermore, this isomorphism induces an isomorphism between N ∗

S and N ∗OS

, and thus an isomorphism ofsheaves of OS-modules χ: IS,NS

/I2S,NS

→ IS/I2S , where IS,NS

is the ideal sheaf of S in NS .By definition, an isomorphism between SN (1) and S(1) is given by an isomorphism of sheaves of rings

ψ:ONS/I2

S,NS→ OM/I2

S such that θ1 ψ = θN1 , where θN

1 :ONS/I2

S,NS→ OS is the canonical projection.

If SN (1) and S(1) are isomorphic, we can define a morphism of sheaves of rings ρ:OS → OM/I2S by

setting ρ = ψ ρN , where ρN is the first order lifting induced by the splitting of S in NS described above.Then it is easy to see that θ1 ρ = id, and thus S splits in M by Theorem 1.5.

Conversely, assume that S splits in M , and let ρ:OS → OM/I2S be a first order lifting. Then we can

define a morphism ψ:ONS/I2

S,NS→ OM/I2

S by setting

ψ = ρ θN1 + ι1 χ ρN ,

where ρN is the θN1 -derivation associated to the first order lifting ρN . Then it is not difficult to check that

ψ is an isomorphism of sheaves of rings such that θ1 ψ = θN1 , and thus SN (1) and S(1) are isomorphic.


We end this section with a list of examples of splitting submanifolds.

Definition 1.11: A (reduced, globally irreducible) subvariety S of a complex manifold M is a localholomorphic retract if there are an open neighbourhood U of S in M and a holomorphic map p:U → S suchthat p|S = idS ; the map p is a (local) holomorphic retraction of U onto S.

Example 1.1: A local holomorphic retract is always split in the ambient manifold (and thus it isnecessarily non-singular). Indeed, if p:U → S is a local holomorphic retraction, then a first order liftingρ:OS → OM/I2

S is given by ρ(f) = [f p]2. In particular, the zero section of a vector bundle always splits,as well as any slice S × x in a product M = S × X (with both S and X non-singular, of course).

Example 1.2: Let φ:M → M be a non-trivial involution of a complex manifold M such that its fixedpoint set S is non-singular; then S splits into M . Indeed, the decomposition of TM |S into the sum of the1-eigenspace TS and the (−1)-eigenspace of dφ yields a left splitting morphism τ∗:TM |S → TS, and thusa splitting morphism τ∗:NS → TM |S .

Example 1.3: If S is a Stein submanifold of a complex manifold M (e.g., if S is an open Riemannsurface), then S splits into M . Indeed, we have H1(S, TS ⊗ N ∗

S) = (O) by Cartan’s Theorem B, and theassertion follows from Proposition 1.8. In particular, if S is a singular curve in M then the non-singular partof S always splits in M .

Example 1.4: Let M be the blow-up of a point in a complex manifold M . Then the exceptionaldivisor E ⊂ M splits in M : indeed, it is easy to check that the atlas of M induced by the atlas of M is asplitting atlas.

Example 1.5: A smooth closed irreducible subvariety of Pn splits into P

n iff it is a linear subspace (see[VdV], [MP]).

Example 1.6: Let S be a connected submanifold of a complex torus M . Then S splits into M iff it isa local holomorphic retract of M iff it is a (complex) subtorus of M (see [MR]).

Example 1.7: Let S be a non-singular, compact, irreducible curve of genus g on a surface M . IfS · S < 4 − 4g then S splits into M . In fact, the Serre duality for Riemann surfaces implies that

H1(S, TS ⊗N ∗S) ∼= H0(S, ΩS ⊗ ΩS ⊗NS),

and the latter group vanishes because the line bundle T ∗S ⊗ T ∗S ⊗ NS has negative degree by assumption.The bound S · S < 4 − 4g is sharp: for instance, a non-singular compact projective plane conic S hasgenus g = 0 and self-intersection S · S = 4, but it does not split in the projective plane (see Example 1.5).

2. Comfortably embedded submanifolds

In the previous section we have seen that a complex submanifold S of a complex manifold M splits into Miff the sequence

O −→ IS/I2S

ι1−→OM/I2S

θ1−→OM/IS −→ O (2.1)

splits as a sequence of sheaves of rings, and in this case there is a (unique up to isomorphisms) structure oflocally free OS-module on OM/I2

S so that (2.1) splits as sequence of locally free OS-modules. Furthermore,we have seen that this is equivalent to the existence of an adapted atlas (Uα, zα) such that

z′′β = φβα(z′′α) + R2

with R2 ∈ I2S .

In this section we shall discuss other, more stringent conditions on the way S sits into M , beginningwith a natural generalization of splitting:


Definition 2.1: Let S be a submanifold of a complex manifold M , and k ≥ 1. We shall say thatS k-splits into M if there is an infinitesimal retraction of S(k) onto S, that is if there is a k-th orderlifting ρ:OS → OM/Ik+1

S , or, in other words, if the exact sequence

O −→ IS/Ik+1S −→ OM/Ik+1

S

θk−→OM/IS −→ O (2.2)

splits as sequence of sheaves of rings. Furthermore, we shall say that an atlas (Uα, zα) adapted to S isadapted to a k-th order lifting ρ:OS → OM/Ik+1

S if

ρ(f)(zα) = f(z′′α) + Ik+1S

for all f ∈ OS |Uα∩S .

There is a natural analogue of Propositions 1.9 and 1.10:

Theorem 2.1: Let S be an m-codimensional submanifold of an n-dimensional complex manifold M . ThenS k-splits into M iff there is an atlas U = (Uα, zα) adapted to S such that

∂zpβ

∂zrα

∈ IkS (2.3)

for all r = 1, . . . , m, p = m + 1, . . . , n and indices α, β such that Uα ∩ Uβ ∩ S = ∅. Furthermore, for anyk-th order lifting ρ:OS → OM/Ik+1

S we can find an atlas satisfying (2.3) and adapted to ρ.

Proof : Let us set up a few notations we shall need in the proof. Let (Uα, zα) be a chart adapted to S. Weshall denote by ϕα:Uα → C

n the local chart, so that ϕα(p) =(z1α(p), . . . , zn

α(p))

for all p ∈ Uα. If Uα∩S = ∅,we shall also assume (as we can, up to shrinking Uα) that

(O′, z′′α(p)

)∈ ϕα(Uα) for all p ∈ Uα. So we can

define a local k-th order lifting ρα:OS |Uα∩S → OM/Ik+1S |Uα∩S by setting

ρα(f)(p) = f(ϕ−1

α (O′, z′′α(p)))

+ Ik+1S (2.4)

for all f ∈ OS |Uα∩S and p ∈ Uα.Now let (Uβ , zβ) another chart adapted to S and such that Uα∩Uβ∩S = ∅. We shall need an expression

for ρβ(f) − ρα(f). First of all we have

ρβ(f)(p) − ρα(f)(p) = f(ϕ−1

β (O′, z′′β(p)))− f

(ϕ−1

α (O′, z′′α(p)))

+ Ik+1S

= f(ϕ−1

β (O′, z′′β(p)))− f

(ϕ−1

β

(ϕβ ϕ−1

α

(O′, (ϕα ϕ−1

β )′′(zβ(p)))))

+ Ik+1S .

Now let us assume that∂zp

β

∂zrα

∈ IlS

for all r = 1, . . . , m, p = m + 1, . . . , n, where 0 ≤ l ≤ k, which is equivalent to assuming that

∂lzpβ

∂zr1α · · · ∂zrl

α∈ IS

for all r1, . . . , rl = 1, . . . , m and all p = m + 1, . . . , n. Then it easy to prove by induction on l that we canwrite

zpα(zβ) = (ϕα ϕ−1

β )p(zβ) = φpαβ(z′′β) + hp

r1...rl+1(zβ)zr1

β · · · zrl+1β (2.5)

for suitable φpαβ ∈ O

(ϕβ(Uα ∩ Uβ ∩ S)

)and hp

ra...rl+1∈ O

(ϕβ(Uα ∩ Uβ)

), symmetric in the lower indices.

Clearly, φβα φαβ = id and ϕβ ϕ−1α (O′, z′′α) =

(O′, φ′′

βα(z′′α)). Therefore

ϕβ ϕ−1α

(O′, (ϕα ϕ−1

β )′′(zβ(p)))

=(O′, φ′′

βα

(φ′′

αβ(z′′β) + hpr1...rl+1

(zβ)zr1β · · · zrl+1

β

))


and

φ′′βα

(φ′′

αβ(z′′β) + hpr1...rl+1


β

)= z′′β +

∂φ′′βα

∂zpα

(φ′′

αβ(z′′β))hp

r1...rl+1(zβ)zr1

β · · · zrl+1β + Rl+2

= z′′β +∂z′′β∂zp

α(O′, z′′α)hp

r1...rl+1(zβ)zr1

β · · · zrl+1β + Rl+2,

where, here and elsewhere, Rj will denote a rest with coefficients in IjS . Summing up we get

ρβ(f) − ρα(f) = −∂zq

β

∂zpα

hpr1...rl+1


β

∂f

∂zqβ

+ Rl+2 + Ik+1S

= −hpr1...rl+1


β

∂f

∂zpα

+ Rl+2 + Ik+1S .

In particular, if l = k we get ρα ≡ ρβ , and so if (2.3) holds we get a global k-th order lifting (and our atlasis adapted to it). Furthermore, il l = k − 1 we get

ρβ − ρα = −hpr1...rk

|S∂

∂zpα⊗ [zr1

β · · · zrk

β ]k+1 ∈ H0(Uα ∩ S, TS ⊗ IkS/Ik+1

S ). (2.6)

Conversely, let us assume that we have a k-th order lifting ρ:OS → OM/Ik+1S ; we claim that there

exists an atlas adapted to ρ and such that (2.3) holds.We shall argue by induction on k. For k = 1 the assertion follows from Propositions 1.9 and 1.10. Now

let k > 1. Then ρ1 = θk,k−1 ρ is a (k− 1)-th order lifting; let U = (Uα, zα) be an atlas adapted to ρ1 andsuch that (2.3) holds for k − 1. Define local k-th order liftings ρα as in (2.4), and set σα = ρ − ρα. Now

θk,k−1 σα = ρ1 − θk,k−1 ρα ≡ O,

because the atlas is adapted to ρ1; therefore the image of σα is contained in IkS/Ik+1

S . The latter is an OS-module; we claim that σα:OS |Uα∩S → Ik

S/Ik+1S |Uα∩S is a derivation. Indeed,

σα(fg) = ρ(f)ρ(g) − ρα(f)ρα(g) = ρ(f)(ρ − ρα)(g) + ρα(g)(ρ − ρα)(f) = f · σα(g) + g · σα(f),

beacuse σα(f), σα(g) ∈ IkS/Ik+1

S , and uv = θk(u) · v for all u ∈ OM/Ik+1S and v ∈ Ik

S/Ik+1S .

Therefore we can find (sα)pr1...rk

∈ O(Uα ∩ S), symmetric in the lower indices, such that

σα = (sα)pr1...rk

∂

∂zpα⊗ [zr1

α · · · zrkα ]k+1.

Now, by construction σα − σβ = ρβ − ρα; therefore (2.6) yields

hps1···sk

∂zs1β

∂zr1α

· · ·∂zsk

β

∂zrkα

+ (sα)pr1...rk

− ∂zpα

∂zqβ

(sβ)qs1...sk

∂zs1β

∂zr1α

· · ·∂zsk

β

∂zrkα

∈ IS ,

and then

hps1···sk−1r

∂zs1β

∂zr1α

· · ·∂z

sk−1β

∂zrk−1α

+ (sα)pr1...rk

∂zrkα

∂zrβ

− ∂zpα

∂zqβ

(sβ)qs1...sk−1r

∂zs1β

∂zr1α

· · ·∂z

sk−1β

∂zrk−1α

∈ IS . (2.7)

Let us then consider the change of coordinateszrα = zr

α,zpα = zp

α + (sα)pr1...rk

(z′′α)zr1α · · · zrk

α ,

defined in suitable open sets Uα ⊆ Uα; we claim that (Uα, zα) is the atlas we are looking for.


Indeed, we have

∂zpα

∂zrβ

=∂zp

α

∂zsβ

∂zsβ

∂zrβ

+∂zp

α

∂zqβ

∂zqβ

∂zrβ

=∂zp

α

∂zrβ

+∂zp

α

∂zqβ

∂zqβ

∂zrβ

=∂zp

α

∂zrβ

+ k

[(sα)p

r1...rkzr1α · · · zrk−1

α

∂zrkα

∂zrβ

− ∂zpα

∂zqβ

(sβ)qs1...sk−1rz

s1β · · · zsk−1

β

]+ Rk.

Now, (2.5) with l = k − 1 yields

∂zpα

∂zrβ

= khps1...sk−1rz

s1β · · · zsk−1

β + Rk,

and so

∂zpα

∂zrβ

= k

[hp

s1...sk−1rzs1β · · · zsk−1

β + (sα)pr1...rk

zr1α · · · zrk−1

α

∂zrkα

∂zrβ

− ∂zpα

∂zqβ

(sβ)qs1...sk−1rz

s1β · · · zsk−1

β

]+ Rk

= k

[hp

s1···sk−1r

∂zs1β

∂zr1α

· · ·∂z

sk−1β

∂zrk−1α

+ (sα)pr1...rk

∂zrkα

∂zrβ

− ∂zpα

∂zqβ

(sβ)qs1...sk−1r

∂zs1β

∂zr1α

· · ·∂z

sk−1β

∂zrk−1α

]zr1α · · · zrk−1

α + Rk

= Rk ∈ IkS

thanks to (2.7), where we used the fact that zrkα = (∂zrk

α /∂zrβ)zr

β + R2.Finally, we should check that (Uα, zα) is adapted to ρ. But indeed (2.6) applied with zα instead of zβ

yields

f(z′′α) − f(z′′α) = (sα)pr1...rk

zr1α · · · zrk

α

∂f

∂zpα

+ Rk+1 = σα(f) + Rk+1,

and henceρ(f) = ρα(f) + σα(f) = f(z′′α) + Ik+1

S .

Remark 2.1: It is possible, though not trivial, to prove that if there is an adapted atlas Uα, ϕαsatisfying (2.3), then a k-th order lifting can be defined by the formula

ρ[f ]1 =k∑

l=0

(−1)l 1l!

[∂lf

∂zr1α · · · ∂zrl

αzr1α · · · zrl

α

]k+1

,

for all [f ]1 ∈ OS .

A consequence of the proof of the previous theorem is that there is an infinitesimal retraction of S(k)onto S if and only if there is an atlas (Uα, zα) adapted to S such that the coordinates changes are of theform

zrβ = (aβα)r

s(zα)zsα for r = 1, . . . , m,

zpβ = φp

αβ(z′′α) + Rk+1 for p = m + 1, . . . , n,

which, roughly speaking, says that a neighbourood of M is a fiber bundle over S up to order k.This is the kind of information we can extract from the existence of a k-th order lifting, that is from

the splitting of (2.2), which is a natural generalization of (2.1). But there is another natural generalizationof (2.1) that we can consider, that is

O −→ I2S/I3

S −→ OM/I3S

θ2,1−→OM/I2S −→ O; (2.8)

the splitting (as sequence of sheaves of rings) of this exact sequence is equivalent to the existence of aninfinitesimal retraction of S(2) onto S(1). Surprisingly enough, this cannot ever happen:


Proposition 2.2: Let S be a reduced, globally irreducible subvariety of a complex manifold M , and takek > h ≥ 1. Then if (k + 1)/2 ≥ h + 1 (where x is the smallest integer greater or equal to x) there areno infinitesimal retractions of S(k) onto S(h). In particular, there are no infinitesimal retractions of S(k)onto S(1) for any k ≥ 2.

Proof : For any 1 ≤ l ≤ h consider the following commutative diagram with exact rows and columns

O O O −→ Ih+1

S /Ik+1S −→ Il

S/Ik+1S

θk,h−→ IlS/Ih+1

S −→ O∥∥∥ O −→ Ih+1

S /Ik+1S −→ OM/Ik+1

S

θk,h−→ OM/Ih+1S −→ O

θk,l−1

θh,l−1

OM/Il

S == OM/IlS

O O

.

Assume, by contradiction, that we have a morphism of sheaves of rings ρ:OM/Ih+1S → OM/Ik+1

S suchthat θk,h ρ = id. Composing with θh,l−1 on the left we get

θh,l−1 = θh,l−1 θk,h ρ = θk,l−1 ρ.

This implies that ρ(IlS/Ih+1

S ) ⊆ IlS/Ik+1

S : indeed, if u ∈ IlS/Ih+1

S we have θk,l−1

(ρ(u)

)= θh,l−1(u) = O,

and hence ρ(u) ∈ IlS/Ik+1

S .Now, if u ∈ Il

S/Ih+1S we have ur = O as soon as r ≥ (h + 1)/l. Therefore if r ≥ (h + 1)/l we have

O = ρ(ur) = ρ(u)r ∈ IlS/Ik+1

S

for all u ∈ IlS/Ih+1

S . But since S is reduced, we have vr = O in IlS/Ik+1

S if and only if v ∈ IpS/Ik+1

S

with p ≥ (k + 1)/r. Therefore if (k + 1)/r ≥ h + 1 we have ρ(IlS/Ih+1

S ) ⊆ Ih+1S /Ik+1

S , and thusθk,h ρ|Il

S/Ih+1

S≡ O, contradiction.

Now, the largest value of (k + 1)/r is attained for the lowest value of r; and since r ≥ (h + 1)/l, thelowest value of r is 2, attained taking l = h. Therefore we get a contradiction if (k + 1)/2 ≥ h + 1, asclaimed.

The proof of this result depends on the fact that rings like IlS/Ik+1

S are nilpotent: there is an r ≥ 1such that vr = O for all v ∈ Il

S/Ik+1S . This suggests that the structure to consider in sequences like

O −→ IhS/Ih+1

S −→ IS/Ih+1S

θh,h−1−→ IS/IhS −→ O (2.9)h

is not the ring structure. It turns out that if S k-splits in M then there is another natural structure toconsider:

Proposition 2.3: Let S be a complex submanifold of codimension m of a complex manifold M , and letρ:OS → OM/Ik+1

S be a k-th order lifting, with k ≥ 0. Then for any 1 ≤ h ≤ k + 1 the lifting ρ induces a

structure of locally OS-free module on IS/Ih+1S so that (2.9)h becomes an exact sequence of locally OS-free

modules.

Proof : We shall work by induction on k. For k = 1 there is nothing to prove; so let us assume that theassertion holds for k − 1. First of all notice that every Ih

S/Ih+1S has a natural structure of locally free OS-

module, because IhS/Ih+1

S∼= Symh(N ∗

S). The k-th order lifting ρ induces a (k−1)-order lifting ρ1 = θk,k−1ρ;


therefore by induction for 1 ≤ h ≤ k we have a structure of locally free OS-module on IS/IhS so that all the

(2.9)h become exact sequences of locally free OS-modules. Now, IS/Ik+2S naturally is a OM/Ik+1

S -module;we can then endow it with the OS-module structure obtained by restriction of the scalars via ρ:

v · [h]k+2 = ρ(v) · [h]k+2,

for all v ∈ OS and h ∈ IS , where in the right-hand side we are using the OM/Ik+1S -module operation.

Since ρ is a ring morphism and ρ[1]1 = [1]k+1, by Remark 1.6, we get a well-defined structure of OS-moduleon IS/Ik+2

S . We must verify that the inclusion ι: Ik+1S /Ik+2

S → IS/Ik+2S and the projection θk+1,k are OS-

module morphisms when Ik+1S /Ik+2

S has its own OS-structure and IS/Ik+1S has the OS-structure induced

by ρ1 by induction.Given v ∈ OS , choose f ∈ OM such that v = [f ]1, and fI ∈ IS so that ρ(v) = [f + fI ]k+1. Then for all

h ∈ Ik+1S we have

ι(v · [h]k+2) = ι([fh]k+2) = [fh]k+2 = [(f + fI)h]k+2 = ρ(v) · [h]k+2 = v · ι[h]k+2,

and ι is an OS-morphism.Analogously, if h ∈ IS we have

θk+1,k(v · [h]k+2) = θk+1,k[fh + fIh]k+2 = [fh + fIh]k+1 = ρ1(v) · [h]k+1 = v · θk+1,k[h]k+2,

and θk+1,k is an OS-morphism.Finally, since Ik+1

S /Ik+2S and (by induction) IS/Ik+1

S are locally OS-free, Ik+1S /Ik+2

S is locally OS-freetoo, and we are done.

Remark 2.2: If (Uα, zα) is an atlas adapted to a k-th order lifting ρ:OS → OM/Ik+1S , it is easy to

see that [zrα]h+1, . . . , [zr1

α · · · zrhα ]h+1 is a local free set of generators of IS/Ih+1

S over OS for h = 1, . . . , k+1.

Thus if S k-splits into M we have an extension of IS/Ik+1S by Ik+1

S /Ik+2S . It is then natural to compute

the cohomology class associated to this extension:

Lemma 2.4: Let S be a complex submanifold of codimension m of a complex manifold M , and letρ:OS → OM/Ik+1

S be a k-th order lifting. Choose an atlas U = (Uα, zα) adapted to ρ. Then the

1-cocycle hρβα ∈ H1

(US ,Hom(IS/Ik+1

S , Ik+1S /Ik+2

S ))

given by

hρβα([zt1

α · · · zthα ]k+1) =

1(k + 1)!

∂zs1β

∂zr1α

· · ·∂z

sk+1β

∂zrk+1α

∂k+1(zt1α · · · zth

α )∂zs1

β · · · ∂zsk+1β

∣∣∣∣∣S

[zr1α · · · zrk+1

α ]k+2 (2.10)

for 1 ≤ t1, . . . , th ≤ m and 1 ≤ h ≤ k, represents the cohomology class hρ associated to the extension

O → Ik+1S /Ik+2

S → IS/Ik+2S → IS/Ik+1

S → O. (2.11)

Proof : We can define local splittings να: IS/Ik+1S |Uα → IS/Ik+2

S |Uα by

να[zt1α · · · zth

α ]k+1 = [zt1α · · · zth

α ]k+2

and extending by OS-linearity; since U is adapted to ρ, each να is a well-defined morphism of OS-modules.Now, for any f ∈ OM we can write

f(z′β , z′′β) = fβ(O′, z′′β) +k+1∑j=1

1j!

∂jf

∂zs1β · · · ∂z

sj

β

(O′, z′′β)zs1β · · · zsj

β + Rk+2, (2.12)

where Rk+2 ∈ Ik+2S . In particular,

[h]k+2 =k+1∑j=1

1j!

ρ

([∂jh

∂zs1β · · · ∂z

sj

β

]1

)· [zs1

β · · · zsj

β ]k+2,


for all h ∈ IS , and

[f ]k+1 = ρ([f ]1) +k∑

j=1

1j!

ρ

([∂jf

∂zs1β · · · ∂z

sj

β

]1

)[zs1

β · · · zsj

β ]k+1, (2.13)

for all f ∈ OM .Using these formulas we easily see that

hρβα([zt1

α · · · zthα ]k+1) = να([zt1

α · · · zthα ]k+1) − νβ([zt1

α · · · zthα ]k+1) = [zt1

α · · · zthα ]k+2 − νβ([zt1

α · · · zthα ]k+1)

=1

(k + 1)!

[∂k+1(zt1

α · · · zthα )

∂zs1β · · · ∂z

sk+1β

]1

[zs1β · · · zsk+1

β ]k+2,

=1

(k + 1)!∂zs1

β

∂zr1α

· · ·∂z

sk+1β

∂zrk+1α

∂k+1(zt1α · · · zth

α )∂zs1

β · · · ∂zsk+1β

∣∣∣∣∣S

[zr1α · · · zrk+1

α ]k+2,

as claimed.

We are thus led to the following

Definition 2.2: A submanifold S of a complex manifold M is k-comfortably embedded in M if itk-splits into M and the sequences (2.9)h split as sequence of locally free OS-modules for 1 ≤ h ≤ k + 1.

We can characterize 1-comfortably embedded submanifolds using adapted atlases.

Definition 2.3: Let S be a k-comfortably embedded submanifold of codimension m of a complexmanifold M . Let ρ:OS → OM/Ik+1

S a k-th order lifting, and ρk,k+1: IS/Ik+1S → IS/Ik+2

S a splittingmorphism of (2.11) considered with the OS-structure induced by ρ. We shall say that an atlas (Uα, zα)adapted to S is adapted to the pair (ρ, ρk,k+1) if it is adapted to ρ and

ρk,k+1([zr1α · · · zrj

α ]k+1) = [zr1α · · · zrj

α ]k+2

for all 1 ≤ j ≤ k and 1 ≤ r1, . . . , rj ≤ m.

Then we recover the original definition of comfortably embedded submanifold introduced in [ABT]:

Proposition 2.5: Let S be an m-codimensional submanifold of an n-dimensional complex manifold M .Then S is 1-comfortably embedded into M iff there exists an atlas U = (Uα, zα) adapted to S such that

∂zpβ

∂zrα

∈ IS and∂2zr

β

∂zsα∂zt

α

∈ IS (2.14)

for all r, s, t = 1, . . . , m, p = m + 1, . . . , n and indices α, β such that Uα ∩ Uβ ∩ S = ∅. Furthermore, forany first order lifting ρ:OS → OM/I2

S and every related splitting morphism ρ1,2: IS/I2S → IS/I3

S of (2.9)2we can find an atlas satisfying (2.14) and adapted to (ρ, ρ1,2).

Proof : If we have an atlas satisfying (2.14), from Proposition 1.8 and Lemma 2.4 follows that S is 1-comfortably embedded.

Conversely, assume that S is 1-comfortably embedded; in particular, we have a first order liftingρ:OS → OM/I2

S and a related splitting morphism ρ1,2: IS/I2S → IS/I3

S of (2.9)2. Furthermore, Lemma 2.4implies that h

ρ = O. Therefore we can find an atlas U adapted to ρ and a 0-cochain cα of

Hom(IS/I2S , I2

S/I3S) = NS ⊗ Sym2(IS/I2

S)

such that hρβα = cα − cβ . Writing

cα = (cα)rst ∂α,r ⊗ [zs

αztα]3,


we define zα by setting zrα = zr

α + (cα)rst(z

′′α) zs

αztα for r = 1, . . . , m,

zpα = zp

α for p = m + 1, . . . , n,

on a suitable Uα ⊆ Uα. Then U = (Uα, zα) clearly still is a splitting atlas adapted to ρ; we claim that italso satisfies (2.14). Indeed, if we write zr

β = (aβα)rsz

sα, by definition the functions

(aβα)rs = [δr

u + (cβ)ruv(aβα)v

t ztα](aβα)u

u1(dα)u1

s

satisfy zrβ = (aβα)r

szsα, where the (dα)u1

s ’s are such that zu1α = (dα)u1

s zsα. Hence

(∂(aβα)r

s

∂ztα

+∂(aβα)r

t

∂zsα

)∣∣∣∣S

= 2(cβ)ruv(aβα)u

s (aβα)vt |S +

(∂(aβα)r

s

∂ztα

+∂(aβα)r

t

∂zsα

)∣∣∣∣S

+ (aβα)ru

(∂(dα)u

s

∂ztα

+∂(dα)u

t

∂zsα

)∣∣∣∣S

.

Now, differentiatingzuα = (dα)u

v

(zvα + (cα)v

rszrαzs

α

)we get

δut =

∂(dα)uv

∂ztα

(zvα + (cα)v

rszrαzs

α

)+ (dα)u

v

(δvt + 2(cα)v

rtzrα

)and

0 =(

∂(dα)us

∂ztα

+∂(dα)u

t

∂zsα

)∣∣∣∣S

+ 2(cα)ust.

Recalling that hρβα = cα − cβ we then see that U satisfies (2.14), as desired.

Finally, if necessary we argue as in the proof of Proposition 1.9.(iii) to adapt U to (ρ, ρ1,2).

Definition 2.4: An atlas satisfying (2.14) will be said a 1-comfortable atlas adapted to ρ.

Remark 2.3: In other words, S is 1-comfortably embedded into M if and only if there is an at-las (Uα, zα) adapted to S such that the changes of coordinates are of the form

zrβ = (aβα)r

s(z′′α)zs

α + R3 for r = 1, . . . , m,zpβ = φp

αβ(z′′α) + R2 for p = m + 1, . . . , n. (2.15)

We have a third way of generalizing the notion of splitting submanifold:

Definition 2.5: Let S be a complex submanifold of a complex manifold M . We shall say that Sis k-linearizable if its k-th infinitesimal neighbourhood S(k) in M is isomorphic to its k-th infinitesimalneighbourhood SN (k) in NS , where we are identifying S with the zero section of NS .

We have seen that S splits into M if and only if it is 1-linearizable (Proposition 1.11). In general,however, k-split does not imply k-linearizable (while the converse hold). The missing link is provided by thenotion of comfortably embedded:

Theorem 2.6: Let S be a complex submanifold of a complex manifold M , and k ≥ 2. Then S is k-linearizable if and only if it is k-split and (k − 1)-comfortably embedded.

Proof : We shall denote by IS,N the ideal sheaf of S in NS , and by θNh,k:ONS

/Ih+1S,N → ONS

/Ik+1S,N the canonical

projections in NS . Notice that S is k-split and k-comfortably embedded in NS for any k ≥ 1, by, for instance,Theorem 2.1 and Lemma 2.4; we shall denote by ρN

k :OS → ONS/Ik+1

S,N and ρNk−1,k: IS,N/Ik

S,N → IS,N/Ik+1S,N

the corresponding morphisms.We shall work by induction on k. We have already seen that SN (1) ∼= S(1) implies that S is 1-split.

Suppose now that SN (k−1) ∼= S(k−1) implies that S is (k−1)-split and (k−2)-comfortably embedded, and


let assume that SN (k) ∼= S(k); let ψ:ONS/Ik+1

S,N → OM/Ik+1S be a ring isomorphism such that θk ψ = θN

k .The gist of the proof is contained in the following commutative diagrams:

O −→ IS,N/Ik+1S,N −→ ONS

/Ik+1S,N

θNk−→

←−ρN

k

OS −→ O

ψ

ψ

∥∥∥O −→ IS/Ik+1

S −→ OM/Ik+1S

θk−→←−ρk

OS −→ O

,

O −→ IkS,N/Ik+1

S,N −→ IS,N/Ik+1S,N

θNk,k−1−→←−

ρNk−1,k

IS,N/IkS,N −→ O

ψ

ψ

ψ

O −→ Ik

S/Ik+1S −→ IS/Ik+1

S

θk,k−1−→←−

ρk−1,k

IS/IkS −→ O

.

First of all, we define ρk = ψ ρNk . This is a ring morphism such that

θk ρk = θk ψ ρNk = θN

k ρNk = id,

and so S is k-split in M .Now, θk ψ = θN

k implies that ψ(Ker θNk ) ⊆ Ker θk, that is ψ(IS,N/Ik+1

S,N ) ⊆ IS/Ik+1S . Since ψ is an

isomorphism, it then induces a ring isomorphism (still denoted by ψ) between IS,N/Ik+1S,N and IS/Ik+1

S andthus, by restriction, a ring isomorphism between Ik

S,N/Ik+1S,N and Ik

S/Ik+1S . Therefore it also induces a quotient

ring isomorphism ψ between ONSIk

S,N and OM/IkS such that θk−1ψ = θN

k−1, and thus SN (k−1) ∼= S(k−1).Furthermore, ψ sends IS,N/Ik

S,N and IS/IkS so that ψ θN

k,k−1 = θk,k−1 ψ. This restriction of ψ is anisomorphism of OS-modules: indeed

ψ(u · [h]k) = ψ(ρN

k−1(u)[h]k)

= ψ(θN

k,k−1(ρNk (u)[h]k+1)

)=

[ψ

(ρn

k (u))ψ[h]k+1

]k

=[ρk(u)ψ[h]k+1

]k

= ρk−1(u)ψ([h]k) = u · ψ([h]k)

for all u ∈ OS and h ∈ IS,N , where ρNk−1 = θN

k,k−1 ρNk and ρk−1 = θk,k−1 ρk.

We then define ρk−1,k = ψ ρNk−1,k ψ−1; we claim that ρk−1,k is a morphism of OS-modules such that

θk,k−1 ρk−1,k = id. Indeed,

ρk−1,k(u · [h]k) = ψ ρNk−1,k ψ−1(u · [h]k) = ψ ρN

k−1,k

(u · ψ−1([h]k)

)= ψ

(u · (ρN

k−1,k ψ−1)([h]k))

= ψ(ρN

k (u)(ρNk−1,k ψ−1)([h]k)

)= ψ

(ρN

k (u))(ψ ρN

k−1,k ψ−1)([h]k) = ρk(u)ρk−1,k([h]k)

= u · ρk−1,k([h]k)

for all u ∈ OS and h ∈ IS . Finally, θk,k−1 ρk−1,k = ψ θNk,k−1 ρN

k−1,k ψ−1 = id, and hence S is(k − 1)-comfortably embedded in M , as claimed.

Conversely, assume that S is k-split and (k − 1)-comfortably embedded. Since we shall use differentmaps, let us rewrite the involved commutative diagrams:

O −→ IS,N/Ik+1S,N −→ ONS

/Ik+1S,N

θNk−→

←−ρN

k

OS −→ O

ψk

ψ

∥∥∥O −→ IS/Ik+1

S −→ OM/Ik+1S

θk−→←−ρk

OS −→ O

,

O −→ IlS,N/Il+1

S,N−→←−ρN

l−1,l

IS,N/Il+1S,N

θNl,l−1−→←−ρN

l−1,l

IS,N/IlS,N −→ O

χl

ψl

ψl−1

O −→ Il

S/Il+1S

−→←−ρN

l−1,l

IS/Il+1S

θl,l−1−→←−ρl−1,l

IS/IlS −→ O

.


In the proof of Proposition 1.11 we defined an OS-module isomorphism χ: IS,N/I2S,N → IS/I2

S ; sinceIl

S/Il+1S = Syml(IS/I2

S), and likewise for IlS,N/Il+1

S,N , we get for all l ≥ 1 an OS-module isomorphismχl: Il

S,N/Il+1S,N → Il

S/Il+1S . We claim that for all 1 ≤ l ≤ k we can define a ring and OS-module isomorphism

ψl: IS,N/Il+1S,N → IS/Il+1

S so that the above diagram commutes.We argue by induction on l. For l = 1, it suffices to take ψ1 = χ. Assume now that we have defined ψl−1,

and letψl = ρl−1,l ψl−1 θN

l,l−1 + χl ρNl−1,l,

where ρNl−1,l is the left splitting morphism associated to ρN

l−1,l. It is easy to check that ψl is invertible, withinverse given by

(ψl)−1 = ρNl−1,l (ψl−1)−1 θl,l−1 + (χl)−1 ρl−1,l,

where ρl−1,l is the left splitting morphism associated to ρl−1,l. Since all the maps involved are OS-modulemorphisms, ψl is a OS-module morphism; we are left to show that ψl is a ring morphism. Using the definition,it is easy to see that ψl is a ring morphism if and only if

ρl−1,l

(ψl−1(u)

)ρl−1,l

(ψl−1(v)

)− ρl−1,l

(ψl−1(u)ψl−1(v)

)= χl ρN

l−1,l

(ρN

l−1,l(u)ρNl−1,l(v)

),

that is if and only if

ρl−1,l

((ρl−1,l ψl−1)(u)(ρl−1,l ψl−1)(v)

)= χl ρN

l−1,l

(ρN


), (2.16)

for all u, v ∈ IS,N/IlS,N .

To prove (2.16), we work in local coordinates. Let (U, z) be a chart in a (k − 1)-comfortable atlas, sothat we have ρi−1,i[zr1 · · · zra ]i = [zr1 · · · zra ]i+1 for all i = 1, . . . , l, a = 1, . . . , l−1 and r1, . . . , ra = 1, . . . , m.The chart (U, z) induces a chart (U , z) in a (k − 1)-comfortable atlas of NS , with z = (v, z′′), wherev = (v1, . . . , vm) are the fiber coordinates. Then we have χ[vr]2 = [zr]2, and thus

χi[vr1 · · · vri ]i+1 = [zr1 · · · zri ]i+1

for all i = 1, . . . , k. From this and the fact that ρNi−1,i([v

r]i+1) = O for all i = 1, . . . , k and r = 1, . . . , m, itfollows easily that ψi([vr]i+1) = [zr]i+1 for all i = 1, . . . , l − 1 and r = 1, . . . , m.

Now take u, v ∈ IS,N/IlS,N ; we can write

u =l−1∑a=1

αr1...ra· [vr1 · · · vra ]l and u =

l−1∑b=1

βs1...sb· [vs1 · · · vsb ]l

for suitable αr1...ra, βs1...sb

∈ OS . Then

χl ρNl−1,l

(ρN


)=

∑a+b=l

αr1...raβs1...sb· [zr1 · · · zrazs1 · · · zsb ]l+1.

Using the fact that ψl−1 is a ring morphism and an OS-morphism we also find

(ρl−1,l ψl−1)(u) =l−1∑a=1

αr1...ra· [zr1 · · · zra ]l+1 and (ρl−1,l ψl−1)(v) =

l−1∑b=1

βs1...sb· [zs1 · · · zsb ]l+1,

and hence

ρl−1,l

((ρl−1,l ψl−1)(u)(ρl−1,l ψl−1)(v)

)=

∑a+b=l

αr1...raβs1...sb· [zr1 · · · zrazs1 · · · zsb ]l+1,

as claimed.


So in particular we have proved that ψk: IN,S/Ik+1N,S → IS/Ik+1

S is a ring and OS-module isomorphism.Let us then define ψ:ONS

/Ik+1S,N → OM/Ik+1

S by

ψ = ρk θNk + ψk ρN

k ,

where as usual ρNk is the left splitting morphism. It is easy to check that θk ψ = θN

k , and that ψ is invertible;we are left to show that it is a ring morphism.

If u, v ∈ ONS/Ik+1

S,N we can write u = ρNk (uo)+ ρN

k (u), with uo = θNk (u); so ψ(u) = ρk(uo)+ψk

(ρN

k (u));

and analogously for v. Therefore

uv = ρNk (uo)ρN

k (vo) +[ρN

k (uo)ρNk (v) + ρN

k (vo)ρNk (u) + ρN

k (u)ρNk (v)

]= ρN

k (uo)ρNk (vo) +

[uo · ρN

k (v) + vo · ρNk (u) + ρN

k (u)ρNk (v)

],

and thus

ψ(uv) = ρk(uovo) + ψk(uo · ρN

k (v) + vo · ρNk (u) + ρN

k (u)ρNk (v)

)= ρk(uo)ρk(vo) + uo · ψk

(ρN

k (v))

+ vo · ψk(ρN

k (u))

+ ψk(ρN

k (u))ψk

(ρN

k (v))

= ψ(u)ψ(v).

Remark 2.4: Thus a submanifold S is 2-linearizable if and only if there is an atlas (Uα, zα) adaptedto S such that the change of coordinates are of the form

zrβ = (aβα)r

s(z′′α)zs

α + R3 for r = 1, . . . , m,zpβ = φp

αβ(z′′α) + R3 for p = m + 1, . . . , n.

Therefore the 2-linearizable curves of [CMS] are 2-linearizable in our sense too; our definition is slightlyweaker because we do not require the z′′β coordinates to depend on the z′′α coordinates only.

The following result will be crucial in the sequel:

Corollary 2.7: Let S be a 1-comfortably embedded submanifold of codimension m of a complex manifold M ,and let U = (Uα, zα) be a 1-comfortable atlas adapted to a first order lifting ρ. Then

ρ

(∂zs

β

∂zrα

∣∣∣∣S

)=

[∂zs

β

∂zrα

]2

for all r, s = 1, . . . , m and indices α, β such that Uα ∩ Uβ ∩ S = ∅. In other words,

(∂zs

β

∂zrα

∣∣∣∣S

)τ∗

α

−∂zs

β

∂zrα

∈ I2S

for all r, s = 1, . . . , m and indices α, β such that Uα ∩ Uβ ∩ S = ∅, where we are using the notationsintroduced in Proposition 1.10, and τ∗:NS → TM,S is the splitting morphism associated to ρ.

Proof : The assertion follows from formula (2.13) applied to f = ∂zsβ/∂zr

α recalling that we are using a1-comfortable atlas.

We end this section with a couple of examples of k-split and comfortably embedded submanifolds.

Example 2.1: A local holomorphic retract is always k-split in the ambient manifold. Indeed, if p:U → Sis a local holomorphic retraction, then a k-th order lifting ρ:OS → OM/Ik+1

S is given by ρ(f) = [f p]k+1.

Example 2.2: If S is a Stein submanifold of a complex manifold M (e.g., if S is an open Riemannsurface), then S is k-split and k-comfortably embedded into M for any k ≥ 1. Indeed, by Cartan’s Theorem Bthe first cohomology group of S with coefficients in any coherent sheaf vanishes, and the assertion follows


from Proposition 1.7. In particular, if S is a singular curve in M then the non-singular part of S alwayssplits in M .

Example 2.3: Let M be the blow-up of a point in a complex manifold M . Then the exceptionaldivisor E ⊂ M is comfortably embedded in M : indeed, it is easy to check that the atlas of M induced bythe atlas of M is a comfortable atlas.

Example 2.4: Let S be a non-singular, compact, irreducible curve of genus g on a surface M such thatS · S < 4 − 4g, so that S splits into M . Now, the Serre duality for Riemann surfaces implies that

H1(S,NS ⊗N ∗S ⊗N ∗

S) ∼= H1(S,N ∗S) ∼= H0(S, ΩS ⊗NS);

the latter group vanishes if the line bundle T ∗S⊗NS has negative degree, that is if S ·S < 2−2g. Thereforeif g > 0 we have that S · S < 4 − 4g implies that S is comfortably embedded; if g = 0 we need S · S < 2 tohave S comfortably embedded.

3. Holomorphic actions

Let S be a complex manifold (for a while we forget the embedding of S in M). We refer to [BB] and [CL]for more details on holomorphic actions.

We recall that a subsheaf G ⊂ TS is integrable if G is coherent and, for each x ∈ S, the fiber Gx is closedunder the bracket operation for vector fields.

Let F be the sheaf of germs of holomorphic sections of a holomorphic vector bundle πF : F → S on S.A section X ∈ H0(S, TS ⊗ F∗) can be interpreted as a morphism X : F → TS and it induces a dual mapX∗ : Ω1

S → F∗ and a derivation X# : OS → F∗ by setting

X#(g) = X∗(dg) ∀x ∈ S, g ∈ (OS)x.

Definition 3.1: Assume that the image of X is an integrable subsheaf of TS . A holomorphic action ofF on E along X (or a X-connection on E) is a C-linear map X : E → F∗ ⊗ E such that:

X(gs) = X#(g) ⊗ s + gX(s) ∀ local sections g of OS and s of E .

If F = TS and the section X corresponds to the identity TS → TS , then X#(g) = dg, and a holomorphicaction of TS on E along X is just a (1, 0)-connection. If F is an integrable subbundle of TS and the sectionX corresponds to the inclusion map F → TS , an X-connection can be considered as a “partial” (1, 0)-connection. If F and G are integrable subbundles of TS with F ⊂ G, we denote by XF ∈ H0(S, TS ⊗ F∗)and XG ∈ H0(S, TS ⊗ G∗) the sections corresponding to the inclusions F → TS and G → TS .

Definition 3.2: Assume that F ⊂ G ⊂ TS are integrable subbundles and denote by ι : F → G theinclusion. We say that XG extends XF if the following diagram is commutative:

E XG−→ G∗ ⊗ E↓ XF ↓ ι∗⊗1

F∗ ⊗ E =−→ F∗ ⊗ E .

If F = TS , then TS admits an X-connection via Lie bracket. For general F and X, the bundle TS doesnot admit an X-connection; however, if one defines Q by the exact sequence O → F X−→TS

φ−→Q → O, thenone obtains a natural X-connection X : Q → F∗ ⊗ Q on Q by taking X(q) to be the map s → φ([X(s), q]),for a local section q of Q and any lifting q of q under φ.

Remark 3.1: Let E be a locally free sheaf on S. Given X ∈ H0(S, TS ⊗ F∗), the obstruction toconstructing X is given by a class δX(E) ∈ H1(End(E) ⊗ F∗) (cf., e.g., [CL], section 1). Tensorizing byEnd(E) the dual map X∗ : Ω1

S → F∗ and taking the map induced in cohomology, one gets a map

(id ⊗ X∗) : H1(S, End(E) ⊗ Ω1S) → H1(End(E) ⊗F∗).


Carrell and Lieberman ([CL], Prop. (1.1)) show that the obstruction δX(E) to the existence of an action Xand the Atiyah class c(E) of E are related by:

δX(E) = (id ⊗ X∗)(c(E)).

We recall that the Atiyah class c(E) ∈ H1(S, End(E) ⊗ Ω1S) of E is, by definition, the obstruction to the

splitting of the exact sequenceO → Ω1

S ⊗ E → J1(E) → E → Ogiven by J1(E) the E-valued one jets. According to [At], the vanishing of c(E) is equivalent to the existenceof a (1, 0)-connection on E .

Let us now assume that S ⊂ M is a connected, reduced, pure-dimensional subvariety of codimension min M , and we denote by S′ = S \ Sing(S) the regular part of S. Moreover we assume that S splits into M .

If F is a holomorphic foliation of M (we refer to [BB] or to [S2] for definitions and properties offoliations), and σ: TM,S → TS is a splitting morphism of S, we define

Fσ := σ(F ⊗OMOS),

and we setSing(Fσ) = (Sing(F) ∩ S) ∪ x ∈ S | rankFx = rankFσ

x .Definition 3.3: We say that the splitting morphism σ is F-faithful if Sing(Fσ) = S.

Notice that if the splitting morphism σ is F-faithful, the set So = S \(Sing(Fσ)∪ Sing(S)

)is open and

dense in S. By definition, no splitting morphism is F-faithful in case dimF > dimS.

Remark 3.2: If F has dimension either 1 or n − m, then Fσ is involutive on S′ \ Sing(Fσ), and thusit induces a foliation on S′. On the contrary, if F has dimension different from 1 or n − m then Fσ mightbe no involutive, as the following example shows.

Example 3.1: Let M = C4 with coordinates (z1, . . . , z4), let S = z1 = 0 and F be the (involu-

tive) non-singular foliation defined by (z2 − z1) ∂∂z3

+ ∂∂z4

and ∂∂z1

+ ∂∂z2

. Consider the natural splittingσ : (z1, z2, z3, z4) → (z2, z3, z4). Then Fσ is generated by z2

∂∂z3

+ ∂∂z4

and ∂∂z2

. Thus Fσ is a non-singulardistribution in S which is however non-involutive.

Remark 3.3: The splitting morphism σ is F-faithful if and only if kerσx ∩ Fx = (O) for at least onex ∈ S′ \ Sing(F).

A first condition which guarantees faithfulness is the following “no nowhere transversality condition”:

Proposition 3.1: Let S be a (connected, reduced, pure-dimensional) subvariety of a complex manifold M ,splitting in M , and let F be a foliation on M . If there exists x ∈ S \ (Sing(S) ∪ Sing(F)) such that S istangent to F at x, i.e., Fx ⊆ TS,x, then any splitting morphism σ is F-faithful.

Proof : We need to show that if σ is a splitting morphism then kerσx ∩Fx = (O). Suppose v ∈ Kerσx ∩Fx.Let ι : TS → TM,S be the natural injection. Since v ∈ Fx ⊆ TS,x and σ ι = idTS

by hypothesis, it followsthat σ(v) = 0 if and only if v = 0 and we are done.

Notice that there are topological obstructions for a foliation to be everywhere non-tangential to S. Forinstance, in [Bru], [Hon] it is proved that for S a curve in a surface M the number of points of tangency,counted multiplicity, is given by S2 − S · F . Therefore generically, fixed a splitting σ, this is F-faithful.On the other hand one can fix the foliation F and try to change the splitting. At least for one-dimensionalfoliations most splittings are faithful:

Proposition 3.2: Let S be a connected non-singular hypersurface splitting in a complex manifold M , andlet F be a one dimensional holomorphic foliation on M . Assume that S is not contained in Sing(F). Thenthere is at most one splitting morphism σ which is not F-faithful.

Proof : Suppose σ is a splitting morphism of S in M which is not F-faithful. By Lemma 1.2 any other splittingmorphism is given by σ + ϕ π for ϕ ∈ H0(S, Hom(NS , TS)) (and π : TM |S → NS the natural morphism).


If ϕ = 0 the set A(ϕ) = x ∈ S : ϕx = 0x is open and dense in S. In particular W = A(ϕ) ∩ (S \ Sing(F))is an open and dense subset of S. Since S has codimension one in M then NS has rank one and thus ϕ isinjective on W , i.e., kerϕx = (0) for all x ∈ W . Since σ is not F-faithful for any v ∈ Fx it follows thatv ∈ ker σx. Let 0 = v ∈ Fx ⊂ ker σx for x ∈ W . We have

(σ + ϕ π)x(v) = σx(v) + (ϕ π)x(v) = (ϕ π)x(v) = 0

for πx is injective on kerσx and ϕx is injective at x ∈ W . Therefore ker(σ +ϕπ)x∩Fx = (0) for any x ∈ Wand then the splitting morphism (σ + ϕ π) is F-faithful. Since this holds for any ϕ ∈ H0(S, Hom(NS , TS))it follows that σ is the unique non-faithful splitting morphism.

Corollary 3.3: Let S be a connected non-singular hypersurface comfortably embedded in a complex man-ifold M , and let F be a one dimensional holomorphic foliation on M . Assume that S is not contained inSing(F). If H0(S, TS ⊗N ∗

S)) = 0 then there exists a splitting morphism σ which is F-faithful.

Proof : If H0(S, Hom(NS , TS)) = H0(S, TS ⊗N ∗S)) = 0 then by Lemma 1.2 there exist at least two different

splitting morphisms. Then the result follows from Proposition 3.2.

Now we present few instances of holomorphic actions, which will be used later in the proof of our indicestheorems. First, assume S is a splitting submanifold of M , let σ a splitting of S in M and let F a foliationon M . We have the natural map X:Fσ → TS . This map induces an injective morphism of vector bundlesoutside Sing(Fσ).

Proposition 3.4: Let S be comfortably embedded submanifold of a complex manifold M , and let F be afoliation on M . Assume there is an F-faithful splitting morphism σ and Fσ is integrable. Then there existsa holomorphic action X of Fσ on NS on So = S \ Sing(Fσ) along X.

Proof : We are going to define X : Ns → (Fσ)∗ ⊗NS = Hom(Fσ,NS) in terms of adapted coordinates. Welet σ : TM,s → TS be the splitting morphism and π : TM,S → NS the projection.

Let (Uα, zα) be a chart centered at a point x ∈ So belonging to a σ-comfortable atlas, and take v ∈ Fσx .

Let

vα = ajα

∂

∂zjα

be the unique element in Fx such that σ(vα|S) = v. In adapted coordinates we have

v = apα|S

∂

∂zpα

.

Then we define a (local) extension vσα ∈ (TM,S)x of v by setting

vσα := ap

α

∂

∂zpα

+ [arα − (ar

α|S)σ]∂

∂zrα

, (3.1)

where (arα|S)σ is defined as in Proposition 1.10. For v ∈ Fσ and s ∈ NS (on So) define

X(s)(v)x = π([vσα, s]|S,x),

where s ∈ TM,S,x is such that π(s|S) = s and vσα is a local extension of v at x as previously defined. We

claim that X(s)(v) depends only on v, s and it is a holomorphic action of Fσ on NS along X.We first show that X does not depend on the local extension vσ

α chosen. To see this, let (Uβ , zβ) beanother chart centered at x belonging to the same σ-comfortable atlas. We claim that

vσβ = vσ

α + T1 + R2, (3.2)

where T1 ∈ TS ⊗OSIS and R2 ∈ TM,S ⊗OS

I2S .


First of all, we remark that we have

asα|S = aj

β |S∂zs

α

∂zjβ

∣∣∣∣∣S

= arβ |S

∂zsα

∂zrβ

∣∣∣∣∣S

+ apβ |S

∂zsα

∂zpβ

∣∣∣∣∣S

= arβ |S

∂zsα

∂zrβ

∣∣∣∣∣S

,

because ∂zsα/∂zp

β ∈ IS for any adapted atlas. In particular, Proposition 1.10 yields

(asα|S)σ = (ar

β |S)σ

(∂zs

α

∂zrβ

∣∣∣∣∣S

)σ

modI2S . (3.3)

Therefore

vσβ = ap

β

∂

∂zpβ

+ [arβ − (ar

β |S)σ]∂

∂zrβ

= ajβ

∂zkα

∂zjβ

∂

∂zkα

− (asα|S)σ ∂

∂zsα

+ (asα|S)σ ∂

∂zsα

− (arβ |S)σ ∂zk

α

∂zrβ

∂

∂zkα

= vσα +

[(as

α|S)σ − (arβ |S)σ ∂zs

α

∂zrβ

]∂

∂zsα

+ T1 = vσα + (ar

β |S)σ

[(∂zs

α

∂zrβ

∣∣∣∣∣S

)σ

− ∂zsα

∂zrβ

]∂

∂zsα

+ T1 + R2

= vσα + T1 + R2.

Therefore[vσ

β , s] = [vσα + T1 + R2, s] = [vσ

α, s] + T0 + R1,

where T0 represents a local section whose restriction is tangent to S and R1 is a local section vanishing onS. Thus π([vσ

β , s]|S) = π([vσα, s]|S). Now we are left to show that X(u)(s) is independent of s and it is a

holomorphic action. This is a straightforward calculation as in the proof of Theorem 5.1 in [ABT] and weleave it to the reader.

Remark 3.4: The integrability hypothesis of Fσ in the previous proposition is not used in the proof.However it is necessary in order to get vanishing of given characteristic classes.

Proposition 3.5: Let S be a 2-splitting submanifold of a complex manifold M , and let F be a foliationon M . Assume there is an F-faithful splitting morphism σ and Fσ is integrable. Let L be an integrablelocally OS-free submodule of TS . Assume that [L,Fσ] ⊆ L. Let NL := TM,S/L. Then there exists a

holomorphic action X of Fσ on NL on So = S \ Sing(Fσ) along X.

Proof : let U = (Uα, zα) be an atlas satisfying (2.3) with k = 2. Since L is involutive, we can find newcoordinates zr

α = zrα and zp

α = ϕpα(z′′α) in such a way that the new atlas (Uα, zα) satisfies

(i) for any α such that S ∩ Uα = ∅ it follows that L is generated by ∂∂zp

αfor p = n − + 1, . . . , n; where

≤ n − m is the rank of L;(ii) for all α, β such that Uα ∩ Uβ ∩ S = ∅ it follows

∂zqα

∂zrβ

∈ I2S ,

where q = m + 1, . . . , n − and r = 1, . . . , m.Using this coordinates the proof goes similarly to the one of Proposition 3.4. Indeed, for v ∈ Fσ and

s ∈ NL we consider a local extension vσα given in (3.1) and s such that ρ(s|S) = s, where ρ : TM,S → NL is

the natural projection. Then we define

X(v)(s) := ρ([vσα, s]|S).

The same calculations as in Proposition 3.4 give

vσβ = vσ

α + TL1 + R2,

where TL1 ∈ L ⊗ IS . As before, using [Fσ,L] ⊆ L one can see then that X(v)(s) depends only on v and s

and it defines a holomorphic action of Fσ on NL.

Remark 3.5: The main application of Proposition 3.5 is when L = Fσ (outside Sing(Fσ)), for thenthe hypothesis [Fσ,L] ⊂ L is satisfied as soon as Fσ is involutive (for instance when F has dimension one).In this case the holomorphic action is related to the variation of [LS2].


4. Transversal indices theorems

A first transversal index theorem is a trivial consequence of the Baum-Bott Theorem (see [BB] or Theo-rem III.7.6 in [Su2]): if F has dimension one and σ is F-faithful then Fσ is a one-dimensional foliation of S′

and hence

Theorem 4.1: Let S be a nonsingular compact submanifold of codimension m of a complex n-dimensionalmanifold M , and let F be a one-dimensional foliation of M . Assume that S splits in M and that there existsan F-faithful splitting morphism σ: TM,S → TS . Let Sing(Fσ) = ∪Σλ be the decomposition of Sing(Fσ) inconnected components. Let Qσ = TS/Fσ be the quotient sheaf representing the virtual normal bundle ofthe foliation Fσ. Then for any homogeneous symmetric polynomial ϕ of degree n − m there exist complexnumbers Resϕ(σ,Qσ,Σλ) ∈ C, depending only on the local behavior of Fσ near Σλ, such that

∑λ

Resϕ(σ,Qσ,Σλ) =∫

S

ϕ(Qσ).

Remark 4.1: If Σλ reduces to one point, the number Resϕ(σ,Qσ,Σλ) can be calculated by means ofa Grothendieck residue as in [BB] (see also [Su2]).

The other indices theorems that we have are based on the holomorphic actions defined in Proposition 3.4and Proposition 3.5. We need one more definition:

Definition 4.1: Let S be a subvariety of M and let Q be a OS-locally free module on So, an opendense subset of S \Sing(S). We say that Q is extendable to M if there exists a coherent sheaf of OM -modulesQ on M such that Q ⊗OM

OS = Q. We say that Q is an extension of Q.

Example 4.1: If S is nonsingular then NS is always extendable to M . If S is singular but has codi-mension one in M then the line bundle O([S]) associated to the divisor [S] provides an extension of the sheafof holomorphic sections of the normal bundle of S \ Sing(S) in M . More generally, if S is a local completeintersection defined by a section, or a strongly locally complete intersection, then the sheaf of holomorphicsections of the normal bundle of S \ Sing(S) in M is extendable to M (see [LS1] and [LS2]).

Using the very general Lehmann-Suwa theory (see, e.g., Chapter VI in [Su2], or [LS2]), Proposition 3.4gives us

Theorem 4.2: Let S be a compact, complex, reduced, irreducible, possibly singular, subvariety of codi-mension m of an n-dimensional complex manifold M , such that the regular part S′ of S is comfortablyembedded in M . Assume that there exists an extension N of NS′ . Let F be a singular holomorphic foliationof dimension p on M , and assume that the splitting σ is F-faithful and that Fσ is involutive. Let Σλbe the decomposition in connected components of the singular set Sing(Fσ) ∪ Sing(S). Finally, let ϕ be ahomogeneous symmetric polynomial of degree d > n − p − m. Then we can associate to each connectedcomponent Σλ a residue

Resϕ(Fσ,N ; Σλ) ∈ H2n−2(m+d)(Σλ; C),

depending only on the local behavior of Fσ near Σλ, such that

∑λ

(iλ)∗Resϕ(Fσ,N ; Σλ) = [S] ϕ(N ) in H2n−2(m+d)(S; C), (4.1)


Remark 4.2: Explicit calculations of the residue Resϕ(Fσ,N ,Σλ) when p = 1 and Σλ reduces to asingle point x ∈ S′ can be done exactly as in [LS]. Indeed, assume F is defined by the holomorphic vectorfield vα on the domain Uα of a chart centered at x (and belonging to a σ-comfortable atlas). Consider the(local) foliation Fσ on Uα defined by the vector vσ

α given by (3.1). Then S ∩Uα is Fσ-invariant and one hasan explicit expression of the residue as in Theorem 1 of [LS2].

Using Proposition 3.5 we obtain


Theorem 4.3: Let S be a compact, complex, reduced, irreducible, possibly singular, subvariety of codimen-sion m of an n-dimensional complex manifold M , such that the regular part S′ of S is 2-splitting in M . LetL be an integrable locally OS-free module on S′. Let F be a singular holomorphic foliation of dimension pon M , and assume that the splitting σ is F-faithful and that Fσ is involutive. Assume also that [L,Fσ] ⊆ Land that TM,S′/L has an extension NL. Let Σλ be the decomposition in connected components of the sin-gular set Sing(Fσ)∪Sing(S). Finally, let ϕ be a homogeneous symmetric polynomial of degree d > n−p−m.Then we can associate to each connected component Σλ a residue

Resϕ(Fσ,NL; Σλ) ∈ H2n−2(m+d)(Σλ; C),depending only on the local behavior of Fσ near Σλ, such that∑

λ

(iλ)∗Resϕ(Fσ,NL; Σλ) = [S] ϕ(NL) in H2n−2(m+d)(S; C), (4.2)

where iλ: Σλ → S is the inclusion, and ϕ(NL) denotes the class obtained evaluating ϕ in the Chern classesof NL.

In particular, we have the following corollary which generalizes the Camacho [C] and Camacho-Movasati-Sad [CMS] index theorems:

Theorem 4.4: Let S be a compact (reduced) irreducible curve comfortably embedded in a two dimensionalcomplex manifold M . Let F be a (one-dimensional) holomorphic foliation on M , and assume there is anF-faithful splitting morphism σ. Let Σ = Sing(Fσ)∪Sing(S) be the finite set of singular points of S and Fσ.Then for any p ∈ Σ there exists a complex number Res(σ,F , p) ∈ C, depending only on the local behaviorof F and σ nearby p, such that ∑

p∈Σ

Res(σ,F , p) = S · S.

Furthermore, if (U, (x, y)) is a chart centered at p such that S ∩U = θ = 0 and dθ∧ dy = 0 on S ∩U \ p,and F is generated on S ∩ U \ p by a ∂

∂θ + b ∂∂y then

Res(σ,F , p) =1

2πi

∫Γ

1b

∂a

∂θdy,

where Γ is the link of the singularity of S at p, i.e., Γ = S3 ∩ S for some small real 3-dimensional sphere S3

centered at p.

Proof : The first part of the theorem follows from Theorem 4.2, taking N = O([S]), ϕ = c1 the first Chernpolynomial, and recalling that [S] c1(O([S])) = S · S.

Thus we are left with the computation of the residue. First, we remark that dθ ∧ dy = 0 on S ∩U \ pimplies that, up to shrinking U if necessary, y is a coordinate on S ∩ U \ p. Also, there exists a open setVp ⊂ U \ p such that Vp ∩ S = U \ p ∩ S and dθ ∧ dy = 0 on Vp.

Furthermore, as explained in Remark 6.5 of [ABT] we can assume that, up to replace θ with a functionof the form

(1 + c(y)θ

)θ, the chart

(Vp, (θ, y)

)belongs to a σ-comfortable atlas.

Then, according to [LS] (see also [Su2] and [ABT], section 6) the residue is given by an expression ofthe form

− 12πi

∫Γ

η

where η is the 1-form representing the holomorphic action X of Fσ on NS |Γ along X as given by Proposi-tion 3.4. Note that the dual of [θ] ∈ IS/I2

S is a holomorphic frame for O([S]) on U , which we denote by ξ,that is ξ = π(∂/∂θ) outside p. From the very definition we have that (η ⊗ 1)( ∂

∂y ⊗ ξ) = X( ∂∂y )(ξ), and this

latter is defined as

X(∂

∂y)(ξ) = π

([λvσ,

∂

∂θ

]∣∣∣∣S

),

where π: TM,S → NS is the canonical projection, vσ = (a − a|σS) ∂∂θ + b ∂

∂y and λ = 1/b (if necessary, we canshrink Γ so that b = 0 on Γ). A direct calculation gives then

X(∂

∂y)(ξ) = − ∂

∂l

(a − a|σS

b

)|S ⊗ ξ = −1

b

∂a

∂l|S ⊗ ξ,

and then the result.


Remark 4.3: If S is a singular curve in M then its regular part is always comfortable embedded in Mand thus Theorem 4.4 always applies in this situation.

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Documents

tovena/transpisa.pdfIndex theorems for subvarieties transversal to a holomorphic foliation Marco Abate, Filippo Bracci, Francesca Tovena April 2004 0. Introduction In 1982, C. Camacho