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Global stucture of webs in codimension one.
Vincent Cavalier, Daniel Lehmann
To cite this version:
Vincent Cavalier, Daniel Lehmann. Global stucture of webs in codimension one.. 2008. �hal-00264473�
hal-00264473, version 1 - 17 Mar 2008
GLOBAL STRUCTURE OF WEBS IN CODIMENSION ONE
(´etat du 17/03/2008) Vincent Cavalier and Daniel Lehmann
Abstract
In this paper, we describe the global structure of webs in codimension one, and study in particular their singularities (thecaustic). We define and study different concepts which have no interest locally near regular points, such as thetype, thereducibility, thequasi-smoothness, theCI- property (complete intersection), thedicriticity.... As an example of application, we explain how the algebraicity of a global holomorphic web in codimension one on the complex n-dimensional projective space Pn (algebraicity that we prove by the way to be equivalent to the linearity) depends only on the behaviour of this web near its caustic, at least for quasi-smooth webs with CI irreducible components.
1- Introduction
2- Background on the manifold of contact elements 3- Webs of codimension one on a holomorphic manifoldM 4- Global polynomial partial differential equations
5- Complete and locally complete intersection webs (CI and LCI) 6- Webs on the complex projective spacePn
7- Dicriticity and algebraicity of global webs onPn References
A.M.S. Classification : 57 R 20, 57 R 25, 19 E 20.
Key words: Global webs, caustics, quasi-smoothness, dicriticity, linearizability, algebraicity.
Global structure of webs in codimension one
Vincent Cavalier Daniel Lehmann
1 Introduction
In this paper, we want for instance to explain how the algebraicity of a global holomorphic web in codimension one on the complexn-dimensional projective space Pn reflects on the behaviour of this web near its singularities, and -in some cases- can be readen on this behaviour, generalizing a theorem already proved in [CL1] forn= 2. By the way, we shall need to describe the global structure of webs in codimension one, which may be interesting in itself. We shall define in particular thequasi-smoothness and thedicriticity, both concepts which can be readen on the singular part of the web (itscaustic). In fact, we shall see easily that any linear web onPn is dicritical (in fact, any globally defined linear web is algebraic). Conversely, at least for webs whose any irreducible component is acomplete intersection (CI), we shall prove that the quasi-smoothness (a generically satisfied condition) and the dicriticity (generically not satisfied) imply the algebraicity. The interest of such a result is to provide a situation where properties of the web on the caustic imply properties everywhere else.
The general method will be to define a web of codimension one on an-dimensional manifold M by the data of a n-dimensional subvarietyW of the manifold Mfof the contact elements of M, the tautological contact form on Mfbeing integrable on the regular part W′ of W, and inducing conse- quently a foliationFeof codimension one. This foliation is nothing else but a kind of desingularization (“decrossing”) of the web: above the regular partM0 of the web (the points of M where the leaves of the local foliations defined by the web are not tangent),the leaves of Fe project locally onto the leaves of the local foliations inM, the partW0 ofW aboveM0 being simply a covering space ofM0. But we shall be also interested by looking what happens above the part ΓW of W which projects onto the caustic M \M0. The quasi-smoothness means that any irreducible component C of W is smooth (C′ =C), so that we may use onCthe general tools for foliations on smooth manifold, and in particular the vanishing theorem of Bott : dicriticity means in fact that the foliationF|eCinduced by Fe onC′ has no singularity ; in this case, the square (c1)2 N(F|eC)
of the Chern class of the bundle N(F|eC) normal to the foliationF|eC, must vanish. In particular, whenM =Pn, this will imply that all partial degreesδαof the CI-web are necessarily zero : this characterizes algebraic CI-webs.
Let us be now more precise. Let d be an integer ≥ 1. Locally, on an open set U of Cn with coordinatesx= (xλ)λ, a codimensional one holomorphicd-web onUis a family ofdone codimensional holomorphic foliationsFi, (1≤i≤d), onU. The regular part of the web is the open subsetU0 ofU where these foliations are all non-singular and where their leaves are not tangent. We shall not require in this definition ofU0that anykof thesedfoliations are in general position fork≤n. Each foliation Fi may be given by a holomorphic 1-formωi =Pn
λ=1aλi(x) dxλ which is integrable (ωi∧dωi ≡0), defined up to multiplication by a unit (holomorphic nowhere-vanishing functionu), and whose kernel is the Lie algebra of vector fields tangent toFi. If we assumeani(x)6= 0, the leaves ofFi will be the hypersurfaces which are the graphsxn = ϕ(x1,· · ·, xn−1) of the common solutions ϕ to the n−1 partial differential equationsaαi(x) +ani(x)∂x∂ϕα = 0, (1≤α≤n−1).
We can still say that a locald-web is defined by the unionW =Sd
i=1Ui of then-dimensionalUi’s, eachUi being defined by the n−1 equations aαi(x) +ani(x)pα = 0 (1 ≤α ≤ n−1) in U ×Cn−1 with coordinates x= (xλ), p= (pα)
, (1≤λ≤n , 1≤α≤n−1). Observe that the local contact form dxn−P
αpα dxα has a restriction to W which is integrable, and that the foliationFe defined
by it onW has a restriction to Ui which projects onto Fi by the map (x, p) 7→x. Of course, this transcription inU×Cn−1 may seem pedantic and needless while we remain locally onU. But when we wish to define a d-web globally on a holomorphic n-dimensional manifold M which is no more necessarily equal to an open setU ofCn, it may happen, as already observed in the casen= 2 (see [CL1]), that the local foliationsFi are not globally distinguishable. Thus, it will be useful to handle them all together, (x, p) becoming local coordinates on the manifoldMfof contact elements ofM, and W a n-dimensional subvariety of Mf, on the smooth partW′ of which the tautological contact form ofMfbecomes integrable, W0 →M0 becoming a d-fold covering space which is no more necessarily trivial.
In the next section 2, we recall some basic facts, more or less known, on the 2n−1-dimensional manifoldMfof contact elements inM.
In section 3, we give the general definition of a globald-web onM as andimensional subvarietyW ofMf, globalizing the previous considerations, and precising thecritical set ΓW and its projection onto M which is the caustic. Various concepts such as thetype of ad-web, the reducibility (decomposition of the web as the juxtaposition of a finite number of global “irreducible” webs), the “smoothness” and the “quasi-smoothness”, the “dicriticity”, etc... are defined.
After defining a global PDE in section 4 as a particular hypersurfaceS inMf, we study in section 5 the particular case of the so-calledCI-webs (whenW is the complete intersectionW =∩n−1α=1Sα of n−1 distinct global PDE’s Sα), and LCI-webs (locally complete intersections). Many calculations are much easier in these cases. In particular, after defining the critical scheme, whose underlying analytical set is ΓW, we give a practical criterium in this case for a web to be dicritical.
In section 6, we study some properties specific to the caseM =Pn. LetP′
n be the dual projective space of hyperplanes inPn. Both Pfn and Pf′n are equal to the sub-manifold ofPn×P′n whose points are the pairs (m, H) of a point m ∈ Pn and a hyperplane H such that m ∈ H. An interesting duality exists between webs onPn and onP′
n. In particular, some sub-varietyW’s ofPfn may define simultaneously a non-dicritical web on Pn and a non-dicritical web on P′
n (the so called bi-webs).
We shall study also the degree of a web and the multi-degree of a CI-web: in particular, the CI- webs which are algebraic are those of muti-degree 0. Their leaves are also the hyperplanes tangent to an algebraic developable hypersurface (hypersurface equal to the union of a one-parameter family of (n−2)-dimensional projective subspaces ofPn, along which the tangent hyperplane remains the same). Thelinear webs are those whose all leaves are hyperplanes ofPn: they all are dicritical. The cohomology ofPfn is then computed.
We have now all the needed tools for giving in section 7 the properties of the dicritical webs onPn. In a second part of our study which will be published somewhere else (see preprint [CL2]), we focus on abelian relations. We study there the webs satisfying to a condition which is generically satisfied, the so calledregularity:
- the rank of such a regular web at a non-singular point (i.e. the dimension of the vector space of germs of abelian relations at that point) is upper-bounded by some number π′(n, d) which, for n≥3, is strictly smaller than the number π(n, d) of Castelnuovo (the maximal geometrical genus of an algebraic irreducible and non-degenerate curve of degreed in Pn, which Chern proved to be the upper-bound of the rank in the general case ([C]));
- we define onM0, whendis equal for somer to the the dimension c(n, r) of the vector space of all homogeneous polynomials of degreerinnvariables with scalar coefficients, a generalization of the Blaschke curvature; this is the curvature of some connection. The non-vanishing of this curvature is an obstruction for the rank of the web to reach the maximal valueπ′(n, d).
[In the casen = 2, any web is regular, any d may be written c(2, d−1), the numbers π(2, d) and π′(2, d) are equal; we then recover, globalizing them onM0, the Blaschke curvature defined locally in [H] forn= 2, d≥3, generalizing the casen= 2, d= 3 of Blaschke ([B])].
2 Background on the manifold of contact elements
LetM be a holomorphic (non-singular) manifold, not necessarily compact. Denote bynits complex dimension,T M its complex tangent bundle, andMfthe 2n−1-dimensional manifold equal to the total space of the grassmanian bundleGn−1M →π M (a point ofMfover a pointm∈Mis an−1-dimensional sub-vector space ofTmM, and is calleda contact elementofM atm=π(m).e
In the sequel, indices such asλ, µ,· · · will denote integers runing from 1 ton, whileα, β,· · · will denote integers runing from 1 ton−1.
For all family v = (vα) of n−1 vectors linearly independant in TmM, [v] or [v1,· · ·, vn−1] or [(vα)α] will denote the contact element generated byv in Mfm.
Since anyme overmis the kernel of a non-vanishing 1-form onTmM well defined up to multiplication by a scalar unit, we can also identifyMfwith the total space ofPT∗M →M (the projectivisation of the complex cotangent bundleT∗M).
2.1 Canonical bundles and tautological contact form on M f :
LetT ⊂π−1(T M) be the tautological vector-bundle overMf(whose fiber over [v1,· · ·, vn−1] is the sub-vector space ofTmM generated by (v1,· · ·, vn−1), identified with the line inTm∗M of all 1-forms vanishing on thevα’s (1≤α≤n−1).
Lemma 2.1 The dual L∗ of the quotient bundle L= π−1(T M)/T is the tautological complex line- bundle ofP(T∗M).
The proof is obvious.
Let (∗) and (∗∗) the two obvious exact sequences of vector bundles (∗) 0→ T →π−1(T M)→ L →0.
and
(∗∗) 0→ V →TMf→π−1(T M)→0,
whereV denotes the sub-bundle of tangent vectors toMfwhich are ”vertical” (i.e. tangent to the fiber ofπ).
By composition of the projectionTMf→π−1(T M) of (∗∗) with the projectionπ−1(T M)→ Lof (∗), we get a canonical holomorphic 1-form onMf, with coefficients inL
ω:TMf→ L,
which is calledthe tautological contact form. This terminology will be justified below by lemma 2-3.
Remark 2.2 We imbed L∗ intoπ−1(T∗M)by duality of(∗), and π−1(T∗M)intoT∗Mfby duality of (∗∗). We may also imbed T intoTMf, when we identify it to the kernel ofω.
2.2 Local coordinates on M f
Letx= (x1, x2,· · ·xn) be local holomorphic coordinates on an open setU ofM, andma point ofU. Letu= (u1,· · ·, un) denote the coordinates inTm∗M with respect to the basis (dx1)m,· · ·,(dxn)m; let [u] = [u1,· · ·, un] denote the point in PTm∗M of homogeneous coordinates u with repect to the same basis in Tm∗M, and let p = (p1, p2,· · ·pn−1) be the system of affine coordinates on the affine subspaceun6= 0 ofPTm∗M defined by pα=−uuα
n (1≤α≤n−1).
Therefore, (x, p) = (x1, x2,· · ·xn, p1, p2,· · ·pn−1) is a system of local holomorphic coordinates on the open set Ue of contact elements above U which are not parallel to ∂x∂
n
: the point of coordinates (x, p) is the (n−1)-dimensional subspace of Tm∗M which is the kernel of the 1-form η = (dxn)m−Pn−1
α=1pα(dxα)m ; it is generated by the the vectors (Xα)m =
∂
∂xα
m+pα
∂
∂xn
m, (1≤α≤n−1).
Obviously, the vector fieldsXα=
∂
∂xα
+pα
∂
∂xn
, (1≤α≤n−1), define a holomorphic local trivializationσT ofT aboveUe, while the imageσL= ∂
∂xn
of ∂x∂n by the projectionπ−1(T M)→ L of (∗) defines a holomorphic local trivialization ofL.
Remark 2.3 By permutation of the order of the coordinates xi, such that the new last one be no more the former last one(writing for examplex′n=x1, x′1=xn andx′λ=xλ for λ6= 1, n), we get a new system of local coordinates on some open setfU′ containing contact elements parallel to ∂x∂
n
m. Lemma 2.4
(i)The local formη =dxn−Pn−1
α=1 pα dxα is a contact form on U, and defines a local holomorphice trivialization ofL∗. Moreover, the trivializations η and σL= [∂x∂n]are dual to each other.
(ii)The 1-formη⊗σL is equal to the restriction ofω toUe.
Proof : Both forms η⊗σL and ω vanish when applied to the vertical vectors and to the vectors
∂
∂xα+pα ∂
∂xn (withα≤n−1), and take the valueσL when applied to ∂x∂n, hence are equal. Moreover η belongs to the dual L∗ (identified to the set of 1-forms onMfvanishing onT). Since η(∂x∂
n)≡1, σL andη are dual to each other. We check easily thatη∧(dη)n−1 is a volume form.
QED
2.3 Change of local coordinates and transition functions
Let (x′, p′) (or (x′, u′)) be the local coordinates associated to x′ = (x′1,· · ·, x′n) above some open set U′ of M such that U ∩U′ 6= ∅. Let J (or J(x, x′) in case of ambiguity) be the jacobian matrix J = D(xD(x′11,···,···,x,x′nn)) with coefficientsJµλ = ∂x∂x′λµ. Denote by ∆Mn (or ∆Mn (x, x′)) its determinant, and let K (or K(x, x′) be the matrix K = ∆Mn . J−1, with coefficients Kµλ = ∆Mn .∂x∂xλ′
µ. In the sequel u= (u1, u2,· · ·, un) will also be understood as a line-matrix (u1 u2 · · · un).
Proposition 2.5
(i)The coordinates uandu′ are related by the formulae u′= (1/∆Mn )u◦K andu=u′ ◦J), while for homogeneous coordinates [u′] (resp. [u]), defined up to multiplication by a non-zero scalar, we may write[u′] = [u◦K] (resp. [u] = [u′ ◦J]). The following formulae hold :
p′α=− P
βKβα pβ−Knα P
βKβn pβ−Knn andpα=− P
βJβα p′β−Jnα P
βJβn p′β−Jnn. (ii) The expressions ∆Tn−1(x, x′) =− P
βKβn pβ−Knn
constitute a system of transition functions for the bundleVn−1
T :
X1∧ · · · ∧Xn−1= ∆Tn−1(x, x′) X1′ ∧ · · · ∧Xn−1′ .
(iii)Let η′=dx′n−P
α p′α dx′α. The following formula holds : η′= ∆Mn (x, x′)/∆Tn−1(x, x′)
η,
and the functions∆Mn /∆Tn−1 constitute a system of transition functions forL :
Proof :
In fact the formulau=u′ ◦J is equivalent to the definition of the jacobian matrix. Interchanging xandx′, we getu′= (u◦J−1) ; but, at the level of the projective spaceJ−1 andKinduce the same automorphism, and [u′] is also equal to [u◦K].
We then deduce the following formula, by writingun =u′n =−1,uα =pα and u′α=p′α. Inter- changingJ andJ−1, we get also thepα’s in function of thep′α’s.
Let ∆Tn−1 be a transition function for the bundle Vn−1
T. From the exact sequence (∗), we deduce the isomorphismL ∼=Vn
π−1(T M)⊗Vn−1
T∗. Therefore, ∆Mn /∆Tn−1is a system of transition functions forL.
Moreover, since both 1-formsη andη′ are colinear, we deduce the equality η= −1
∆Mn X
β
Kβn pβ−Knn η′
from the equality ∆Mn .dxλ =P
µKµλdx′µ. This proves that ∆Tn−1=− P
βKβn pβ−Knn
and achieves the proof of the proposition.
3 Webs of codimension one on a holomorphic manifold M
LetW be a sub-variety ofMfhaving pure dimensionn. LetπW :W →M be the restriction toW of the projectionπ:Mf→M . Denote by
W′ the regular part ofW,
ΓW the set of pointsme ∈W where - eitherW is singular,
- or the differentialdπW :TmeW′ →Tπ(m)e M is not an isomorphism.
W0 the complementary part (included intoW′) of ΓW in W, M0=π(W0).
Letdbe an integer≥1.
Definition 3.1 We shall say that W is a d-web if (o)The mapπW :W →M is surjective 1.
(i)The restrictionωW toW′ of the tautological contact form ω:TMf→ L satisfies to the integra- bility condition: this means that, given a local trivialization σL of L, the restriction ηW toW′∩Ue of the local contact formη such thatω=η⊗σL satisfies to the conditionηW ∧dηW ≡0 (this condition not depending on the local trivializationσL).
(ii)The restriction πW :W0→M0 of πW toW0 is a d-fold covering [The integerd will be called theweight of the web].
(iii)The analytical setΓW has complex dimension at mostn−1, or is empty.
(iv)For any m∈M, the set(πW)−1(m) =W ∩(π)−1(m) is an algebraic subset of degreed and dimension 0 inPTm∗(M).
The analytical setΓW is called the critical setof the web, its projection π(ΓW) the causticor the singular part, and is complementary partM0 the regular partof the web.
1This condition is automatic whenM is compact, because of (ii) and (iii).
Remark 3.2
(i)Anyd-web W on M induces ad-webW|U =W∩π−1(U)on any open set U ofM.
(ii) In this definition, we do not require for d ≥ n that any n-uple of points in a same fiber of W0 →M0 be in general position : it may exist some point m ∈M0 and some n-uple(me1,· · ·,men) of points in (πW)−1(m) which are in a same (n−2)-dimensional sub-vector space of the (n−1)- dimensional vector space Gn−1M
m.
Definition 3.3 The irreducible components C of the variety W in Mfare called the components of the web. They all are webs. The web is said to be irreducibleif it has only one component. It is said to besmoothif W is smooth(W′ =W, which implies that it is irreducible), and quasi-smoothif any irreducible component is smooth.
3.1 The foliation F e
The integrability condition (i) of the definition of a web implies that the distribution of vector fields on W′ belonging to the kernel of ωW : T W′ → L is involutive: the integral submanifolds of this distribution defines therefore a foliationFe on W′. This foliation may have singularities, but not on W0: in fact, the restrictionsexi ofxito W0(1≤i≤n) define local coordinates on every sheet of the coveringW0 →M0. With respect to these coordinates,ηW0 is written dexn−Pn−1
α=1pα(x)e dexα and does not vanish.
By projectionπW of the restriction of this foliation toW0, we get locally, near any pointm∈M0, d distinct one-codimensional regular foliations Fi, and Fe may be understood as a “decrossing” of thesedfoliations.
Definition 3.4 We call leaf of the webany hypersurface inM whose intersection withM0 is locally a leaf of one of the local foliationsFi.
Conversely, givenddistinct regular foliationsFi(1≤i≤d) of codimension one on some open set U ofM, assume moreover that these foliations are mutually transverse to each over at any point ofU, and that there exists holomorphic coordinates (x1,· · · , xn) onU such that ∂x∂
n be tangent to none of the foliationsFei: we may defineFiby an integrable non-vanishing 1-formηi=dxn−Pn−1
α=1piα(x)dxα. The sub-manifoldUi ofMfdefined by then−1 equationspα=piα(x) (1≤α≤n−1) does not depend on the choice of the local coordinates. The unionWU =`d
i=1Uiis then ad-fold trivial covering space of U, and defines a d-web (everywhere regular, with no critical set). Denoting by πi : Ui →U the restriction ofπto the sheetUi, the form (πi)∗(ηi) is equal to the restriction ofdxen−Pn−1
α=1pα(x)e dexα
toUi.
3.2 Dicriticity :
SinceT W0is naturally isomorphic to (πW)−1(T M0), the natural injectionT →(π)−1(T M) defines an obvious injective mapT |W0 →T W0, and the image of this map is the tangent bundle to the foliation F|eW0 since it is annihilated byωW : the foliationFeis non-singular onW0.
Let now C be a irreducible component of W and C′ its regular part. In general, Fe will have singularities onC′.
Definition 3.5 We shall say that the web is dicritical onC if the restriction of the foliationFe toC′ is non-singular. We shall say that the web isdicriticalif it is dicritical on every irreducible component.
The dicriticity on C is equivalent to the possibility of extending T |W0 → T W0 as an injective morphismT |C′ →T C′. We shall give below a practical criterium for that, in the case of LCI-webs.
We shall see also that any totally linearizable web on a manifoldM (for example any linear web onPn) is dicritical.
3.3 Irreductibility of webs and indistinguishability of the foliations :
The results of this subsection will not be used below.
Let’s consider
- the numberN1 (≥1) of irreducible components ofW, - the numberN2 (≥1) of connected components of W0,
- the numberN3(≥0) of global distinct foliations onM0whose leaves coincide locally near a point with a leaf of one of thedlocal foliationsFi near this point.
It is clear that N3≤N2. [We may haveN3 < N2 ; consider for instance, the 2-web onM =P2 whose leaves are the tangent lines to a given proper conicX. Then, the caustic isX, and the 2-fold coveringW0→M0is not trivial ; however,N3= 0, whileN1=N2= 1].
In particular, whenM0 is connected, thed-foliationsFi are globally distinguishable (i.e. N3=d) iff thed-fold coveringW0→M0is trivial (i.e. N2=d).
On the other hand, N2 ≥ N1, and we can prove in fact that N2 = N1 if W is compact and quasi-smooth. (see [CL1] in the casen= 2).
4 Global first order homogeneous polynomial partial differen- tial equations
In view of the the next section where we shall study the CI-webs whose solutions satisfy ton−1 distinct global first order homogeneous polynomial partial differential equations, we want first to precise in this section what is a global first order homogeneous polynomial PDE (sometimes, we shall say in short PDE in the sequel of this paper).
Above some open set U of Cn, a local first order polynomial PDE homogeneous of degreed is a differential operatorD which is a holomorphic section of Sd(T U) (the dth-symetric power of the tangent spaceT U), which can always be written
D= X
|I|=d
AI(x)∂I,
whereI= (i1,· · · , in,) denotes a multi-index of non-negative integers,|I|=i1+· · ·+in, the coefficients AI are holomorphic functions ofx, and∂I denotes the symetric productQ
λ
∂
∂xλ
iλ
.
Solutions of this PDE are hypersurfaces Σ of equationf(x1,· · ·, xn) = 0 inU, such thatDf= 0, i.e.:
X
|I|=d
AI(x).∂If = 0.
For avoiding extra-solutions defined by the vanishing of a common factor to all of the coefficientsAI, we shall also assume that, at any point ofU, the germs of theAI’s are relatively prime.
In particular, if Σ is a graph-hypersurface of equationϕ(x1,· · ·, xn−1)−xn = 0,ϕis solution of the equation
F
x1,· · · , xn−1, ϕ(x1,· · ·, xn−1), ∂ϕ
∂x1
, . . . , ∂ϕ
∂xn−1
= 0,
where F : Ue → C is the holomorphic function of 2n−1 variables (x1, . . . , xn−1, xn, p1, . . . , pn−1) defined as
F(x, p) = X
|I|=d
(−i)in AI(x)pIb,
where Ib= (i1,· · ·, in−1) will be written shortlyIb=I\ {in}, andpIb= (p1)i1. . .(pn−1)in−1. Setting
Fe(x, u1,· · ·, un−1, un) = (−un)dF(x,−uu1
n,· · · ,−uun−1
n ),
= P
|I|=dAI(x).uI, whereuI =Q
λ(uλ)iλ, we still can write
F(x, p) =F(x, pe 1,· · · , pn−1,−1).
Let S be the hypersurface of equation F(x, p) = 0 in Ue. We shall assume that this equation is reduced. Hence, ifG(x, p) = 0 is another equation ofS, obtained from another differential operator D1, there is a unique unitρ:U →C∗, such thatD1=ρ D. The two equationsDf = 0 andD1f = 0 having the same solutions, we are mainly interested by the surfaceS.
If we get G and Ge from G associated to D1 = ρ D when using other local coordinates (x′) on another open setU′, we must haveG(xe ′, u′) =ρ(x)F(x, u) abovee U∩U′. Since uu′nn =∆∆Tn−M1(x,x′)
n(x,x′) after proposition 2-5, we get:
Lemma 4.1 There exists some unitρ:U∩U′→C∗, uniquely defined, such that G(x′, p′) =
∆Tn−1(x, x′)
∆Mn (x, x′) −d
. ρ(x) . F(x, p).
We shall therefore define a global first order homogeneous polynomial PDE of degree d on a holomorphicn-dimensional manifoldM by the data of a family (Ua, Da) such that:
- theUa’s are open sets inM making a covering ofM,
- for anya,Da is a local homogeneous polynomial PDE’s of degreedoverUa,
- for any pair (a, b) such that Ua ∩Ub be not empty, there exists a (necessarily unique) unit ρab:Ua∩Ub→C∗such thatDb=ρabDa.
Because of their uniqueness, the functionsρab satisfy to the cocycle condition, and are therefore a system of transition functions for a holomorphic line-bundleE overM. Hence, the sectionsDa glue together to define a holomorphic sectionD of the vector bundleE⊗Sd(T M) overM.
Let (axbe a system of holomorphic local coordinates onUa. From the previous lemma, we deduce that the corresponding functionsFa adFb are related onUea∩Ueb by the formula
Fb =
∆Tn−1(ax,bx)
∆Mn (ax,bx) −d
. ρab. Fa.
Hence, the functionsFa glue together, defining a sectionsof the bundleπ−1(E)⊗ Ld overMf. This justifies the following
Definition 4.2 A holomorphic line-bundle E on M and an integerd≥1 being given, a global first order polynomial PDE homogeneous of degreedand typeEon the holomorphicn-dimensional manifold M is a holomorphic sectionD ofE⊗Sd(T M), the corresponding local equationsF= 0being reduced, and having coefficients AI(x) not depending2 on the coordinates p, and having germs at any point relatively prime(these conditions not depending on the local holomorphic coordinates on M, and on a local holomorphic trivializationσE of E). The integer dis also called the weightof the PDE.
Remark 4.3 It is equivalent to giveS, or(up to multiplication by a global unit onM, i.e. a constant ifM is compact)the section s, or the differential operatorD with coefficients inE.
2Notice that any section ofπ−1(E)⊗ LdonUemay be written locallyP
|I|=d AI(x, p)pI (σL)d⊗π−1(σE), once given a local trivializationσE ofE and local coordinates, with the conventionpn=−1. Their coefficientsAI depend in general onp, but here they don’t.
4.1 Linearizability and co-critical set of a polynomial PDE
Let (D, s, S) be a global polynomial PDE on M as above.
Definition 4.4 This PDE is said to be linearizable if it satisfies to the following assumption : For any point me0∈S above a pointm0∈M, there exists local coordinates x= (x1,· · · , xn)near m0, such that if me0 has coordinates (x0, p0) = (x01,· · ·, x0n, p01,· · ·, p0n−1), then the hypersurface of equation
xn−x0n=X
α
p0α(xα−x0α) is a solution of the PDE.
Definition 4.5 We call co-critical set of the previous PDE the analytical set ∆S of points me ∈ S where either S is singular, or Tme (seen as a subspace ofTmeMf) is included intoTmeS. Locally,∆S is defined by the equationsF = 0and ∂x∂F
α +pα ∂F
∂xn = 0 for anyα.
Proposition 4.6 Assume a global polynomial PDE to be linearizable. Then its co-critical set∆S is all of S. This condition may be written locally, relatively to any system of local coordinates, in the following way:
For anyα, there exists a(local)holomorphic function Cα such that
∂F
∂xα
+pα
∂F
∂xn
=Cα F.
Proof: Assume the web on M to be locally defined by the equation F(x, p) = 0 and let (x0, p0) = (x01,· · ·, x0n, p01,· · · , p0n−1) be such thatF(xo, po) = 0. If the hypersurface of equation
xn−x0n=X
α
p0α(xα−x0α) is a solution of the PDE, then :
F
x1,· · ·, xn−1 , x0n+X
α
p0α(xα−x0α), p01,· · · , p0n−1
≡0.
By derivation of this identity, with respect to eachxα, we get : ∂x∂Fα+pα ∂F
∂xn = 0 at the point (x0, p0).
Hence
∂F
∂xα
+pα
∂F
∂xn
≡0 on S for anyα.
But this means precisely that the vector fields Xα = ∂x∂α +pα ∂
∂xn, which generate T locally are tangent toS at any point ofS. Since this property has an intrinsic meaning, not depending on the local coordinates, the same equations will be satisfied when written relatively to any other system of local coordinates.
QED Proposition 4.7 Let (D, s, S) be a global polynomial PDE on M as above. Let ∇ be any con- nection on π−1(E)⊗ Ld, and Ds : T → π−1(E)⊗ Ld the restriction of the covariant derivative
∇s : TMf → π−1(E) ⊗ Ld to the sub-bundle T of TMf. Then, the morphism Ds|S : T |S → π−1(E)⊗ Ld|S does not depend on the connection ∇, and the cocritical set of the PDE is exactly the set of pointsme ∈S whereDs|S vanishes.
Proof : The sectionsmay be written locally F.σfor a convenient trivializationσ of the line-bundle π−1(E)⊗ Ld, hence∇s=dF σ+F ∇σ. This formula proves already that the restriction∇s|S of∇s toS (whereF vanishes) does not depend on ∇. Moreover, since T is generated by the vector fields Xα=∂x∂α +pα ∂
∂xn, the condition ∂x∂Fα +pα ∂F
∂xn = 0 for anyαorDs|S = 0 are equivalent.
QED
5 Complete intersection webs
These webs are those whose leaves are the solutions of a system of n−1 global polynomial partial differential equations: W =Tn−1
α=1Sα.
Definition 5.1 A d-web W on M is said to be a complete intersection web(CI-web) if there exists a family (Sα)α of n−1 global polynomial PDE’s, of respective type Eα and weight dα, such that d =Q
αdα, W =Tn−1
α=1Sα, the sheaf of ideals of germs of holomorphic functions vanishing on W being generated by the germs of functions vanishing on one of theSα’s.
If sα is a holomorphic section of π−1(Eα)⊗ Ldα, such that Sα = (sα)−1(0), sα may be locally writtensα=Fα. σα, whereσαdenotes a local trivialization ofEα(holomorphic section nowhere zero), Fα denoting a holomorphic function of the shape Fα(x, p) = P
|I|=dαAαI(x)pI (with the convention pn=−1)), the germs of the coefficientsAαI(x) being relatively primes, and the equationFα= 0 being reduced. Thus the ideal of germs of functions vanishing onW is generated by theFα’s.
Definition 5.2 Ad-webW onM is said to be a locally complete intersection web(LCI-web)if there exists a covering ofM by open sets Uλ such that every induced web W|Uλ is a CI-web.
Remark 5.3 The property for a web to be CI (resp. LCI) is stronger than the property fo the underlying analytical space W to be CI (resp. LCI) in Mf. For instance, given d distinct points (pi, qi) on a proper conic in the (p, q)-plane, letFibe the the regular foliation on C3(with coordinates (x, y, z)) defined by the 1-formdz−pidx−qidy. Ifdis odd, the regulard-web C3defined by thesed foliations cannot be a CI-web, not even a LCI-web, while the corresponding manifoldW is LCI since it has no singularity.
5.1 Critical scheme of a CI web :
In view of the study below of the dicriticity, we shall be interested in defining ΓW not only as an analytical set, but as a scheme. This will be done now in the case of a CI web.
If a LCI-webW is locally defined by then−1 equationsFα(x, p) = 0, (1≤α≤n−1), the points of W′ are those at which the jacobian matrix D(F1 , ... , Fn−1 )
D(x1,...,xn,p1,...,pn−1) has rank n−1, andW0 is the subset ofW′ where the determinant ∆pof D(FD(p11, ... , F, ... , pn−1)
n−1) does not vanish.
If we change local coordinates and trivializations of theEα’s, we get for anyα, after lemma 4-1 : Gα
x′,−
P
βKβ1 pβ−Kn1 P
βKβn pβ−Knn,· · ·,− P
βKβn−1 pβ−Knn−1 P
βKβn pβ−Knn
=
∆Tn−1(x, x′)
∆Mn (x, x′) −dα
. ρα(x). Fα(x, p).
Denoting by ∆′p′ the determinant of D(GD(p1′ , ... , Gn−1)
1 , ... , p′n−1), and byv the determinant of D(pD(p′11 , ... , p, ... , p′n−1)
n−1), we deduce the equality
∆p=Y
α
ρα
−1
.
∆Tn−1(x, x′)
∆Mn (x, x′) −P
αdα
. v .∆′p′ + terms vanishing onW.
Hence, for a CI web, the ∆p’s glue together aboveW, defining a sectionsΓ overW of the bundle π−1 ⊗αEα
⊗ L
Pα dα
⊗
n−1^ V∗, the zero set of which is ΓW.
While the system (Fα = 0) of local equations forW is reduced, we cannot at all assert that the system (Fα= 0 , ∆p= 0) of local equations for ΓW has the same property.
Definition 5.4 We call critical schemethe scheme defined by the vanishing of the above section sΓ.
5.2 Dicriticity of a CI web :
Still assuming the CI-webW locally defined by the n−1 equationsFα(x, p) = 0, we first want to precise the natural morphismT |W0 →T W0defining FeonW0 : hence, we are looking for the matrix ((Rβα)) such that, for anyα, the vector field ∂x∂
α +pα ∂
∂xn+P
βRβα∂p∂
β be tangent toW0. The Rβα’s are then given by the solution of the linear system
X
β
Rαβ∂Fγ
∂pβ
=−∂Fγ
∂xα
+pα
∂Fγ
∂xn
, for anyα, γ.
This can still be written :
D(F1 , . . . , Fn−1) D(p1 , . . . , pn−1) ◦
tR=−Θ, where Θ denotes the matrix with coefficients Θγα=
∂Fγ
∂xα +pα∂Fγ
∂xn
, hence the solution
tR=−L◦Θ, whereL= ∆p.
D(F1 , . . . , Fn−1) D(p1, . . . , pn−1)
−1
, the matrix D(FD(p11 , ... , F, ... , pn−1)
n−1) being invertible onW0.
Assuming for the moment W to be irreducible, the only case where it is possible to extend T |W0 →T W0 as an injective morphismT |W′ →T W′ is the case where all coefficients of the matrix L◦Θ contain ∆pas a common factor, i.e. vanish on the critical scheme. In the general case, the same can be done for any irreducible component. We thus get the
Proposition 5.5 For a LCI-web W to be dicritical (resp. dicritical on an irreducible component C of W), it is necessary and sufficient that the matrixL◦Θvanishes on the critical scheme (resp. on the critical scheme ofC).
Remark 5.6 We shall see below that, in case of a linear web on the complex projective spacePn, the matrixΘ itself vanishes on the critical scheme : such a property will be called hyper-dicriticity. In fact, in this case,sΓ can be defined on all ofPfn,Θvanishes on all ofW, on the condition to use only affine coordinates as local coordinates.
Definition 5.7 A linearizable web on M is a web locally isomorphic to a linear web : this means that there exists local holomorphic coordinates near any point ofM0, with respect to which the leaves of the web have an affine equation. If there exists local holomorphic coordinates near any point of M (not only ofM0), with respect to which the leaves of the web have an affine equation, we shall say that the web is totally linearizable.
Proposition 5.8
Any totally linearizable web on a manifoldM is dicritical.
Proof: Assume a web on M to be locally defined by the equationsFα(x, p) = 0, (1 ≤α≤ n−1), and let (x0, p0) = (x01,· · ·, x0n, p01,· · ·, p0n−1) be such that, for anyα,Fα(xo, po) = 0. If the leaves of the web have an affine equation with respect to the local coordinatesx, the hypersurface of equation xn−x0n=P
αp0α(xα−x0α) is a leaf, so that, for anyα, Fα(x1,· · · , xn−1, x0n+X
α
p0α(xα−x0α), p01,· · ·, p0n−1)≡0.
By derivation of this identity with respect to each xβ, we get Θβα = 0 at the point (x0, p0), hence Θ≡0; a fortioriL◦Θ≡0. Since, on the critical scheme, this equation does not depend on the local coordinates, the web is dicritical as far as the local coordinates are available on ΓW, i.e. if the web is hyperlinearizable.
QED
6 Webs on the complex projective space
Pn.
6.1 The manifold e
Pnof contact elements :
Denote by (X0,· · ·, Xn) the homogeneous coordinates on Pn, and (u0,· · · , un) the homogeneous coordinates on the dual projective spaceP′
n of the projective hyperplanes in Pn : the hyperplane of coordinates (u0,· · ·, un) is the hyperplane of equationu0X0+u1X1+· · ·+unXn= 0 inPn.
Lemma 6.1 The manifold ePn is naturally bi-holomorphic to the submanifold of points [X0,· · · , Xn],[u0,· · ·, un]
in Pn×P′
n such that u0X0+u1X1+· · ·+unXn = 0 : a contact ele- ment is a pair (m, h) made of a pointm ∈Pn and of a hyperplane h∈P′
n going through this point (m ∈ h). According to this identification, the projection π becomes the restriction to Pfn of the first projection of Pn×P′
n.
Proof : We identify obviously the pair (m, h) (m∈h) to the sub vector-spaceTmhofTmPn(which is a point inGn−1Pn).
Remark 6.2
(i) If m has affine coordinates (xλ = XXλ0 on the open set X0 6= 0, and h has for equation xn −Pn−1
α=1pαxα = 0 with respect to these coordinates, we identify (xλ)λ,(pα)α
to the point [X0,· · · , Xn],[u0,· · ·, un]
withX0= 1,Xλ=xλ,u0=xn−Pn−1
α=1pαxα,uα=pα andun=−1.
(ii)Conversely, we identify the point [X0,· · ·, Xn],[u0,· · · , un]
in the open setX0.un6= 0ofPn×P′n, such thatu0X0+u1X1+· · ·+unXn= 0, with the point of coordinates xλ= XXλ0, pα=−uuα
n
.
The spaces Pn et P′
n play the same role, so that ePn = eP′
n : the second projection π′ : Pfn → P′
n is also a fiber space with fiber Pn−1, with which we can do the same constructions as withπ. LetV′,T′ andL′ be the bundles built fromπ′ in the same way asV,T and L are built from π. Writting pn = xn −P
αpαxα, notice that, the 1-form dxn −P
αpαdxα may be written dpn−P
αxαdpαon the open setX0.wn6= 0 ofPen, with respect to the coordinates (−xα, pn, pα). The tautological contact formω is then the same when Pfn is seen as the manifold of contact elements of f
Pn or ofPf′
n. Therefore, when a n-dimensional subvarietyW ofPfn is simutaneously a web onPn and onP′
n (“bi-web”), the foliationFe on the regular part ofW is the same for both webs.
Denote respectively byO(−1) and O′(−1) the tautological line bundles ofPn andP′n, O(1) and O′(1) their dual bundle. Setℓ=π−1 O(−1)
and ℓ′=π′−1 O′(−1)
. More generally,ℓk (resp. ℓ−k or ˘lk) will denote, for any integerk≥0,thekthtensor power ofℓ(resp. of its dual bundle ˘ℓ), and the same goes for (ℓ′)k and (ℓ′)−k= (˘l′)k.
Lemma 6.3
(i)The line bundleL is isomorphic toℓ˘⊗ℓ˘′, which is also the normal bundleNPfn ofPfn inPn×P′
n. (ii)The line bundleπ−1Vn
TPn
is isomorphic to(n+ 1)˘ℓ.
(iii)The sub-bundles V andT′ of TePn the same goes , as well as V′ andT. Proof :
On the open set (X0 6= 0)∩(Xn 6= 0) of Pn, we get p′α = x −pα
n−P
αpαxα by the change of affine coordinates (x′α= xxα
n, x′n= x1
n), hence the formula dxn−X
α
pα xα=−xn xn−X
α
pα xα
dx′n−X
α
p′αx′α .
