LOGARITHMIC FORMS AND SINGULAR PROJECTIVE FOLIATIONS

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ANNALES DE

L’INSTITUT FOURIER

LesAnnales de l’institut Fouriersont membres du

Javier Gargiulo Acea

Logarithmic forms and singular projective foliations Tome 70, n^o1 (2020), p. 171-203.

<http://aif.centre-mersenne.org/item/AIF_2020__70_1_171_0>

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

http://creativecommons.org/licenses/by-nd/3.0/fr/

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LOGARITHMIC FORMS AND SINGULAR PROJECTIVE FOLIATIONS

by Javier GARGIULO ACEA (*)

Abstract. —In this article we study polynomial logarithmicq-forms on a projective space and characterize those that define singular foliations of codimension q. Our main result is the algebraic proof of their infinitesimal stability whenq= 2 with some extra degree assumptions. We determine new irreducible components of the moduli space of codimension two singular projective foliations of any degree, and we show that they are generically reduced in their natural scheme structure.

Our method is based on an explicit description of the Zariski tangent space of the corresponding moduli space at a given generic logarithmic form. Furthermore, we lay the groundwork for an extension of our stability results to the general case q>2.

Résumé. —Dans cet article nous étudions desq-formes logaritmiques polyno- miales sur un espace projectif et nous caractérisons celles qui définissent des feuilletages singuliers de codimensionq. Notre principal résultat est la preuve algébrique de leur stabilité infinitésimale lorsqueq= 2 avec quelques hypothèses supplémen- taires sur leurs degrés. Nous donnons des nouvelles composantes irréductibles des espaces de modules des feuilletages projectifs de codimension deux et de degré quel- conque, et nous montrons que ces composantes sont génériquement réduites selon leur structure naturelle de schéma. Notre méthode est basée sur le calcul explicite de l’espace tangent de Zariski de l’espace de modules en une forme logarithmique générique. Nous posons aussi les bases pour l’extension de nos résultats de stabilité au cas généralq>2

1. Introduction

This article is concerned with the study of complex projective logarithmic forms of arbitrary degrees in the setting of the theory of algebraic foliations.

The problem which motivates this paper is the analysis of the irreducible components of moduli spaces of algebraic singular projective foliations, and the description of their geometry. For some classical results on this

Keywords:logarithmic forms, singular projective foliations, moduli spaces.

2020Mathematics Subject Classification:14D20, 37F75, 14B10, 32S65.

(*) The author was fully supported by CONICET, Argentina.

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classification problem for codimension one foliations we refer to e.g. [1], [2]

and [11]. More in general, we recommend [6] and [7] for some remarkable facts about the higher codimensional case.

Fix q∈N, n, d∈N>q and m∈N>q+1. Also, setd= (d1, . . . , dm) such thatd=Pm

i=1di. A polynomial logarithmicq-form of typedis a projective twisted differential form

(1.1) ω= X

I⊂{1,...,m}

|I|=q

λIFˆIdFi₁∧ · · · ∧dFi_q,

where λ ∈ Vq

C^m satisfies that its interior product by d vanishes (i.e., id(λ) = 0∈Vq−1

C^m),F1, . . . , Fm are homogeneous polynomials inn+ 1 variables of degreesd1, . . . , dmand ˆFI =Q

j /∈IFjfor everyI⊂ {1, . . . , m}.

If we assume thatλis totally decomposable (i.e.,λ=λ¹∧ · · · ∧λ^q), then ω is an element ofH⁰(Pⁿ,Ω^q_Pn(d)) that also satisfies the Plücker’s decomposability equation (2.1) and the so called Frobenius’s integrability equation (2.2) (see Section 2.1 for more details). These conditions ensure that the distribution associated to its kernel at each point determines a singular complex foliation of codimensionqonPⁿ.

In addition, the forms as in (1.1) are related to the classical sheaf of logarithmic forms in the following sense. With the notation above, if we set DF as the divisor defined by the zero locus ofF=Qm

i=1Fi, and assume it has simple normal crossings, then all the elementsη∈H⁰(Pⁿ,Ω^q

Pⁿ(logDF)) can be described in homogeneous coordinates by the formula

η= ω

F = X

I⊂{1,...,m}

|I|=q

λI

dFi₁

F_i₁ ∧ · · · ∧dFi_q

F_i_q .

Remember that the sheaf Ω^•_Pn(logDF) is defined by all the rational forms ηsuch thatη and dη have at most simple poles alongDF. In other words, the global sections of the sheaf of logarithmic forms over D_F are in 1- 1 correspondence with the vectors λ ∈ Vq

C^m such that id(λ) = 0. See Proposition 2.6 for more details.

On the other hand, denote by Fq(d,Pⁿ)⊂PH⁰(Pⁿ,Ω^q_Pn(d)) the moduli space of algebraic singular projective foliations of degree don Pⁿ, which corresponds to the algebraic space of twisted q-forms of total degree d satisfying the announced equations (2.1) and (2.2).

We consider ρ as a rational map parametrizing those logarithmic q- forms of type d that define foliations (see Definition 2.21), and we set L_q(d, n) ⊂ F_q(d,Pⁿ) as the Zariski closure of its image. We refer to this projective irreducible variety as the logarithmic variety associated to the

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partition d. Our purpose is to show that these logarithmic varieties are irreducible components of the corresponding moduli spaces of foliations.

In this paper, we solve this problem for q = 2 and with some condition among the vector of degreesd(see the 2-balanced assumption from Def- inition 3.18). Furthermore, we expect that, up to some technical details, our results and proofs can be adapted for anyq6n−2 at least assuming an extended q-balanced condition. When q = n−1 there is no hope to establish a stability result even for a generic logarithmic form. In this case the corresponding moduli space coincides withPH⁰(Pⁿ,TPⁿ).

In order to put our work in context, we shall explain some related articles.

It was proved in [1] by Omegar Calvo Andrade that the logarithmic varieties L₁(d, n) are irreducible components ofF₁(d,Pⁿ), and so this type of forms satisfy a sort of stability condition among all the integrable forms (see integrability condition (2.1) forq = 1). Actually, his article is essentially based on analytical and topological methods, and the main results are proved for logarithmic foliations on a general complex manifoldM with the assumptionH¹(M,C) = 0. Later, in [5] it was developed a new proof of the stability result for polynomial logarithmic one-forms in projective spaces.

Here the methods are completely algebraic and provide further information about the irreducible componentsL1(d, n). For example, it is deduced that the schemeF1(d,Pⁿ) is generically reduced along these components.

Moreover, in [7] the authors proved that the varieties Lq(d, n) are irreducible components whenm=q+ 1 (lower possible value for m). In this case, the corresponding differentialq-forms determine projective foliations which are tangent to the fibers of quasi-homogeneous rational maps. From now, we refer to this type of forms as rationalq-forms of typed. It is also remarkable that there are not many known irreducible components of the moduli space Fq(d,Pⁿ) for general q. See for instance the introductions of [6] and [7].

In some sense, this article is concerned with obtaining a common gen- eralization of the definitions and results from [1] and [7], using the same algebraic methods as in [5]. Notice that formula (1.1) coincides with that given for a general rationalq-form of typed in [7] whenq =m−1, and with that given for a polynomial logarithmic 1-form in [1] whenq= 1.

The article is organized as follows. In Section 2 we present a brief sum- mary of the objects involved in the definitions of Fq(d,Pⁿ), Lq(d, n) and the corresponding parametrization map, with a particular attention to the caseq= 2. We suggest to consult Definitions 2.12 and 2.21, and also Propo- sitions 2.14 and 2.17 that contain our definition of logarithmic q-forms of

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typed, and a characterization of those that determine singular projective foliations of codimension q.

Since our methods are based on Zariski tangent space calculations, in Section 2 we also present a description of these spaces forF2(d,Pⁿ) and the space of parametersP₂(d) of the natural parametrizationρ. In addition, a formula for the derivative of the parametrization map is given at the end of the section. Our main tool is based on the comparison of the image of this derivative with the tangent space ofF2(d,Pⁿ) at a generic logarithmic form of typed. See our Subsections 2.3 and 2.4 for a complete treatment of these tasks.

Our Section 3 contains a development of the main result: Theorem 3.1, which establishes that the logarithmic varieties L₂(d,Pⁿ) are irreducible components of the moduli space F2(d,Pⁿ). In addition, we deduce that these components are generically reduced according to its induced scheme structure. Our result assumes thatm, n >3 and that the vector of degrees dis 2-balanced, that is

di+dj < X

k6=i,6=j

dk ∀i, j∈ {1, . . . , m}.

The formal statement of the theorem is the following:

Theorem. — Fix natural numbersn, m∈N>3, and a 2-balanced vector of degrees d = (d1, . . . , dm) with d = P

idi. The variety L2(d, n) is an irreducible component of the moduli space F2(d,Pⁿ). Furthermore, the schemeF2(d,Pⁿ)is generically reduced along this component, in particular at the points ofρ(U2(d)).

The proof of Theorem 3.1 is supported on Proposition 3.2 (surjectivity of the derivative of the natural parametrization). This method had been originally used in [6] and [7], and was also applied in [5]. However, the proof of this proposition is quite technical, and it is obtained through various steps and lemmas, including certain results of independent interest. See for instance Steps 1 to 7.

Finally, we would like to observe the followings facts about the general caseq >2. The definition of our natural parametrization ρ, the varieties Lq(d, n) and some of our lemmas are still valid for larger q (see Subsec- tion 3.2.1). However, most of the steps of the proof of Proposition 3.2 are quite technical, and the combinatorics behind them is hard to extend for q >2. See also Remark 3.28 for more details.

Acknowledgements.The author is grateful with Fernando Cukierman for suggesting the problem and for his further valuable help. Also gratitude

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is due to Federico Quallbrunn, Cesar Massri and the anonymous Referee for their useful comments and corrections.

The content of this paper is part of the author’s doctoral thesis, for the degree of Doctor de la Universidad de Buenos Aires.

Remark 1.1. — After completing and having posted this article we learned about the article [3] by D. Cerveau and A. Lins Neto. Some of our results for logarithmicq-forms turn out to be contained in this paper, but the methods of proof are completely different. In particular, the authors obtain a stability result for the general case 1 6 q 6 n−2, after reducing the problem to the case of foliations of dimension two (see [3, Theorems 5 and 6]). We would like to observe that our methods, based on computing the Zariski tangent space at a generic point, are completely algebraic and provide further information in the particular case of logarithmic foliations of codimension two. Especially, the fact that the moduli space of singular projective foliations results generically reduced along the logarithmic irreducible components that we obtained.

2. Definitions and constructions

In this article we shall use the following basic notation concerning polynomials and forms.

S⁽ⁿ⁾ = C[x0, . . . , xn] for the graded ring of polynomials with complex coefficients inn+ 1 variables. Whennis understood we writeS⁽ⁿ⁾=S.

S_d⁽ⁿ⁾ = H⁰(Pⁿ,O_Pⁿ(d)) for the space of homogeneous polynomials of degreedinn+ 1 variables. Whennis understood we denoteS_d⁽ⁿ⁾=S_d.

Ω^q_X(L) = Ω^q_X⊗ O(L) for the sheaf of differential forms twisted by a line bundleLonX.

iv : Ω^•_X → Ω^•−1_X for the interior product or contraction with a vector fieldvon a varietyX. In addition, ifm, q∈N,d= (d₁, . . . , dm)∈C^mand λ∈Vq

C^m, we also denote byid(λ)∈Vq−1

C^m the interior product ofλ byd, that is

i_(d₁_,...,d_m₎ X

I:|I|=q

λIei₁∧ · · · ∧ei_q

!

= X

I:|I|=q q

X

j=1

λI(−1)^j+1di_jei₁∧ · · · ∧ˆei_j∧ · · · ∧ei_q, for{e1, . . . , em} the canonical base ofC^m.

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2.1. Singular projective foliations and logarithmic forms

We writeFq(d,Pⁿ) for the moduli space of singular projective foliations of codimensionqand degreed. Recall from [13] (or see also [7]) that this space is naturally described by projective classes of twisted projective formsω∈ H⁰(Pⁿ,Ω^q

Pⁿ(d)) which satisfy both the Plücker’s decomposability condition

(2.1) iv(ω)∧ω= 0 ∀v∈

q−1

^ Cⁿ⁺¹

and the so called Frobenius’s integrability condition (2.2) iv(ω)∧dω= 0 ∀v∈

q−1

^ Cⁿ⁺¹.

The elements like v can be considered as local frames on the affine cone overPⁿ, or alternatively as rational multi-vector fields.

The first equation ensures that ω is locally decomposable outside its singular set, so this section of Ω^q_Pn(d) belongs to the corresponding Grass- mannian space at that points. The second equation is a condition to guar- antee that the singular distribution associated to the kernel of ω is also integrable. See for instance [13, Proposition 1.2.2]. We also recall that the singular set of the foliation induced byωcorresponds to its vanishing points Sω={p∈Pⁿ:ω(p) = 0}.

Remark 2.1. — Each elementω∈H⁰(Pⁿ,Ω^q_Pn(d)) can be represented in homogeneous coordinates by homogeneous affine q-forms of total degreed like

ω= X

I={i1...,i_q}

AI(z)dzi₁∧ · · · ∧dzi_q,

where{AI}I ⊂S_d−q are selected in order to satisfy the so calleddescend condition

(2.3) iR(ω) = 0∈H⁰(Cⁿ⁺¹,Ω^q−1

Cⁿ⁺¹).

HereRdenotes the radial Euler fieldP

izi ∂

∂z_i.

Remark 2.2. — Forq= 2 the decomposability equation (2.1) is slightly simpler than in the general case because is equivalent to i_v(ω∧ω) = 0

∀v∈Cⁿ⁺¹,so it can be replaced by

(2.4) ω∧ω= 0.

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In conclusion, we defineFq(d,Pⁿ), the algebraic space of codimension q singular foliations onPⁿ of degreed>q, as the following set

(

[ω]∈PH⁰(Pⁿ,Ω^q_Pn(d)) :ω satisfies (2.1), (2.2) and codim(Sω)>2 )

. Thus each [ω]∈ Fq(d,Pⁿ) determines a singular holomorphic foliation on Pⁿ whose leaves are of codimensionq, i.e., a regular holomorphic foliation outside the singular setSω. Moreover, the degree of the divisor of tangencies of such leaves with a generic linearly embedded P^q in Pⁿ is in this case d−q−1 (sometimes also used to refer to the degree of the corresponding foliation).

On the other hand, we want to define correct formulas for those polynomial logarithmicq-forms inH⁰(Pⁿ,Ω^q

Pⁿ(d)) that define foliations according to the above-mentionned equations.

First, we recall the principal definitions and properties of the well-known sheaf of meromorphic logarithmic forms and its classical residues in our particular case of interest. For a more extensive development of this task see [8].

Let D=Pm

i=1(Fi= 0) be a projective divisor defined by homogeneous polynomialsFi ∈Sd_i fori= 1, . . . , m. The sheaf Ω^q

Pⁿ(logD) of meromorphic logarithmicq-forms is defined as a subsheaf of the sheaf Ω^q_Pn(∗D) of meromorphic forms with arbitrary poles alongDaccording to

Ω^q_Pn(logD) ={α∈Ω^q_Pn(∗D) :αand dαhave simple poles alongD}.

IfDhas simple normal crossings, it is a well known fact that (Ω^•_Pn(logD), d),→(Ω^•_Pn(∗D), d)

determines a subcomplex and a quasi-isomorphism (see [14, Proposition 4.3]).

We consider the following usual filtration

(2.5) Wk(Ω^q_Pn(logD)) =











0 ifk <0

Ω^q−k_Pn ∧Ω^k_Pn(logD) if 06k6q Ω^q

Pⁿ(logD) ifk>q, and use the notation

• Xi= (Fi= 0) fori= 1, . . . , m,

• XI =Xi₁∩ · · · ∩Xi_k forI={i1. . . ik} ⊂ {1, . . . , m},

• D(I) =P

j6∈IX_I∩X_j as a divisor onX_I,

• jI =XI ,→Pⁿ,

• X_D^k =`

I:|I|=kX_I,

• jk=X_D^k ,→Pⁿ.

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The desired formula for polynomial logarithmic forms will be supported on the characterization of the global sections of the sheaf Ω^q

Pⁿ(logD). Our arguments will rely on the following known constructions and results re- garding this classical sheaf of forms.

Proposition 2.3. — Assume thatDhas simple normal crossings.

(1) There is a well defined residue map

ResI : Ω^•_Pn(logD)−→Ω^•_X_I(logD(I))[−k]

for each multi-indexIwith|I|=k.

(2) The residue map restricts to

ResI :Wk(Ω^•_Pn(logD)−→(jI)_∗(Ω^•_X_I[−k]),

is surjective and also well defined on the quotientGr_k^W(Ω^•_Pn(logD)).

(3) The total residue map Resk := M

I:|I|=k

ResI :Gr^W_k (Ω^•_Pn(logD))−→(jk)_∗(Ω^•_Xk D

[−k]) is an isomorphism.

Proof. — See [14, Lemma 4.6], or [8] for an extended overview.

Now we are able to compute H⁰(Pⁿ,Ω^q

Pⁿ(logD)).

Lemma 2.4. — With the notation above, assume q < min{m, n−1}.

Then the following computations hold:

(1) H⁰(Pⁿ, Wk(Ω^q_Pn(logD))) = 0 ∀k= 0, . . . , q−1, (2) H¹(Pⁿ, W_k(Ω^q

Pⁿ(logD))) = 0 ∀k= 0, . . . , q−2, (3) H¹(Pⁿ, W_q−1(Ω^q

Pⁿ(logD)))has a natural injective map toVq−1

C^m. Proof. — First, recall that the Hodge numbers associated to the classical projective space areh^p,q_Pn =δpq(Kronecker delta). Due to the normal crossings condition and the Lefschetz hyperplane theorem (see for instance [12, Section 3.1]) it is also true that

h^p,q_X

I =δ_pq ∀p, q, I:p+q < n− |I|,

for every complete intersection subvarietyXI. Now, we proceed by induction on the numberk >0. The base case is trivial. Then, the results (1) and (2) are immediate consequences of considering the long exact sequence in cohomology associated to the exact sequence

(2.6) 0−→Wk−1(Ω^q_Pn(logD))−→Wk(Ω^q_Pn(logD))−→(jk)∗(Ω^q−k_Xk D

)−→0,

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where the exactness follows from Proposition 2.3. Finally, consider the long sequence on cohomology fork=q−1, that is

0→H¹(Pⁿ, W_q−1(Ω^q_Pn))→ M

I:|I|=q−1

H¹(XI,Ω¹_X_I)

→H²(Pⁿ, Wq−2(Ω^q_Pn(logD))). . . , and use the knowledge of the Hodge numberh^1,1_X

I for eachIof sizeq−1 to

show our assertion (3).

With a similar proof and the same inductive argument, we can also state the following result.

Corollary 2.5. — If we assume that D has simple normal crossings andq <min{m, n−j}, then

H^j(Pⁿ, Wk(Ω^q_Pn(logD))) = 0 ∀k= 1, . . . , q−j−1.

Proposition 2.6. — Assume q < min{m, n−1}. Let F1, . . . , Fm be homogeneous polynomials with simple normal crossings and respective de- greesd1, . . . , dm, defying a divisorDas before. For every global sectionη∈ H⁰(Pⁿ,Ω^q

Pⁿ(logD)),there exist unique constantsλ={λ_I}_I:|I|=q ∈Vq

C^m such thatη can be written in homogeneous coordinates as

η= X

I⊂{1,...,m}

|I|=q

λ_IdF_i₁ Fi1

∧ · · · ∧ dF_i_q Fiq

,

where also λsatisfies id(λ) = 0 to ensure that iR(η) = 0. Recall that id

denotes the interior product by the constant vectord= (d₁, . . . , d_m).

Proof. — We use once more the exact sequence of sheaves (2.6) considered at the proof of Lemma 2.4, but now in the particular casek=q, and we get

0−→Wq−1(Ω^q_Pn(logD))−→Ω^q_Pn(logD)−−−→^Res^q (jq)∗(O_X^q

D)−→0.

The result follows from considering again the associated long exact sequence in cohomology, and applying the previous lemma to deduce the injectivity of the total residue map Resq on global sections. Notice that the normal crossing assumption ensures that the image of the total residue map described in homogeneous coordinates can be considered as a subset ofC(^m_q)'Vq

C^m. Hence, the residues of the formP λI

dFi1

F_i₁ ∧ · · · ∧^dF_F^iq

iq are the numbers (λ_I). In addition, the kernel of the first connection morphism of the long exact sequence is exactly determined by the descend condition

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among the previous form, that is i_d(λ) = 0. This equation characterizes

the image of Resq and implies our statement.

Remark 2.7. — In fact, last proposition can be proved for a general complex projective algebraic varietyX of dimensionnwith the assumption

h^p,0_X = dim(H⁰(X,Ω^p_X)) = 0, for 16p6n−1.

Corollary 2.8. — Select λ ∈ Vq

C^m and homogeneous polynomials Fi ∈Sd_i having simple normal crossings. ThenP

λI dFi1

F_i₁ ∧ · · · ∧^dF_F^iq

iq = 0 if and only ifλ= 0.

The last statement is a sort of extension of [11, Lemma 3.3.1] to the case of higher degree logarithmic forms, with a more restrictive hypothesis.

Corollary 2.9. — For every simple normal crossings divisorDset as before we have

H⁰(Pⁿ,Ω^q

Pⁿ(logD))' (

λ∈

q

^

C^m:id(λ) = 0 )

.

Moreover, notice that for each q ∈N<m the right set corresponds to the degreeqcycles of the Koszul complex associated to the vectord∈C^m, i.e., the exact complex

K(d) : 0→

m

^ C^{m i}−→^d

m−1

^

C^{m i}−→^d . . .−→ⁱ^d ^

C^{m i}−→^d C→0.

Definition 2.10. — Fix natural numbers d and m. A partition of d into m parts is an m-tuple of degrees d = (d₁, . . . , d_m) ∈ N^m satisfying Pm

i=1di=d.

For future convenience, we will use the following notation.

Definition 2.11. — ForFi∈Sd_i as above we denote F = (F1, . . . , Fm),

F =

m

Y

j=1

F_j, Fˆ_i=Y

j6=i

F_j, Fˆ_ij = Y

k6=i,k6=j

F_k, (i6=j),

or, more generally, forI⊂ {1, . . . , m}we writeFˆI =Q

j /∈IFj.

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Definition 2.12. — Fix d as a partition of d ∈ N>q into m parts, and assume m > q + 1. A twisted projective differential q-form ω ∈ H⁰(Pⁿ,Ω^q

Pⁿ(d)) is a polynomial logarithmic q-form of type d if it is defined by the following formula

(2.7) ω=F



 X

I:|I|=q

λ_IdFi₁

F_i₁ ∧ · · · ∧dF iq

F_i_q



= X

I:|I|=q

λ_IFˆ_IdF_I, whereF_i∈S_d_i and λ∈Vq

C^msatisfiesid(λ) = 0.

Remark 2.13. — From Corollary 2.9 we deduce that for every logarithmic q-formω as in (2.7), there existγ∈Vq+1

C^msuch thatid(γ) =λand ω=iR



 X

J:|J|=q+1

γJFˆJdFJ



.

Hence the above formula could be an alternative definition for logarithmic q-forms of typed. See also [7] to compare our formulas with the definition presented there for rationalq-forms.

At first, we want to determine when this type of forms define foliations of codimension q, i.e., when the forms as in (2.7) satisfy the equations of Fq(d,Pⁿ). For simplicity, we start with the Plücker’s decomposability condition (2.1) (or equivalently (2.4)) forq= 2.

Proposition 2.14. — If ω is a logarithmic 2-form of type d and the polynomials involved have simple normal crossings, then the following conditions are equivalent:

(1) ω∧ω= 0 (2) λ∧λ= 0.

Proof. — Notice that ω∧ω=F²



 X

i6=j6=k6=l

(λ∧λ)ijkl

dFi

Fi

∧dFj

Fj

∧dFk

Fk

∧dFl

Fl



.

The result then follows from Corollary 2.8.

Remark 2.15. — For ω as above, the condition λ∧λ= 0 implies that there existλ¹, λ² ∈C^msuch that λ=λ¹∧λ². In this case, if we consider the meromorphic logarithmic forms defined by

η_j=

m

X

i=1

(λ^j)i

dF_i Fi

j= 1,2,

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then the meromorphic logarithmic 2-form η = _F^ω = η₁ ∧η₂ is globally decomposable.

Remark 2.16. — If we consider a totally decomposableq-vectorλ=λ¹∧

· · · ∧λ^q ∈Vq

C^m withi_d(λ) = 0, then for every selection of homogeneous polynomials, the induced logarithmicq-form of type das in (2.7) satisfies the Plücker’s decomposability equation (2.1). Using the above notation, the reason is that the meromorphicq-formη= ^ω_F =η1∧ · · · ∧ηq is globally decomposable.

Let us recall that the grassmannian space Gr(q,C^m) of q-dimensional subspaces of C^m can be considered as a projective algebraic subset of P(Vq

C^m) via the Plücker embedding ι:Gr(q,C^m)−→P

q

^ C^m

!

span(λ¹, . . . , λ^q)7−→[λ¹∧ · · · ∧λ^q].

With a slight abuse of notation, we only write λ or λ¹∧ · · · ∧λ^q for the elements of Gr(q,C^m) and (λI) for its corresponding antisymmetric coordinates.

Proposition 2.17. — Fordas above,λ∈Gr(q,C^m)such thatid(λ) = 0and(Fi)^m_i=1 ∈Qm

i=1Sd_i, the logarithmic q-form of typed

ω= X

I:|I|=q

λIFˆIdFI

is a twisted projective form of total degreed, i.e., ω∈H⁰(Pⁿ,Ω^q

Pⁿ(d), and satisfies(2.1)and (2.2). Hence[ω]∈ Fq(d,Pⁿ).

Proof. — If we take into consideration Remark 2.16, then it only re- mains to prove that ω satisfies the integrability equation iv(ω)∧dω = 0, for allv ∈ Vq−1

Cⁿ⁺¹. This follows by a straight forward calculation

using that dω= ^dF_F ∧ω andiv(ω)∧ω= 0.

The next result characterizes in an useful way the condition i_d(λ) = 0 forλ∈Gr(q,C^m).

Lemma 2.18. — For λ=λ¹∧ · · · ∧λ^q ∈Vq

(C^m)− {0}, the following equations are equivalent:

(1) id(λ) = 0

(2) id(λⁱ) = 0, ∀i= 1, . . . , q.

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Proof. — The equivalence follows from i_d(λ) =

m

X

j=1

(−1)^j i_d(λ^j)λ¹∧ · · · ∧λˆ^j∧ · · · ∧λ^q,

and the fact that{λ¹∧ · · · ∧ˆλ^j∧ · · · ∧λ^q}j are linearly independent.

Corollary 2.19. — The following equality holds:

{λ∈Gr(q,C^m) :i_d(λ) = 0}=Gr(q,C^md),

where C^md denotes the linear space of vectors µ ∈ C^m such that idµ = µ·d=Pm

i=1µ_id_i= 0.

Remark 2.20. — Notice that we have not proved that ifωis a logarithmic q-form of type d as in Definition 2.12, then the moduli equations (2.1) and (2.2) are equivalent to the condition λ ∈ Gr(q,C^md), i.e., ω globally decomposable outsideD_F = (F = 0). Proposition 2.17 only states a partial answer to that problem. However, when q = 2 the equivalence is in fact true according to Proposition 2.14. It is also remarkable that the caseq >2 is an open question in [3, Problem 1].

2.2. The logarithmic varieties and the natural parametrization

Consider again natural numbers q∈ N, n∈ N>q+1, d, m ∈N>q+1 and a partition dofd into mparts. According to Proposition 2.17 we denote by l_q(d, n) the algebraic set of logarithmic q-forms of type d which are globally decomposable outside their corresponding divisorDF = (F = 0), and whose projective classes determine codimensionqfoliations. Actually, this set also coincides with the image of the multi-linear map

φ: (C^md)^q×

m

Y

i=1

Sdi −→H⁰(Pⁿ,Ω^q_Pn(d)) ((λ¹, . . . , λ^q),(F1, . . . , Fm))7−→ X

I:|I|=q

(λ¹∧ · · · ∧λ^q)IFˆIdFI.

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Definition 2.21. — We introduce the following map as our natural parametrization:

ρ:Pq(d) :=Gr(q,C^md)×

m

Y

i=1

P(Sd_i)− → Fq(d,Pⁿ)⊂PH⁰(Pⁿ,Ω^q

Pⁿ(d)) (2.8)

(λ= [(λ¹∧ · · · ∧λ^q)], F = ([F1], . . . ,[Fm]))7−→[ω] =



 X

I:|I|=q

λIFˆIdFI



.

From now on and when there is no confusion, we avoid the notation [·] for the corresponding projective classes of the elements involved in the definition ofρ.

Definition 2.22. — We define the logarithmic varietyLq(d, n)as the Zariski closure of the image ofρ, i.e.,L_q(d, n) = imρ⊂ F_q(d,Pⁿ).

Remarks 2.23.

(1) ρis only a rational map and so it is not well defined on the whole space of parametersPq(d). In particular, on the parameters which give rise to forms that vanish completely (base locus of the parametrization).

(2) The space Pq(d) is an algebraic irreducible projective variety of dimensionPm

i=1 n+d_i

di

−m+q(m−1−q).

(3) Notice that the image ofρalso coincides with Plq(d, n).

(4) The spaceLq(d, n) is a projective irreducible subvariety ofFq(d,Pⁿ).

With some conditions among d, we will prove that L₂(d, n) is an irreducible component of the space F2(d,Pⁿ). From now on, in general, we assumeq= 2.

For our prompt purposes we need to assume some generic conditions on the parameters in P2(d). The polynomials {Fi} and the constants {λij} are general according to the following definition.

Definition 2.24. — We take the non-empty algebraic open subset of P2(d)defined by

U₂(d) ={(λ, F)∈ P₂(d) : λ_ij 6= 0, λ_ij−λ_ik+λ_jk6= 0, λ_ij−λ_ik−λ_jk 6= 0 andF₁, . . . , F_mare smooth irreducible with normal crossings}.

Remark 2.25. — It follows from Corollary 2.8 thatU₂(d) does not inter- sect the base locus ofρ.

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A more detailed analysis of the base locus of ρ, including a description of its irreducible components and scheme structure will be pursued in [4].

2.3. The Zariski tangent space of F2(d,Pⁿ)

For a schemeXand a pointx∈X we denote byTxXthe Zariski tangent space ofX at x. Remember that for a classical projective space X =PV, associated to a finite dimensional vector spaceV, it is common to identify its Zariski tangent space at a given pointx=π(v) with

TxX =V/_hvi.

Also recall that F2(d,Pⁿ) is the closed subscheme of the projective space P(H⁰(Pⁿ,Ω²_Pn(d))) defined by equations (2.1) and (2.4). With a slight abuse of notation, the Zariski tangent spaceTωF2(d,Pⁿ) can be represented by the formsα∈H⁰(Pⁿ,Ω²_Pn(d))/_hωi such that

(ω+α)∧(ω+α) = 0 and

(ivω+ivω)∧(dω+dα) = 0 ∀v∈Cⁿ⁺¹, with ²= 0.

Remark 2.26. — If we fix an elementω ∈ F2(d,Pⁿ), then a simple calculation shows thatα∈H⁰(Pⁿ,Ω²_Pn(d))/_hωibelongs to the Zariski tangent space ofF2(d,Pⁿ) atω if and only if it fulfills the following equations:

α∧ω= 0 (2.9)

(ivω∧dα) + (ivα∧dω) = 0 ∀v∈Cⁿ⁺¹. (2.10)

We refer to the first equation as the decomposability perturbation equa- tionand to the second one as the integrability perturbation equation. In conclusion, we have

TωF2(d,Pⁿ) ={α∈H⁰(Pⁿ,Ω²_Pn(d))/_hωi:αsatisfies (2.9) and (2.10)}.

2.4. The derivative of the natural parametrization

One of our main purposes is to show that the derivative ofρis surjective (see for instance Section 3.1). In order to carry out this plan we need to set some notation.

As it was explained in the previous section, for everye∈Nwe have Tπ(F)P(S_e) =S_e/_hFi.

From now on we will writeF⁰ for a general element of this space.

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Moreover, let λ = [λ¹∧λ²] be an element in the grassmannian space Gr(2,C^md). The Zariski tangent space of Gr(2,C^md) at λ has a natural identification with the space of antisymmetric vectors λ⁰ ∈ V2

(C^md)/_hλi such that

(2.11) λ⁰∧λ= 0.

Even more, the vectorsλ⁰ satisfying the above condition can be written as (2.12) λ⁰ = (λ⁰)¹∧λ²+λ¹∧(λ⁰)²,

where (λ⁰)¹,(λ⁰)² ∈ C^md/_hλ1,λ²i. In general, we will write λ⁰ for the elements ofTλGr(2,C^md), i.e., for the vectorsλ⁰ ∈V2

(C^md)/_hλifulfilling equation (2.11) or equivalently of the type (2.12).

Remark 2.27. — Let (λ¹∧λ²,(Fi)^m_i=1) = (λ, F)∈ P2(d) be a fixed pa- rameter in the domain ofρand writeω=ρ((λ, F)) =P

i6=jλijFˆijdFi∧dFj. From the multilinearity ofφand the definition ofρ, we obtain the following formula for the derivative ofρ:

(2.13) dρ(λ, F) :T_λGr(2,C^md)×

m

Y

i=1

T_F_iPS_d_i −→ T_ωF₂(d,Pⁿ) (λ⁰,(F₁⁰, . . . , F_m⁰ ))7−→X

i6=j

λ⁰_ijFˆ_ijdF_i∧dF_j

+ X

i6=j6=k

λijFˆijkF_k⁰dFi∧dFj+ 2X

i6=j

λijFˆijdF_i⁰∧dFj. We will use the following notation for the forms in the image of the partial derivatives ofρ:

α1= dρ(λ, F)(λ⁰,(0, . . . ,0)) withλ⁰∈ TλGr(2,C^md), α2= dρ(λ, F)(0,(F₁⁰, . . . , F_m⁰ )) withF_i⁰∈Sd_i/_hF_i_i.

Remark 2.28. — For each (λ, F)∈ P2(d) we have an inclusion of vector spaces im(dρ(λ, F))⊂ TωF2(d,Pⁿ). In particular, the forms likeα1andα2

satisfy the perturbation equations (2.9) and (2.10).

Now, we want to distinguish the two introduced partial derivatives. With the notation of Section 2.1 (see our definitions in (2.5)), we will see they vanish on different strataX_D^k

F associated to the divisorDF = (F = 0) = Sm

i=1X_i = (F_i = 0). Under the assumptions of Definition 2.24, eachX_I is a smooth complete intersection of codimension |I|. Thus, the stratum X_D^k

F has codimensionkand is singular alongX_D^k+1

F . We have the following results concerning these spaces.

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Proposition 2.29. — For everym >1 andk∈ {1, . . . , m}, the satura- ted homogeneous idealI^(k)associated toX_D^k

F is generated by{FˆJ}_|J|=k−1. Hence we have a surjective map L

|J|=k−1O_Pn(−dˆ_J) I_Xk DF

, where I_Xk

DF

⊂ O_Pn denotes the ideal sheaf of regular functions vanishing on each stratum, anddˆJ=P

j /∈Jdj.

Proof. — We will proceed by induction on the total number of polynomi- alsm, wherem>kis required. The base case is trivial, left to the reader.

It is also clear thatI^(k)=T

J:|J|=khFj1, . . . , Fj_ki,and then we are reduced to prove that

\

J⊂{1,...,m}:|J|=k

hFj₁, . . . , Fj_ki=hFˆJiJ⊂{1,...,m}:

|J|=k−1

.

One inclusion is always clear and does not require the inductive argument.

Next, if we separate in the left term the multi-indexes of the intersection which do not containmand use the inductive hypothesis, then we obtain

(2.14) I^(k)= \

J⊂{1,...,m−1}:

|J|=k−1

hFˆ_J∪{m}i ∩ hFj₁, . . . , Fjk−1, Fmi.

As a consequence, for everyP ∈I^(k) there exists a family of polynomials {HJ}_J:|J|=k−1 such that

(2.15) P = X

J:|J|=k−1

HJFˆ_J∪{m}.

For eachJ0={j1, . . . , jk−1} ⊂ {1, . . . , m−1}the class [P] in the quotient ring C[z₀, . . . , z_n].

hFj₁, . . . , Fj_k−1, Fmi is equal to zero. This is a consequence of formula (2.14). Hence we have [H_J₀][ ˆF_J₀_∪{m}] = 0.Finally, since the ring is integral andFl ∈ hF/ j₁, . . . , Fjk−1, Fmi, for every index l /∈ J0

distinct ofm, we get [H_J₀] = 0. This fact applied to equality (2.15) allows

us to show the other needed inclusion.

Remark 2.30. — Ifω=ρ(λ, F) is a projective logarithmicq-form of type d, thenX_D^q+1

F is contained in its singular set.

Remark 2.31. — Assumem >3. With notation as in Remark 2.27, notice that α₁ vanishes on X_D³

F andα₂ onX_D⁴

F, and so every element αin the image of dρvanishes onX_D⁴_F.

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3. Infinitesimal stability of a generic logarithmic 2-form 3.1. Main results

Our desired result can be summarized in the next theorem. For some technical reasons that will be explained later, the vector of degrees d is assumed to be 2-balanced (see Definition 3.18).

Theorem 3.1. — Fix natural numbers n, m ∈N>3, and a 2-balanced vector of degreesd= (d1, . . . , dm)withd=P

idi. The variety L2(d, n)is an irreducible component of the moduli spaceF2(d,Pⁿ). Furthermore, the schemeF2(d,Pⁿ)is generically reduced along this component, in particular at the points ofρ(U2(d)).

The proof of the above theorem is implied by the surjectivity of the derivative of the natural parametrizationρ(Proposition 3.2 below), combined with some arguments of scheme theory. This method is the same as the one used to prove [7, Theorem 2.1] and [6, Theorem 1]. Moreover, it was also used in [5, Theorem 8.2] for an alternative algebraic proof of the stability of projective logarithmic one forms. As a consequence, we will only present a proof of the corresponding proposition.

Proposition 3.2. — With the notation of Theorem 3.1, let (λ, F) ∈ U2(d)andω=ρ(λ, F). Then the derivative

dρ(λ, F) :T_λGr(2,C^md)×

m

Y

i=1

T_F_iPS_d_i−→ T_ωF₂(d,Pⁿ) is surjective.

Proof. — It will be attained in the following section.

Remark 3.3. — Most of our definitions and constructions concerning logarithmic q-forms of typed= (d₁, . . . , dm) assumem>q+ 1. However, the techniques applied in the proof of Proposition 3.2 (and hence of Theo- rem 3.1) require q = 2 and m >3. Notice that the case m =q+ 1 corresponds to rational q-forms, and it is already known that they determine irreducible and generically reduced components ofFq(d,Pⁿ) (see [7]).

3.2. Surjectivity of the derivative of the natural parametrization Let us now start with several steps towards the proof of Proposition 3.2.