Leray residue theory and Barlet forms on singular complete intersections

A.G.Aleksandrov^{†}, A.K.Tsikh^{††}

We construct a complex Ω_{S}^{•}(log C) of sheaves of the so-called multi-logarithmic differential forms on a complex analytic manifold S with respect to a reduced complete intersection C ⊂ S, and define a residue map res.: Ω_{S}^{•}(log C) → ω_{C}^{•} from this complex onto the Barlet complex ω_{C}^{•} of regular meromorphic differential forms on C. The residue map is proved to be a natural morphism between the two complexes; it follows then that sections of the complex ω_{C}^{•} may be regarded as a generalization of the residue differential forms defined by Leray. Moreover, we show that the map res. can be given explicitly in terms of a certain integration current.

Content

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

§1. Multi-logarithmic differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

§2. Regular meromorphic differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

§3. The Poincaré residue map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

§4. On residues of multi-logarithmic forms and ¯∂-closed currents. . . . . . . . . . . 15

§5. Appendix. Historical remarks and comments . . . . . . . . . . . . . . . . . . . . . . . . . . 20

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

†Partially supported through the RFBR (Russian Foundation of Basic Research) grant no. 99-01-01067 and the NWO-RFBR (the Netherlands Organization for Scientific Research and RFBR) grant no. 047-008-005

††Supported through the RFBR grant no. 99-01-00790 2000 AMS Mathematics Subject Classification: Primary 32S25, 14F40, 14F10, 14J17; Sec-

ondary 32S22, 32S45, 58K65, 14F25

Key words and phrases: complete intersection; multi-logarithmic differential forms; regular meromorphic differential forms; Poincaré residue; logarithmic residue; Grothendieck duality; residue current; principal value integral residue

Introduction

In 1887, Henry Poincaré introduced the notion of a residue 1-form for rational 2-forms on C^{2}; this notion was generalized by Georges de Rham (1954) and Jean Leray (1959) who defined and studied the Poincaré residue of a d-closed meromorphic q-form ω, having poles of the first order along a smooth divisor (see [17]). In 1972 J.-B. Poly [21] proved that the Leray residue is well defined for any (not necessarily d-closed) differential form ω, as soon as ω and dω have a simple pole along a hypersurface.

In the present paper, we introduce a notion of the multi-logarithmic differential form. Such a meromorphic differential form has poles along a divisor D = ⋃_{i=1}^{k} D_{i} such that C = ⋂_{i=1}^{k} D_{i} is a reduced complete intersection. We then define the residue form of a multi-logarithmic differential form, as a generalization of corresponding definitions of J. Leray, J.-B. Poly and K. Saito. Our main theorem asserts that there is a natural morphism res. which maps the complex of the multi-logarithmic differential forms onto the complex of the regular meromorphic forms. The latter complex was defined and studied by D. Barlet [2], [3]. The first author has previously obtained a similar result for logarithmic differential forms along the divisor D. More precisely, it was proved [1] that the residue map defined by K. Saito [25] give a natural quasi-isomorphism between the complexes of sheaves of the logarithmic differential forms along the divisor D and of the regular meromorphic forms on D. In earlier works the second author and his collaborators [27], [28], [26] studied the properties of weakly holomorphic functions and residue currents on complete intersections. Building on these results, we give an explicit description of the natural morphism res. in terms of a certain integration current.

In Section 1, we give the basic notation and definitions and establish some properties of the multi-logarithmic differential forms with respect to a reduced complete intersection in a complex analytic manifold. In Section 2, we give the definition and discuss the properties of the regular meromorphic differential forms. In Section 3, our main theorem is proved. In Section 4, the above mentioned explicit description of this morphism res. is given. Finally, in the appendix we give a brief survey of the history of this subject, which does not seem to be widely known.

Acknowledgment. It is our pleasure to thank the Max-Planck-Institut für Mathematik for its financial assistance and hospitality during the preparation of this paper. The authors are very grateful to Professor B.Z. Moroz for his remarks and help during the preparation of the manuscript, and to Professor A. Yger for useful and stimulating conversations on the theory of residue currents.

1. Multi-logarithmic differential forms

Let U be an open subset of S = C^{m}, and let D_{1}, …, D_{k} be a set of hypersurfaces defined by the equations h_{j}(z) = 0, j = 1, …, k, k ≥ 1, respectively. Here h_{j}(z) = h_{j}(z_{1}, …, z_{m}), j = 1, …, k, is a holomorphic function in U, and z = (z_{1}, …, z_{m}), z_{i} ∈ C. In what follows, we shall suppose that h_{1}, …, h_{k} is a regular sequence in O_{U}, and I = (h_{1}, …, h_{k})O_{U} is a radical ideal, that is I = \sqrt{I}. In other words, the complex analytic variety C = D_{1} ∩ … ∩ D_{k} is a reduced complete intersection. In particular, dim C = m − k and the differential k-form dh_{1} ∧ … ∧ dh_{k} vanishes identically on no irreducible component of C. Let Ω_{U}^{•} = (Ω_{U}^{q}, d)_{q=0,1,...} be the de Rham complex of germs of holomorphic differential forms on U whose terms, locally at the point 0 ∈ U, are defined as follows:

Ω_{U,0}^{q} = O_{U,0}⟨ …, dz_{i_{1}} ∧ … ∧ dz_{i_{q}}, … ⟩ , (i_{1}, …, i_{q}) ∈ [1, m].

Let

D̂_{j} = D_{1} ∪ … ∪ D_{j−1} ∪ D_{j+1} ∪ … ∪ D_{k}, j = 1, …, k,

and let Ω_{U}^{q}(⋆D̂_{j}) be the O_{U}-module of meromorphic differential forms of degree q ≥ 0 consisting of all the differential q-forms with the polar divisor D̂_{j}. Write D̂_{1} = ∅ for k = 1, so that Ω_{U}^{q}(⋆D̂_{1}) = Ω_{U}^{q}.

Proposition 1.1. Let ω be a meromorphic differential q-form on U, q ≥ k, with poles along the divisor D = D_{1} ∪ … ∪ D_{k}. The following conditions are equivalent:

i) h_{j}ω ∈ ∑_{i=1}^{k} Ω_{U}^{q}(⋆D̂_{i}), h_{j}dω ∈ ∑_{i=1}^{k} Ω_{U}^{q+1}(⋆D̂_{i}), j = 1, …, k;

ii) h_{j}ω ∈ ∑_{i=1}^{k} Ω_{U}^{q}(⋆D̂_{i}), dh_{j} ∧ ω ∈ ∑_{i=1}^{k} Ω_{U}^{q+1}(⋆D̂_{i}), j = 1, …, k;

iii)There is a holomorphic function g, not equal identically to zero on every irreducible component of C, a holomorphic (q − k)-form ξ and a meromorphic qform η ∈ ∑_{i=1}^{k} Ω_{U}^{q}(⋆D̂_{i}) on U such that

gω = \frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ∧ ξ + η;

iv)There are analytic subsets A_{j} ⊂ D_{j}, j = 1, …, k, of codimension at least 2 such that the germ of ω at any point x ∈ ⋃_{j=1}^{k}(D_{j} \ A_{j}) belongs to

\frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ∧ Ω_{U,x}^{q−k} + ∑_{i=1}^{k} Ω_{U,x}^{q}(⋆D̂_{i}),

where Ω_{U,x}^{q−k} is the module of germs of holomorphic q-forms on U at x.

Proof. The equivalence of i) and ii) follows directly from the Leibnitz rule:

d(h_{j}ω) = dh_{j} ∧ ω + h_{j}dω.

Let us prove the implication ii) ⇒ iii). Consider the following representation of the form ω :

ω = \frac{∑_{|I|=q} a_{I}(z)·dz_{I}}{h_{1} ⋯ h_{k}} ,

where I := I^{q} = (i_{1}, …, i_{q}), 1 ≤ i_{1}, …, i_{q} ≤ m, is a multi-index, dz_{I} = dz_{i_{1}} ∧ … ∧ dz_{i_{q}}, and a_{I}(z) is a set of holomorphic functions on U, skew symmetric with respect to the index I. It is easy to see that the condition

dh_{j} ∧ ω ∈ ∑_{i=1}^{k} Ω_{U}^{q+1}(⋆D̂_{i})

is equivalent to the condition

dh_{j} ∧ ∑_{I} a_{I}(z)·dz_{I} ∈ ∑_{ℓ=1}^{k} h_{ℓ}Ω_{U}^{q+1}.

For an ordered multi-index I := I^{q+1} = (i_{1}, …, i_{q+1}), the latter gives us the following system of relations between the coefficients a_{I} and the derivatives ∂h_{j}/∂z_{i} :

∑_{ℓ=1}^{q+1}(−1)^{ℓ−1} \frac{∂h_{j}}{∂z_{i_{ℓ}}} a_{I\i_{ℓ}} = b_{jI}^{1}h_{1} + … + b_{jI}^{k}h_{k}, j = 1, …, k,  (1)

with holomorphic coefficients b_{jI}^{1}, …, b_{jI}^{k} ∈ O_{U}.

Let us fix a multi-index J^{p} = (j_{1}, …, j_{p}) ⊂ [1, …, m], 1 ≤ p ≤ k, and denote the corresponding minor of the Jacobian matrix Jac(h_{1}, …, h_{k}) = ∥∂h_{i}/∂z_{j}∥ as follows:

Δ_{J^{p}} = Δ_{j_{1}...j_{p}} = det ∥\frac{∂h_{i}}{∂z_{j_{r}}} ∥_{1≤i, r≤p}.

Let I ⊆ O_{U} be the ideal generated by the sequence of germs (h_{1}, …, h_{k}). We shall prove the following relations:

Δ_{J^{p}}a_{I^{q}} ≡ ∑_{{K⊂I^{q} \atop |K|=p}} sgn \binom{I^{q}}{K, I^{q} \ K}Δ_{K} a_{(J^{p},I^{q}\K)} (mod (I)), p = 1, …, k, (2)

by induction on p. Suppose that p = 1, J^{1} = j_{1}, and set I = (j, I^{q}) = (j_{1}, i_{1}, …, i_{q}) in the formula (1). One gets

\frac{∂h_{1}}{∂z_{j}} a_{I} ≡ ∑_{ℓ=1}^{q+1}(−1)^{ℓ} \frac{∂h_{1}}{∂z_{i_{ℓ}}} a_{I\i_{ℓ}} (mod (I)),

and this coincides with the relation (2) for p = 1.

Assuming the relation (2) to be true for p − 1, one can prove it for p as follows. Making use of the cofactor expansion along the p-th row of the determinant Δ_{J^{p}}, one obtains:

Δ_{J^{p}}a_{I} = ∑_{ℓ=1}^{p}(−1)^{p−ℓ} \frac{∂h_{p}}{∂z_{j_{ℓ}}} Δ_{j_{1}..._{j_{ℓ}}...j_{p}}^{p−1}a_{I}.

We have

Δ_{j_{1}...ℓ̂...j_{p}}a_{I} ≡ ∑_{{K′⊂I^{q} \atop |K′|=p−1}} sgn \binom{I^{q}}{K′, I^{q} \ K′}Δ_{K′}a_{(j_{1}..._{j_{ℓ}}...j_{p}, I\K′)} (mod (I)),

by the induction hypothesis. On substituting this relation into the previous equation and changing the order of summation, one obtains

Δ_{J^{p}}a_{I} ≡ ∑_{{K′⊂I^{q} \atop |K′|=p−1}} sgn \binom{I^{q}}{K′, I^{q} \ K′}Δ_{K′} ∑_{ℓ=1}^{p}(−1)^{p−ℓ} \frac{∂h_{p}}{∂z_{ℓ}} a_{(j_{1}..._{j_{ℓ}}...j_{p}, I\K′)} (mod (I)).

The second sum consists of p summands contained in the formula (1) for j = p, I = (i_{1}, …, i_{p}, I \ K′). Hence one can rewrite it as a sum of other q − p + 1 summands with opposite signs and an element of the ideal (h_{1}, …, h_{k})O_{U}. This yields the following relation moduli I :

Δ_{J^{p}}a_{I} ≡ ∑_{{K′⊂I^{q} \atop |K′|=p−1}}sgn \binom{I^{q}}{K′, I^{q} \ K′}Δ_{K′}(−1)^{p−1} ∑_{i∈I\K′}(−1)^{#(i; I\K′)} \frac{∂h_{p}}{∂z_{i}} a_{(j_{1}...j_{p}, I\K′\i)},

(3)

where #(i; I \ K′) denotes the number of occurrences of the index i in the set I \ K′. Let us order all the pairs (K′, i) in such a way that the multi-index K′ ∪ {i} coincides with a given K ⊂ I. For any such pair, the corresponding coefficient a_{(j_{1}...j_{p}, I\(K′\i))} is equal to a_{(J, I\K)}. Thus the contribution of the set ordered as above to the relation (3) looks as follows:

a_{(J, I\K)}(−1)^{p−1} ∑_{i∈K} sgn \binom{I^{q}}{K \ i, I^{q} \ K, i}(−1)^{#(i; I\(K\i))} \frac{∂h_{p}}{∂z_{i}} Δ_{K\i} =

= sgn \binom{I^{q}}{K, I^{q} \ K}a_{J, I\K} Δ_{K}.

This completes the proof of the relation (2) for p ≥ 1.

It is not difficult to see that the relations (2) imply the following equality

Δ_{i_{1}...i_{k}} · ∑_{|I|=q} a_{I}dz_{I} = dh_{1} ∧ … ∧ dh_{k} ∧ ( ∑_{|I′|=q−k}a_{i_{1}...i_{k}I′} dz_{I′}) + ν,

with ν ∈ ∑_{j=1}^{k} h_{j}Ω_{U}^{q−k}, so that iii) holds, with g = Δ_{i_{1}...i_{k}}, and with ξ, η of the required shape. It remains to show that one can choose the function g in iii) in such a way that g ≢ 0 on each of the irreducible components C_{j} of C. Let {Δ_{i_{1}...i_{k}}}, (i_{1}, ….i_{k}) ∈ [1, …, m], be the set of all k × k-minors, with elements Δ^{1}, …, Δ^{N}, where N = \binom{m}{k}. By assumption, for every non-singular point of C, there is a minor that does not vanish at this point. Let z^{(j)} ∈ C_{j} and suppose that z^{(j)} is non-singular, then the function

Δ_{t}(z) = tΔ^{1} + t^{2}Δ^{2} + … + t^{N}Δ^{N}

regarded as polynomial of t, does not vanish identically for every z = z^{(j)}. Hence, there is a value t_{0} ∈ C such that Δ_{t_{0}}(z^{(j)}) ≠ 0 for all j; thus one can take g(z) = Δ_{t_{0}}(z).

Here is a purely algebraic proof of this result. Consider the ideal G of O_{U} generated by all the minors Δ_{j_{1}...j_{k}} of maximal order of the Jacobian matrix Jac(h_{1}, …, h_{k}). The condition that dh_{1} ∧ … ∧ dh_{k} does not vanish identically on every irreducible component of C implies that the image G̃ of G in the ring O_{C,0} is not equal to Ann O_{C,0}. Thus one can use [4], Th.2.4.(1) which implies that the O_{C,0}-depth of the ideal G̃ is at least 1. Hence there is an element g ∈ O_{C} satisfying the condition in iii).

The implication iii) ⇒ iv) is obvious: take A_{j} = D_{j} ∩ {g = 0}, j = 1, …, k. Let us prove the implication iv) ⇒ ii). It is obvious that h_{j}ω can be expressed

as a sum of meromorphic forms each of which has singularities along at most k − 1 divisors, for example, on D̂_{i}, and on the subset A_{i} ⊂ D_{i} of codimension 2. Hence by the Riemann extension theorem, such a meromorphic form has singularities only on D̂_{i}. As a result, h_{j}ω ∈ ∑_{i=1}^{k} Ω_{U}^{q}(⋆D̂_{i}), j = 1, …, k. Similar considerations are valid for the differential form dh_{j} ∧ ω. This completes the proof of the implication iv) ⇒ ii).

Remark 1.2. In fact, Proposition 1.1 is a generalization of the basic Lemma-Definition of K. Saito for the case k = 1 if C = D is a reduced hypersurface (see [25], (1.1)).

Definition 1.3. A meromorphic differential q-form ω, q ≥ k, on U is called multilogarithmic with respect to the reduced complete intersection C = D_{1} ∩ … ∩ D_{k} if ω satisfies one of the conditions i) - iv) of Proposition 1.1.

Let C be a reduced complete intersection in S, and let h_{1} = 0, …, h_{k} = 0 be a system of defining equations of C in the neighbourhood of x ∈ C. A meromorphic q-form ω, q ≥ k, is logarithmic with respect to C at x, if for all j = 1, …, k the differential forms h_{j}ω and h_{j}dω satisfy either of conditions i) or ii) of Proposition 1.1. We shall denote the O_{S,x}-module of germs of multi-logarithmic q-forms at x and the corresponding sheaf of multi-logarithmic differential q-forms on S by Ω_{S,x}^{q}(log C) and Ω_{S}^{q}(log C), q ≥ 0, respectively. Thus the O_{S}-module Ω_{S}^{q}(log C) is a submodule of Ω_{S}^{q}(⋆D), consisting of all the differential forms with polar singularities along the divisor D, where D = D_{1} ∪ … ∪ D_{k}. Clearly, the sheaves Ω_{S}^{q}(log C) and Ω_{S}^{q} coincide outside D, for all q ≥ k. For the sake of convenience of notation we set

Ω_{S}^{q}(log C) = ∑_{i=1}^{k} Ω_{S}^{q}(⋆D̂_{i}), q = 0, …, k − 1.  (4)

It should be also remarked that Ω_{S}^{q}(log C), q ≥ 0, is a coherent sheaf of O_{S}- modules. Moreover, this sheaf is an O_{S,0}-module of finite type, and the direct sum ⊕_{q=0}^{m}Ω_{S,0}^{q}(log C) is an O_{S,0}-exterior algebra.

In what follows, we let, for simplicity, x = 0 when considering the local problems in the neighbourhood of x. We shall also assume that U is an open subset of S = C^{m} containing the origin.

Definition 1.4. In the notation of iii), the restriction on C = D_{1} ∩ … ∩ D_{k} of the form ξ/g is called the residue form of ω :

res. ω = \frac{ξ}{g} |_{C}.

Proposition 1.5. The residue morphism res. is well-defined.

Proof. Let ω ∈ Ω_{S}^{q}(log C), q ≥ 0. If ω has two different local presentations

g_{ℓ}ω = \frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ∧ ξ_{ℓ} + η_{ℓ}, ℓ = 1, 2,

then

dh_{1} ∧ … ∧ dh_{k} ∧ (g_{1}ξ_{2} − g_{2}ξ_{1}) = h_{1} ⋯ h_{k}(g_{1}η_{2} − g_{2}η_{1}) ∈ (h_{1}, …, h_{k})Ω_{S}^{q}.

Hence

dh_{1} ∧ … ∧ dh_{k} ∧ (g_{1}ξ_{2} − g_{2}ξ_{1}) ≡ 0 (mod (h_{1}, …, h_{k})).

Now one can apply the part i) of the Theorem from [23] for R = O_{C,0}, M = Ω_{S,0}^{q} ⊗ O_{C,0}. It follows then that

G^{e}(g_{1}ξ_{2} − g_{2}ξ_{1}) ∈ dh_{1} ∧ Ω_{S,0}^{q−k} + … + dh_{k} ∧ Ω_{S,0}^{q−k}, e ∈ Z, e ≥ 1,

in Ω_{S,0}^{q} ⊗ O_{C,0}, where the ideal G of O_{S,0} is generated by all minors of maximal order Δ_{j_{1}...j_{k}} of the Jacobian matrix Jac(h_{1}, …, h_{k}). Since C is reduced, the image G̃ of G in the ring O_{C,0} is not equal to Ann O_{C,0}. Thus one can use [4], Th.2.4.(1), which implies that the O_{C,0}-depth of the ideal G̃ is at least 1. Hence there is a maximal minor Δ = Δ_{i_{1}...i_{k}} which is not a zero-divisor in O_{C,0}, and

Δ^{e}(g_{1}ξ_{2} − g_{2}ξ_{1}) ∈ dh_{1} ∧ Ω_{S,0}^{q−1} + … + dh_{k} ∧ Ω_{S,0}^{q−1}

in Ω_{S,0}^{q} ⊗ O_{C,0}. Therefore the class of Δ^{e}(g_{1}ξ_{2} − g_{2}ξ_{1}) in Ω_{C,0}^{q} is equal to zero. Thus

the two elements \frac{1}{g_{1}} ξ_{1} and \frac{1}{g_{2}} ξ_{2} define the same class in M_{C,0} ⊗_{O_{C,0}} Ω_{C,0}^{q}.

Lemma 1.6. The map res. defines a homomorphism of O_{S}-modules

res.: Ω_{S}^{q}(log C) ⟶ M_{C} ⊗_{O_{C}} Ω_{C}^{q−k}, q ≥ k;

moreover, the following sequence of O_{S}-modules

0 ⟶ ∑_{i=1}^{k} Ω_{S}^{q}(⋆D̂_{i}) ⟶ Ω_{S}^{q}(log C) ⟶^{res.} M_{C} ⊗_{O_{C}} Ω_{C}^{q−k}.

is exact, and the morphism res. commutes with the exterior differentiation d.

Proof. Writing gω as in iii) of Proposition 1.1, in view of the Definition 1.4 of the residue map res., one obtains

res. ω = 0 ⇔ gω ∈ ∑_{i=1}^{k} Ω_{S}^{q}(⋆D̂_{i}) ⇔ ω ∈ ∑_{i=1}^{k} Ω_{S}^{q}(⋆D̂_{i}),

and the assertion follows.

Remark 1.7. In particular, for q = k we have

0 ⟶ ∑_{i=1}^{k} Ω_{S}^{k}(⋆D̂_{i}) ⟶ Ω_{S}^{k}(log C) ⟶^{res.} M_{C} ≅ M_{C},

where π: C̃ ⟶ C is the normalization of C. K. Saito has observed that if k = 1 and C = D is a hypersurface which is a normal crossing outside a subvariety of codimension at least 2, then the image Im (res.) coincides with π_{∗}(O_{D}); it consists of the so-called weakly holomorphic functions on D, that is meromorphic functions, whose preimage becomes holomorphic on the normalization (see [25], (2.8)). Some interesting properties of the class of weakly holomorphic functions have been discussed in [27].

Remark 1.8. Thus the image res. Ω_{S}^{q}(log C), q ≥ k, of the residue map is an O_{C}-module. It follows from the definition of multi-logarithmic forms that

h_{j} · (Ω_{S,0}^{q}(log C)/ ∑_{i=1}^{k} Ω_{S}^{q}(⋆D̂_{i})) = 0, j = 1, …, k.

Hence the ideal (h_{1}, …, h_{k}) ⊂ O_{S} annihilates Im (res.).

Remark 1.9. The condition iii) of Proposition 1.1 is equivalent to the following condition:

Δ_{j_{1}...j_{k}} · ω ∈ \frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ∧ Ω_{U}^{•−k} + Ω_{U}^{•}, (j_{1}, …, j_{k}) ∈ [1, …, m],

for any minor Δ_{j_{1}...j_{k}} of maximal order of the Jacobian matrix Jac(h).

2. Regular meromorphic differential forms

Let X be an analytic reduced subspace of a complex manifold S, let dim X = n, and dim S = n + k. The O_{X,0}-module

ω_{X,0}^{n} = Ext_{O_{S,0}}^{k}(O_{X,0}, Ω_{S,0}^{n+k})

is said to be the Grothendieck dualizing module of the germ (X, 0). Definition 2.1. The sheaf ω_{X}^{q}, q ≥ 0, of regular meromorphic differential q-forms is locally given as follows:

ω_{X,0}^{q} = Hom_{O_{X,0}}(Ω_{X,0}^{n−q}, ω_{X,0}^{n});

equivalently, the sheaf ω_{X,0}^{q} consists of all the meromorphic forms ω of degree q on X such that ω ∧ η ∈ ω_{X,0}^{n} for any η ∈ Ω_{X,0}^{n−q}.

We shall study the case when X is taken to be the reduced germ of a complete intersection C = D_{1} ∩ … ∩ D_{k}, where D_{1}, …, D_{k} is a set of hypersurfaces defined by the equations h_{j}(z) = 0, j = 1, …, k, k ≥ 1, respectively (in the notation of Section 1). In this case,

ω_{C,0}^{n} = Ext_{O_{S,0}}^{k}(O_{C,0}, Ω_{S,0}^{n+k}) ≅ O_{C,0} (\frac{dz_{1} ∧ … ∧ dz_{n+k}}{dh_{1} ∧ … ∧ dh_{k}} )

is a free module of rank one, and

ω_{C,0}^{q} ≅ Hom_{O_{C,0}}(Ω_{C,0}^{n−q}, O_{C,0}), 0 ≤ q ≤ n.

Here are some properties of the regular meromorphic differential forms: 1) ω_{C}^{q} = 0 for q < 0 and for q > dim C;

2) The de Rham differential d is extended to the modules ω_{C}^{q}, 0 ≤ q ≤ n, so that (ω_{C}^{•}, d) is a complex;

3) In the case when C is normal, the sheaves ω_{C}^{q} have the following well-known interpretation:

ω_{C}^{q} ≅ j_{∗}j^{∗}Ω_{C}^{q},

where j: C \ Sing C ⟶ C is the canonical embedding. More precisely, in such a case, for any q ≥ 0 there is an exact sequence of local cohomology groups with supports in Z = Sing C :

0 ⟶ H_{Z}^{0}(Ω_{C}^{q}) ⟶ Ω_{C}^{q} ⟶ ω_{C}^{q} ⟶ H_{Z}^{1}(Ω_{C}^{q}) ⟶ 0.

Let k = 1, so that C = D is a hypersurface. Then both the module ω_{D,0}^{0} and the O_{D}-module res. Ω_{S,0}^{1}(log D) contain all the germs of locally bounded meromorphic functions on D (cf. [3], §2, ex. i)). Moreover, these two sets coincide if the local fundamental group of the complement S \ D is abelian (see [23]).

Theorem 2.2 ([3]). The sequence of O_{C}-modules

0 ⟶ ω_{C}^{q} ⟶^{C} Ext_{O_{S}}^{k}(O_{C}, Ω_{S}^{q+k}) ⟶^{α} (Ext_{O_{S}}^{k}(O_{C}, Ω_{S}^{q+k+1}))^{k}, q ≥ 0,

where ω_{C}^{q} ⊂ j_{∗}j^{∗}Ω_{C}^{q}, the map C is multiplication by the fundamental class of C in S, and α(e) = (e ∧ dh_{1}, …, e ∧ dh_{k}), is an exact sequence of O_{C}-modules.

Thus C(v) corresponds to a Čech cocycle w/h such that w = v ∧ dh. It is easy to see that

ω_{C}^{n} ≅ Ext_{O_{S}}^{k}(O_{C}, Ω_{S}^{n+k}).

In particular, ω_{C}^{n} is isomorphic to the Grothendieck dualizing module of the ndimensional complete intersection C ⊂ S. In conclusion, we establish a remarkable property of regular meromorphic differential forms in terms of the trace map associated with the Noether normalization. In fact, this property in a slightly different context has been proved by D. Barlet (see [3]).

Theorem 2.3. The O_{C,0}-module ω_{C}^{q}, q ≥ 0, consists of all the meromorphic forms ω of degree q on C satisfying the following condition: for any η ∈ Ω_{C,0}^{n−q}, the trace

Trace_{C/C^{n}}(ω ∧ η)(z) = ∑_{x∈π^{−1}(z)}(ω ∧ η)(x)

is the germ of a holomorphic form on C.

Proof. Taking the Noether normalization of the germ C, one can suppose, that the germ (S, 0) is isomorphic to the direct product (C^{n}, 0) × (C^{k}, 0) with coordinates (z, w) so that the natural projection π: (C, 0) → (C^{n}, 0) is a finite covering.

Let ω ∈ ω_{C,0}^{q}, so that for any η ∈ Ω_{C,0}^{n−q} one has

ω ∧ η = \frac{α}{dh_{1} ∧ ⋯ ∧ dh_{k}} ∈ ω_{C,0}^{n},

where α ∈ Ω_{S,0}^{n+k}. By definition, one can view the differential form ω ∧ η as an element of a Čech cochain in the open set {Δ(z, w) ≠ 0} :

ω ∧ η = \frac{φ(z, w)}{Δ}dz,

where φ is a holomorphic function and Δ is the determinant of the Jacobian matrix of the sequence of functions h_{1}(z, w), …, h_{k}(z, w) with respect to w, that is,

Δ = det ∥\frac{∂h_{j}}{∂w_{i}} ∥_{1≤i, j≤k}.

Thus

Trace_{C/C^{n}}(ω ∧ η)(z) = ∑_{ℓ} \frac{φ}{Δ}(z, w^{ℓ}(z)) dz.

It is well-known (see [28], Ch.1) that the trace of the meromorphic function φ/Δ, whose denominator is equal to the Jacobian of the map π, is holomorphic. The proof of this fact is based on the integral representation of the trace as

(\frac{1}{2π\sqrt{−1}})^{k}∫_{Γ} \frac{φ(z, w)dw}{h_{1}(z, w) ⋯ h_{k}(z, w)},

where the integrand is holomorphic on the parameters z and

Γ = {w ∈ C^{k}: |h_{1}(0, w)| = … = |h_{k}(0, w)| = ε}

is a compact cycle.

To prove the converse, for simplicity, we consider the case n = 1, so that C is a complete intersection curve. The general case can be studied in the same way. Let ν ∈ Ω_{S,0}^{1}, and suppose that

Trace (φ \frac{ν}{g} )

is holomorphic, for any holomorphic function φ ∈ O_{C,0}. Then one has to prove

\frac{ν}{g} |_{C} = \frac{ψ}{dh_{1} ∧ … ∧ dh_{k}} |_{C},

that is,

ν ∧ dh = ψg + h_{1}f_{1} + … + h_{k}f_{k},  (5)

where f_{1}, …, f_{k} ∈ O_{S,0} are holomorphic functions. By assumption,

Trace (φ \frac{ν}{g} ) (z) = ∑_{ℓ} φ\frac{ν}{g} (z, w^{ℓ}(z))

is holomorphic for any φ. Let us consider the Grothendieck residue

R_{h,g}[φν ∧ dh_{1} ∧ … ∧ dh_{k}] = ∫_{Γ_{ε,δ}} \frac{φν ∧ dh_{1} ∧ … ∧ dh_{k}}{gh_{1} … h_{k}} ,

where Γ_{ε,δ} = {(z, w) ∈ (S, 0) : |h_{1}| = ε, …, |h_{k}| = ε, |g| = δ}. The local variant of the Hilbert Nullstellensatz shows that for some N > 1, one can write

z^{N} = q_{1}h_{1} + … + q_{k}h_{k} + Qg.

Therefore after applying the residue transformation formula one obtains

R_{h,g}[φν ∧ dh_{1} ∧ … ∧ dh_{k}] = R_{h,r}[Qφν ∧ dh_{1} ∧ … ∧ dh_{k}],

where r = z^{N}. The contour of integration in the latter integral

γ(ε, δ) = {(z, w) ∈ (S, 0) : |h_{1}| = ε, …, |h_{k}| = ε, |z| = δ}

is fibered under the projection π over the circle |z| = δ. Hence, integration with respect to w along the classical formula of the logarithmic residue yields

R_{h,g}[φν ∧ dh_{1} ∧ … ∧ dh_{k}] =

= \frac{1}{2π\sqrt{−1}} ∫_{|z|=δ} Trace (φν \frac{q}{r} ) (z) = \frac{1}{2π\sqrt{−1}} ∫_{|z|=δ} Trace (φ \frac{ν}{g} ) (z),

since the restrictions of q/r and 1/g to C are equal. By our assumption the last integral is equal to zero, hence the local duality theorem of Grothendieck implies (5). This completes the proof in the case n = 1.

3. The Poincaré residue map

We start with the definition of the classical Poincaré residue, following [10]. Let ω be a meromorphic differential m-form on an m-dimensional complex analytic manifold S with a polar divisor D ⊂ S. Thus, locally we have a representation:

ω = \frac{f(z)dz_{1} ∧ … ∧ dz_{m}}{h(z)} .

By definition, the Poincaré residue rés_{D}(ω) is a meromorphic (m − 1)-form on D whose singularities are contained in the singular locus Sing D ⊂ D. To define this form explicitly, let us note that at each point x ∈ D \ Sing D one of the derivatives does not vanish:

\frac{∂h}{∂z_{i}} (x) ≠ 0.

Hence, locally one may let

rés_{D}(ω) = (−1)^{m−i} \frac{f(x)dz_{1} ∧ … ∧ d̂z_{i} ∧ … ∧ dz_{m}}{\frac{∂h}{∂z_{i}} (x)} |_{D}.

It is not difficult to verify that this expression does not depend on the index i, on the local coordinates, and on the defining equation of D. Moreover, the Poincaré residue rés_{D}(ω) is holomorphic on the complement S \ D. When D is smooth, one can take h(z) = z_{m}, then

rés_{D}(\frac{f(x)dz_{1} ∧ … ∧ dz_{m}}{z_{m}} ) = f(x)dz_{1} ∧ … ∧ dz_{m−1}

coincides with the usual residue.

For brevity, the Poincaré residue rés_{D}(ω) will be denoted by rés(ω). When D ⊂ S is a nonsingular hypersurface, the Poincaré residue induces an exact sequence

0 ⟶ Ω_{S}^{m} ⟶ Ω_{S}^{m}(D) \underrightarrow{rés} Ω_{D}^{m−1} ⟶ 0 ,

where Ω_{S}^{m}(D) denotes the set of meromorphic forms on S having a pole of the first order along the divisor D. In particular, it follows that the germ of every holomorphic (m − 1)-form on D is a Poincaré residue. It is obvious that this is true globally when the first cohomology group vanishes: H^{1}(S, Ω_{S}^{m}) = 0.

More generally, following J.Leray (see [12]) one can apply the above construction to meromorphic differential q-forms on S with a smooth polar divisor D ⊂ S for any 1 ≤ q ≤ m, and write locally for any d-closed differential q-form ω on S \ D, having poles of the first order along D, the following presentation:

ω = \frac{dh}{h} ∧ ξ + η,

where ξ and η are holomorphic. The restriction of ξ to D is a holomorphic form independent of the choice of the local equation h = 0 of D; we denote it by rés(ω) as before. As a result, one gets (see [19]) an exact sequence

0 ⟶ Ω_{S}^{q} ⟶ Ω_{S}^{q}(D) \underrightarrow{rés} Ω_{D}^{q−1} ⟶ 0, 1 ≤ q ≤ m,

which is equivalent to the sequence

0 ⟶ Ω_{S}^{q} ⟶ Ω_{S}^{q}(log D) \underrightarrow{rés} Ω_{D}^{q−1} ⟶ 0 ,  (6)

since the divisor D is a smooth hypersurface. Remark 3.1. As it has been already remarked by de Rham and Leray (cf. [12]) that for a meromorphic q-form ω with poles of the first order along D the above definition gives us the holomorphic (q − 1)-form rés(ω) in cases when the singular locus Sing D of the divisor D consists of isolated double quadratic points only and either q < m, or q = m, and the coefficient of m-form h · ω vanishes on Sing D. This phenomenon can be readily explained by considerations which we use in the next and previous sections.

An extension of the exact sequence (6) to the case when the divisor D has arbitrary singularities was described in [1]

0 ⟶ Ω_{S}^{q} ⟶ Ω_{S}^{q}(log D) \underrightarrow{res.} ω_{D}^{q−1} ⟶ 0, q ≥ 1.  (7)

The following theorem is devoted to the further generalization of this construction to the case of complete intersection.

Theorem 3.2. In the notation of Section 1 let C ⊂ S be a reduced complete intersection. Then there is an exact sequence

0 ⟶ ∑_{i=1}^{k} Ω_{S}^{q}(⋆D̂_{i}) ⟶ Ω_{S}^{q}(log C) \underrightarrow{res.} ω_{C}^{q−k} ⟶ 0, q ≥ k,

and a natural isomorphism of O_{C}-modules

res. Ω_{S}^{q}(log C) ≅ ω_{C}^{q−k}, q ≥ k.

Proof. The proof goes similarly to [1], §4. Thus, it is sufficient to verify our statements locally. The remark 1.9 implies

Δ_{j_{1}...j_{k}} · res. Ω_{S}^{q}(log C)|_{U} ⊂ Ω_{C}^{q−k}|_{C∩U},

for all maximal minors Δ_{j_{1}...j_{k}}, (j_{1}, …, j_{k}) ∈ [1, …, m] of the Jacobian matrix Jac(\underline{h}). Since ω_{C,0}^{n} ≅ O_{C,0}(dz_{1} ∧ … ∧ dz_{n+k}/dh_{1} ∧ … ∧ dh_{k}) the definition of the modules of regular meromorphic differential forms implies immediately that

res. Ω_{S}^{q}(log C) ⊂ ω_{C}^{q−k}.

Let now K_{•}(\underline{h}) be the usual Koszul complex associated with the sequence \underline{h} = (h_{1}, …, h_{k}) :

0 → O_{S,0}⟨e_{0} ∧ … ∧ e_{k−1}⟩ ⟶^{d_{k−1}} … ⟶^{d_{1}} O_{S,0}⟨e_{0}⟩ + … + O_{S,0}⟨e_{k−1}⟩ →

⟶^{d_{0}} O_{S,0} ⟶^{d_{−1}} O_{C,0} → 0 ,

where

K_{k}(\underline{h}) = O_{S,0}⟨e_{0} ∧ … ∧ e_{k−1}⟩, …, K_{1}(\underline{h}) = O_{S,0}⟨e_{0}⟩ + … + O_{S,0}⟨e_{k−1}⟩,

and K_{0}(\underline{h}) = O_{S,0}, d_{0}(e_{i}) = h_{i+1}, i = 0, …, k − 1, d_{−1}(1) = 1. From the dual exact sequence one obtains, by definition,

Ext_{O_{S,0}}^{k}(O_{C,0}, Ω_{S,0}^{q+1}) ≅ Hom_{O_{S,0}}(K_{k}(\underline{h}), Ω_{S,0}^{q+1})/d^{k−1}(Hom_{O_{S,0}}(K_{k−1}(\underline{h}), Ω_{S,0}^{q+1})).

This means that any element of Ext_{O_{S,0}}^{k}(O_{C,0}, Ω_{S,0}^{q+1}) can be represented as a Čech (k − 1)-cochain (more explicitly, a (k − 1)-cocycle)

\frac{ν}{h_{1} ⋯ h_{k}} ∈ Hom_{O_{S,0}}(K_{k}(\underline{h}), Ω_{S,0}^{q+1}) ≅ C_{(k)}^{k−1}(Ω_{S,0}^{q+1}),

where ν ∈ Ω_{S,0}^{q+1}. Choose now an element ν ∈ Ω_{S,0}^{q+1} such that

\frac{ν}{h_{1} ⋯ h_{k}} ∧ dh_{j} ∈ Ext_{O_{S,0}}^{k}(O_{C,0}, Ω_{S,0}^{q+2}), j = 1, …, k,

corresponds to the trivial element. This means that for any j = 1, …, k the form ν ∧ dh_{j}/h_{1} ⋯ h_{k}, is defined by an element of d^{k−1}(Hom_{O_{S,0}}(K_{k−1}(\underline{h}), Ω_{S,0}^{q+2})). This yields that

ν ∧ dh_{j} ∈ ∑_{i=1}^{k} h_{i}Ω_{S}^{q+2}, j = 1, …, k,

or, equivalently,

ω ∧ dh_{j} ∈ ∑_{i=1}^{k} Ω_{S}^{q+1}(⋆D̂_{i}), j = 1, …, k, where ω = \frac{ν}{h_{1} ⋯ h_{k}} .

In view of Proposition 1.1 this means that ω ∈ Ω_{S,0}^{q}(log C). Now we shall use the notation and exact sequence of Theorem 2.2. Set ν̃ = C^{−1}(ν/h_{1} ⋯ h_{k}). Then C(ν̃) corresponds to the Čech cocycle ν/h_{1} ⋯ h_{k} such that ν = ν̃ ∧ dh_{1} ∧ ⋯ ∧ dh_{k} (take v = ν̃, w = ν in the description of ω_{C}^{q} in terms of multiplication by the fundamental class of C in S). This implies

\frac{ν}{h_{1} ⋯ h_{k}} = ν̃ ∧ \frac{dh_{1}}{h_{1}} ∧ ⋯ ∧ \frac{dh_{k}}{h_{k}} , and res. (\frac{ν}{h_{1} ⋯ h_{k}} ) = ν̃.

Thus, for any element ν̃ ∈ ω_{C}^{q−k} there is a preimage under the multi-logarithmic residue map represented by ν/h_{1} ⋯ h_{k}.

4. On residues of multi-logarithmic forms and ¯∂-closed currents

We first remark, that in view of Theorem 3.2, the sections of ω_{C}^{•} can be considered as “regular meromorphic forms.” More precisely, in complex analysis “regular” or “holomorphic” differential forms are usually described as ¯∂-closed forms (cf. [13]).

The key idea and further considerations in this section are closely related to the methods developed in [27], where the theory of weakly holomorphic functions is studied. Moreover, they lead to another proof of the statement of Theorem 3.2 concerning the representation of sections of Barlet sheaves ω_{C}^{•} as residues of multilogarithmic differential forms. The main result of this section is the following theorem.

Theorem 4.1. There is a natural one to one correspondence between the sets of residues of multi-logarithmic forms and meromorphic ¯∂-closed currents on the complete intersection C = D_{1} ∩ … ∩ D_{k}.

The proof will be given below. We start with the definition of the integration current associated with a meromorphic differential form, and recall the basic properties of residue currents.

Let ψ = ξ/g be a meromorphic differential form on a reduced analytic subset C of a domain U ⊂ C^{m}; here ξ is a holomorphic differential form and g is a holomorphic function. Such a form defines the following current

⟨ [ψ], φ⟩ = ⟨ [\frac{ξ}{g} ] |_{C}, φ⟩ = lim_{ε→0} ∫_{C∩\{|g|>ε\}} \frac{ξ}{g} ∧ φ, φ ∈ D^{•}(U),  (8)

where φ is a test C^{∞}-differential form with compact support on U, and the real dimension of ξ is equal to deg φ + deg ψ. This is an integration current on C in the principal value sense.

A residue current R_{h} associated with a holomorphic map h = (h_{1}, …, h_{k}) : U → C^{k} is is a functional on the space D^{•}(U) of C^{∞}-differential form with compact support on U. It is defined as follows (see [5] or [28]):

R_{h}(φ) = lim_{ε→0} ∫_{T_{ε}(h)} \frac{φ}{h_{1} ⋯ h_{k}} , φ ∈ D^{2m−k}(U),

where the integral is taken over the tube

T_{ε}(h) = {z ∈ S: |h_{j}(z)| = ε_{j}, j = 1, …, k}

with a fixed orientation. Here the approach of the radii ε = (ε_{1}, …, ε_{k}) to zero is required to be along a so-called admissible path: a path ε = ε(τ) is called admissible, if ε_{k} → 0 as τ → 0 and for each j = 1, …, k − 1, the coordinate ε_{j}(τ) approaches zero faster than any power of the next coordinate ε_{j+1}(τ).

The principal value of the residue current R_{h} with respect to a function g: U → C is defined as the limit

P_{g}R_{h}(φ) = lim_{τ→0} ∫_{T_{ε}(h)∩\{|g|>δ\}} \frac{φ}{gh_{1} ⋯ h_{k}}

along the admissible path (ε(τ), δ(τ)): [0, 1] → R^{k+1}.

When the set h^{−1}(0) = {h_{1} = … = h_{k} = 0} is a complete intersection of codimension k then

R_{h} = ¯∂ [\frac{1}{h_{1}} ] ∧ … ∧ ¯∂ [\frac{1}{h_{k}} ] .

Respectively, when g ≢ 0 on every irreducible component of the analytic set h^{−1}(0) then

P_{g}R_{h} = [\frac{1}{g}] ¯∂ [\frac{1}{h_{1}} ] ∧ … ∧ ¯∂ [\frac{1}{h_{k}} ] .

When ω = α/h_{1} ⋯ h_{k} is a meromorphic differential form of degree ℓ then there is the current

R_{h}[ω](φ) := R_{h}(α ∧ φ), φ ∈ D^{2m−k−ℓ}(U).

Analogously for ω = α/g·h_{1} ⋯ h_{k} we set

P_{g}R_{h}[ω](φ) = P_{g}R_{h}(α ∧ φ).

Let us assume that the zero set defined by the system of equations

{g = h_{1} = … = h_{k} = 0}

is a complete intersection. Then the residue currents R_{h} and their principal values P_{g} with respect to g have the following properties:

1) R_{h}[ω] = P_{g}R_{h}[gω];

2) P_{g}R_{h}[\frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ∧ \frac{ξ}{g} ] = [\frac{ξ}{g} ] |_{C} = [\frac{ξ}{g} ] ∧ [C];

where [C] is the integration current (a holomorphic chain) on C. In particular, when ξ and g are identically equal to the constant 1, then

R_{h}[\frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ] = dh_{1} ∧ … dh_{k} ∧ ¯∂ [\frac{1}{h_{1}} ] ∧ … ∧ ¯∂ [\frac{1}{h_{k}} ] = [C];

3) h_{j}P_{g}R_{h} = 0, j = 1, …, k;

4) If U is a domain of holomorphy (or a Stein manifold) then for v ∈ O_{U} one has

vR_{h} = 0 ⟺ v ∈ (h_{1}, …, h_{k})O_{U}.

In particular, this property is valid in the local rings O_{U,x}, x ∈ U.

The properties 1) and 3) are proved in [5], Theorem 4.4.1 and Theorem 1.7.6, respectively, the property 2) is obtained in [26], Theorem 2.1; the property 4) is obtained in [18], Corollary 6.1.2.

Proof of Theorem 4.1. Let ω ∈ Ω_{S,0}^{q+k}(log C), so that

ω = \frac{α}{h_{1} ⋯ h_{k}} , and gω = \frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ∧ ξ + η,

where α, ξ are holomorphic and η satisfies the condition iii) of Proposition 1.1. In view of property 1) one has

R_{h}[ω] = P_{g}R_{h}[gω] = P_{g}R_{h}[\frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ∧ \frac{ξ}{g} ] + P_{g}R_{h}[\frac{η}{g} ].

The second summand corresponds to the current which is trivial by property 3). The property 2) implies

R_{h}[ω] = [\frac{ξ}{g} ] |_{C}.

Hence the current [\frac{ξ}{g} ] |_{C} defined by the residue res. ω = \frac{ξ}{g} |_{C} of the multi-logarithmic

form ω coincides with the residue current R_{h}[ω] which is ¯∂-closed [5].

Conversely, under the same assumptions, let ξ/g be a meromorphic form on C, and the corresponding current (8)

⟨ [ψ], φ⟩ = ⟨ [\frac{ξ}{g} ] |_{C}, φ⟩ = lim_{ε→0} ∫_{C∩\{|g|>ε\}} \frac{ξ}{g} ∧ φ, φ ∈ D^{•}(U)

is ¯∂-closed. Then in virtue of property 2)

0 = ¯∂ [\frac{ξ}{g} ] |_{C} = ¯∂( [\frac{ξ}{g} ] ∧ [C]).

Since ¯∂[C] = 0, applying the Leibnitz rule and the property 2), one obtains

0 = ¯∂ [\frac{ξ}{g} ] ∧ [C] = ¯∂ [\frac{ξ}{g} ] ∧ dh_{1} ∧ … ∧ dh_{k} ∧ ¯∂ [\frac{1}{h_{1}} ] ∧ … ∧ ¯∂ [\frac{1}{h_{k}} ] .

The property 4) yields

dh_{1} ∧ … ∧ dh_{k} ∧ ξ ∈ (g, h_{1}, …, h_{k})Ω_{U,0}^{q+k},

that is,

dh_{1} ∧ … ∧ dh_{k} ∧ ξ = αg − h_{1}η_{1} − … − h_{k}η_{k},  (9)

where α, η_{1}, …, η_{k} ∈ Ω_{U}^{q+k} are holomorphic forms on the domain U ⊂ S. Here the property 4) is used in computing the coefficients of the form ξ ∧dh. Let us consider the meromorphic form ω = α/h_{1} ⋯ h_{k}. We have

g·ω = g· \frac{α}{h_{1} ⋯ h_{k}} = \frac{dh ∧ ξ + h_{1}η_{1} + … + h_{k}η_{k}}{h_{1} ⋯ h_{k}} = \frac{dh_{1}}{h_{1}} ∧ … ∧ \frac{dh_{k}}{h_{k}} ∧ ξ + η,

where

η = ∑_{i=1}^{k} \frac{η_{i}}{h_{1} ⋯ h_{k}} ∈ ∑_{i=1}^{k} Ω_{U,0}^{q+k}(⋆D̂_{i}).

This completes the proof of Theorem 4.1.

One can deduce the main statement of Theorem 3.2 from the following theorem.

Theorem 4.2. A meromorphic differential q-form on the complete intersection C = D_{1} ∩ … ∩ D_{k} defines a meromorphic ¯∂-closed current on C if and only if it is a section of the Barlet sheaf ω_{C}^{q} of regular meromorphic forms.

Proof. Let ω = ξ/g be a meromorphic q-form C. It is not difficult to see that

¯∂(ω) = 0 ⟺ ¯∂(ω ∧ ν) = 0 for any ν ∈ Ω_{C}^{n−q}.

Making use of the property 4) and of similar considerations of the converse part of the proof of Theorem 4.1 we obtain that the property of ¯∂-closedness of ω is equivalent to the existence of the relation

dh ∧ (ξ ∧ ν) = αg − h_{1}η_{1} − … − h_{k}η_{k},

where α, η_{1}, …, η_{k} ∈ Ω_{U}^{m} are holomorphic forms on the domain U ⊂ S. This is equivalent to the condition

\frac{ξ}{g} ∧ ν|_{C} = \frac{α}{dh_{1} ∧ ⋯ ∧ dh_{k}} |_{C} ∈ ω_{C}^{n}.

Hence, by definition,

ω = \frac{ξ}{g} ∈ ω_{C}^{q}

as required.

Remark 4.3. The idea of this proof is the same as that of the proof of Theorem 2 in [13].

Remark 4.4. Thus, any section of the Barlet sheaf ω_{C}^{q} is the residue of a multilogarithmic form. Such meromorphic forms on the complete intersection C are characterized by the following two conditions:

a) they are holomorphic on the set of non-singular points of C (cf. [12]),

b) they have no residues with respect to the singular locus of C (this is the so-called ¯∂-closedness property).

A differential form ψ which satisfies condition a) is not a priori meromorphic. Nevertheless it should define the following integration current (see [26]):

⟨ [ψ], φ⟩ = lim_{δ→0} ∫_{C∩{|dh|>δ}} ψ ∧ φ,  (10)

where dh = dh_{1} ∧ ⋯ ∧ dh_{k}. This integration current differs from (8) by the choice of the “principal value” of the integral. Recall that under our assumptions the singular locus Sing C of C is defined by the condition dh_{1} ∧ ⋯ ∧ dh_{k}|_{C} = 0. Then a natural question arises: Whether the class of differential q-forms ψ which are holomorphic on the non-singular part of C and define the ¯∂-closed currents by means of formula (10), and the class of sections of Barlet sheaf ω_{C}^{q} coincide?

In connection with this problem it should be noted that there are non-exact ¯∂-closed currents corresponding to non-holomorphic differential forms (see [8]).

If one supposes that ψ is meromorphic then Proposition 1 of [26] asserts that it is true. Thus one gets the following elegant description of the Grothendieck dualizing module on a reduced complete intersection in terms of ¯∂-closed currents. We remark that a priori all sections of the dualizing module are meromorphic on C and holomorphic on the non-singular part of C.

Corollary 4.5. Any element ψ of the Grothendieck dualizing module of the reduced complete intersection C = D_{1} ∩ … ∩ D_{k} defines a meromorphic ¯∂-closed current on C by formula (10) and vice versa.

In conclusion we pose a problem which is a stronger version of the above question. Let Ψ^{q} be the set of differential q-forms ψ on U such that they are holomorphic on the non-singular part of C and satisfy the following condition (cf. formula (10)) :

⟨ [ψ], φ⟩ = lim_{δ→0} ∫_{C∩{|dh|=δ}} ψ ∧ φ = 0.  (11)

Conjecture 1. The classes of differential q-forms defined by the set Ψ^{q} and by the sections of the Barlet sheaf ω_{C}^{q} coincide.

Remark 4.6. It is well-known that any pure dimensional analytic set can be regarded locally as a component of some reduced complete intersection (see [9], p.52). In fact, this leads to a generalization of the notion of the multi-logarithmic differential form with respect to an arbitrary pure dimensional analytic set.

5. Appendix. Historical remarks and comments

In 1887, Henry Poincaré [20] introduced the notion of a residue 1-form for rational 2-forms on C^{2}. In the papers of G. de Rham, [22], and J. Leray, [17], this notion was extended to d-closed meromorphic q-form ω having poles of the first order along a smooth divisor. They proved that the residue of such a form is holomorphic. Influenced by the papers of Leray and P. Dolbeault, M. Herrera, P. Liberman, N. Coleff and others have developed theory of the integral representation and residue currents (see [6], [7], [5]).

The next stage the development of the theory starts at the beginning of the seventies. It is closely related to the general theory of logarithmic differential forms. In fact, such forms were first considered in 1972 in a work of J.-B. Poly [21] who defined and studied the Leray residue for semi-meromorphic differential forms. Poly had proved that the Leray residue is well defined for any (not necessarily d-closed) semi-meromorphic differential form ω, as soon as ω and dω have a simple pole along a hypersurface [loc. cite, Remarque 4]. Later, K. Saito further investigated the class of meromorphic differential forms ω satisfying these conditions; in his series of works, these forms were called logarithmic along a hypersurface (see [23], [24]). Saito established the basic properties of meromorphic logarithmic forms, and of their residues, and discussed some important and interesting applications of these objects. The same notion has occurred also in the book by Ph. Griffiths and J. Harris [11], with no references to previous works; in this book, the authors use this notion only in the case of divisors which are normal crossings of smooth hypersurfaces. This case has been also treated in earlier works by P. Deligne, J. Steenbrink, F. Pham, and many others.

Almost at the same time, in 1976, in a work of D. Barlet [2] a special complex ω_{X}^{•} of subsheaves of the sheaves of meromorphic differential forms on an analytic subvariety X of a complex manifold was introduced and investigated (see also [3]).

If dim X = n, the sheaf ω_{X}^{n} is naturally isomorphic to the Grothendieck dualizing module. In a purely algebraic context, a similar notion appeared also in the works of E. Kunz [15], [16], M. Kersken [14], and others on general duality theory. They call ω_{X}^{q} the sheaf of regular meromorphic differential forms of degree q. In a joint paper [13], G. Henkin and M. Passare give an analytic interpretation of regular meromorphic forms. Some of their results are essentially deduced from the basic properties of weakly holomorphic functions, established in a series of earlier works of the second author and his collaborators (see [27], [28], [26]).

In 1990 it was proved (see [1]) that the residue map defined by K. Saito [25] gives a natural quasi-isomorphism between the complexes of sheaves of logarithmic and of regular meromorphic differential forms on a reduced hypersurface in a complex analytic manifold. In our paper, this approach is further developed to treat reduced complete intersections. In fact, the notions of the multi-logarithmic differential form and its residue generalize the corresponding definitions of J. Leray, J.-B. Poly and K. Saito; while a natural morphism between the complexes of multi-logarithmic and of regular meromorphic differential forms on a complete intersection is a generalization of an earlier argument of [1] applicable to the case of hypersurfaces. An explicit description of this morphism in terms of integration currents is given in our paper, based on results obtained in the papers [27], [28], [26], [5] and [18] cited above.

References

[1] Aleksandrov A.G.: Non-isolated hypersurface singularities. In: Theory of singularities and its applications. Advances in Soviet Mathematics (V.I. Arnold, ed.) 1, 211–246. Amer. Math. Soc., Providence, RI, 1990.

[2] Barlet D.: Faisceau ω_{X}^{•}, trace universelle et courants holomorphes sur un espace analytique réduit X de dimension pure, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 11, A579–A582.

[3] Barlet D.: Le faisceau ω_{X}^{•} sur un espace analytique X de dimension pure. Lecture Notes in Mathematics 670, Springer-Verlag, 1978, 187–204.

[4] Buchsbaum D.A., Rim D.S.: A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197–224.

[5] Coleff N.R., Herrera M.E.: Les courants résiduels associés à une forme méromorphe. Lecture Notes in Mathematics 633. Springer-Verlag, Berlin-New York, 1978.

[6] Dolbeault P.: Formes différentielles et cohomologie sur une variété analytique complexe. II, Ann. of Math. (2) 65 (1957), 282–330.

[7] Dolbeault P.: Valeurs principales et opérateurs différentiels semi-holo-morphes. In: Fonctions analytiques de plusieurs variables et analyse complexe (Colloq. Internat. CNRS, No. 208, Paris, 1972), pp. 35–50. “Agora Mathematica”, No. 1, Gauthier-Villars, Paris, 1974.

[8] El Haouzi A.: Sur la torsion des courants ¯∂-fermés sur un espace analytique complexe, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 3, 205–208.

[9] Grauert H., Remmert R.: Coherent analytic sheaves. Grundlehren der Mathematischen Wissenschaften 265, Springer-Verlag, Berlin-New York, 1984.

[10] Griffiths Ph.: Variations on a theorem of Abel, Invent. Math. 35 (1976), 321–390.

[11] Griffiths Ph., Harris J.: Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978.

[12] Henkin G.: Jean Leray and several complex variables. In: Selected papers of Jean Leray. Œuvres scientifiques. Vol. III. Several complex variables and holomorphic partial differential equations/Fonctions de plusieurs variables complexes et équations aux dérivées partielles holomorphes. (P.Malliavin, ed.) Springer-Verlag, Berlin; Société Mathématique de France, Paris, 1998.

[13] Henkin G., Passare M.: Abelian differentials on singular varieties and variations on a theorem of Lie-Griffiths, Invent. Math. 135 (1999), no. 2, 297–328.

[14] Kersken M.: Reguläre Differentialformen, Manuscripta Math. 46 (1984), no. 1, 1–25.

[15] Kunz E.: Holomorphe Differentialformen auf algebraischen Varietäten mit Singularitäten. I, Manuscripta Math. 15 (1975), 91–108.

[16] Kunz E.: Residuen von Differentialformen auf Cohen–Macaulay Varitäten, Math. Zeitschrift 152 (1977), 165–189.

[17] Leray J.: Le calcul différentiel et intégral sur une variété analytique complexe. (Problème de Cauchy. III), Bull. Soc. Math. France 87 (1959), 81–180.

[18] Passare M.: Residues, currents, and their relation to ideals of holomorphic functions, Math. Scand. 62 (1988), no. 1, 75–152.

[19] Peters C., Steenbrink J.H.M.: Infinitesimal variations of Hodge structure and the generic Torelli problem for projective hypersurfaces. Progress in Math. 39. Birkhäuser, 1983, 399–464.

[20] Poincaré H.: Sur les résidus des integrales doubles, Acta Math. 9 (1887), 321–380.

[21] Poly J.-B.: Sur un théorème de J.Leray en théorie des résidus, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A171–A174.

[22] de Rham G.: Sur la division de formes et de courants par une forme linéaire, Comment. Math. Helv. 28 (1954). 346–352.

[23] Saito K.: On a generalization of de-Rham lemma, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 2, 165–170.

[24] Saito K.: On the uniformization of complements of discriminant loci. In: Hyperfunctions and Linear partial differential equations, RIMS Kōkyūroku 287 (1977), 117–137.

[25] Saito K.: Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo, ser. IA 27 (1980), no. 2, 265–291.

[26] Trutnev V. M., Tsikh A.K.: On the structure of residue currents and functionals that are orthogonal to ideals in the space of holomorphic functions, Izv. Ross. Akad. Nauk Ser. Matem. 59 (1995), no. 5, 203–224.

[27] Tsikh A.K.: Weakly holomorphic functions on complete intersections, their holomorphic continuation, Mat. Sb. (N.S.) 133(175) (1987), no. 4, 429–445.

[28] Tsikh A.K.: Multidimensional residues and their applications. Translations of Mathematical Monographs 103, American Mathematical Society, 1992.

A.G.Aleksandrov  A.K.Tsikh
Institute of Control Sciences  Department of Mathematics
Russian Academy of Sciences  Krasnoyarsk State University
Profsoyuznaya ul. 65  pr. Svobodnyi 79
GSP-7, Moscow, 117806  660041 Krasnoyarsk,
Russian Federation  Russian Federation
E-mail: alexandr@ipu.rssi.ru  E-mail: tsikh@lan.krasu.ru