#### The \(\bar{\partial }\) -equation on a non-reduced analytic space

The ∂¯ -equation on a non-reduced analytic space
Mats Andersson 0
Richard Lärkäng 0
0 Division of Algebra and Geometry, Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg , 412 96 Göteborg , Sweden
Let X be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of ∂ -equation on X and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves AqX of (0, q)-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf OX . Our construction is based on intrinsic semi-global Koppelman formulas on X . Mathematics Subject Classification 32A26 · 32A27 · 32B15 · 32C30 Communicated by Ngaiming Mok.
1 Introduction
Let X be a smooth complex manifold of dimension n and let EX0,∗ denote the sheaf of
smooth (0, ∗)-forms. It is well-known that the Dolbeault complex
→ 0
(1.1)
The authors were partially supported by grants from the Swedish Research Council.
is exact, and hence provides a fine resolution of the structure sheaf OX . If X is a reduced
analytic space of pure dimension, then there is still a natural notion of “smooth forms”.
In fact, assume that X is locally embedded as i : X → , where is a pseudoconvex
domain in CN . If Ker i ∗ denotes the subsheaf of all smooth forms ξ in ambient space
such that i ∗ξ = 0 on the regular part Xreg of X , then one defines the sheaf EX of
smooth forms on X simply as
EX := E /Ker i ∗.
It is well-known that this definition is independent of the choice of embedding of X .
Currents on X are defined as the duals of smooth forms with compact support. It is
readily seen that the currents μ on X so defined are in a one-to-one correspondence
to the currents μˆ = i∗μ in ambient space such that μˆ vanish on Ker i∗, see, e.g.,
[
6
]. There is an induced ∂¯ -operator on smooth forms and currents on X . In particular,
(1.1) is a complex on X but in general it is not exact. In [
6
], Samuelsson and the first
author introduced, by means of intrinsic Koppelman formulas on X , fine sheaves AX∗
of (0, ∗)-currents that are smooth on Xreg and with mild singularities at the singular
part of X , such that
i ∂¯
0 → OX → AX0 →∂¯ AX1 →∂¯ · · · → AXn → 0
(1.2)
is exact, and thus a fine resolution of the structure sheaf OX . An immediate
consequence is the representation
H q (X, OX ) =
Ker A 0,q (X ) →∂¯ A 0,q+1(X )
Im A 0,q−1(X ) →∂¯ A 0,q (X )
, q ≥ 1,
(1.3)
of sheaf cohomology, and so (1.3) is a generalization of the classical Dolbeault
isomorphism. In special cases more qualitative information of the sheaves AXq are known,
see, e.g., [
5,23
].
Starting with the influential works [28,29] by Pardon and Stern, there has been a
lot of progress recently on the L2-∂¯ theory on non-smooth (reduced) varieties; see,
e.g., [
15,27,31
]. The point in these works, contrary to [
6
], is basically to determine the
obstructions to solve ∂¯ locally in L2. For a more extensive list of references regarding
the ∂¯ -equation on reduced singular varieties, see, e.g., [
6
].
In [17], a notion of the ∂¯ -equation on non-reduced local complete intersections was
introduced, and which was further studied in [
18
]. We discuss below how their work
relates to ours.
The aim of this paper is to extend the construction in [
6
] to a non-reduced
puredimensional analytic space. The first basic problem is to find appropriate definitions of
forms and currents on X . Let Xreg be the part of X where the underlying reduced space
Z is smooth, and in addition OX is Cohen–Macaulay. On Xreg the structure sheaf OX
has a structure as a free finitely generated OZ -module. More precisely, assume that
we have a local embedding i : X → ⊂ CN and coordinates (z, w) in such that
Z = {w = 0}. Let J be the defining ideal sheaf for X on . Then there are monomials
1, wα1 , . . . , wαν−1 such that each φ in O /J OX has a unique representation
φ = φˆ0 ⊗ 1 + φˆ1 ⊗ wα1 + · · · + φˆν−1 ⊗ wαν−1 ,
(1.4)
where φˆ j are in OZ . A reasonable notion of a smooth form on X should admit a similar
representation on Xreg with smooth forms φˆ j on Z . We first introduce the sheaves
EX0,∗ of smooth (0, ∗)-forms on X . By duality, we then obtain the sheaf CnX,∗ of (n,
∗)n,∗ of pseudomeromorphic
currents. We are mainly interested in the subsheaf PMnX,∗ of such currents with the
currents, and especially, the even more restricted sheaf WX
so-called standard extension property, SEP, on X . A current with the SEP is, roughly
speaking, determined by its restriction to any dense Zariski-open subset.
Of special interest is the sheaf ωnX ⊂ WXn,0 of ∂¯ -closed pseudomeromorphic (n,
0)currents. In the reduced case this is precisely the sheaf of holomorphic (n, 0)-forms
in the sense of Barlet–Henkin–Passare, see, e.g., [
12,16
].
We have no definition of “smooth (n, ∗)-form” on X . In order to define (0,
∗)currents, we use instead the sheaf ωnX in the following way. Any holomorphic function
defines a morphism in Hom(ωn , ωnX ), and it is a reformulation of a fundamental
X
result of Roos [30], that this morphism is indeed injective, and generically
surjec0,∗ induces a morphism in
tive. In the reduced case, multiplication by a current in WX
Hom(ωnX , WXn,∗), and in fact WX0,∗ → Hom(ωnX , WXn,∗) is an isomorphism. In the
non-reduced case, we then take this as the definition of WX0,∗. It turns out that with this
0,∗ admits a unique representation (1.4), where
definition, on Xreg, any element of WX
φˆ j are in WZ0,∗, see Sect. 6 below for details.
0,∗ we say that ∂¯ v = φ if ∂¯ (v ∧ h) = φ ∧ h for all h in ωn .
Given v, φ in WX X
Following [6] we introduce semi-global integral formulas and prove that if φ is a
0,q such that ∂¯ v = φ.
smooth ∂¯ -closed (0, q + 1)-form there is locally a current v in WX
A crucial problem is to verify that the integral operators preserve smoothness on Xreg
so that the solution v is indeed smooth on Xreg. By an iteration procedure as in [
6
] we
0,k and obtain our main result in this paper.
can define sheaves AXk ⊂ WX
Theorem 1.1 Let X be an analytic space of pure dimension n. There are sheaves
AXk ⊂ WX0,k that are modules over EX0,∗, coinciding with EX0,k on Xreg, and such that
(1.2) is a resolution of the structure sheaf OX .
The main contribution in this article compared to [
6
] is the development of a theory
for smooth (0, ∗)-forms and various classes of (n, ∗)- and (0, ∗)-currents in the
nonreduced case as is described above. This is done in Sects. 4–8. The construction of
integral operators to provide solutions to ∂¯ in Sect. 9 and the construction of the fine
resolution of OX in Sect. 11, which proves Theorem 1.1, are done pretty much in the
same way as in [
6
]. The proof of the smoothness of the solutions of the regular part
in Sect. 10 however becomes significantly more involved in the non-reduced case and
requires completely new ideas. In Sect. 12 we discuss the relation to the results in
[
17,18
] in case X is a local complete intersection.
2 Pseudomeromorphic currents
Let s1, . . . , sm be coordinates in Cm , let α be a smooth form with compact support,
and let a1, . . . , ar be positive integers, 0 ≤ ≤ r ≤ m. Then
1 1 α
∂¯ s1a1 ∧ · · · ∧ ∂¯ sa ∧ sa++11 · · · srar
is a well-defined current that we call an elementary (pseudomeromorphic) current.
Let Z be a reduced space of pure dimension. A current τ is pseudomeromorphic on
Z if, locally, it is the push-forward of a finite sum of elementary pseudomeromorphic
currents under a sequence of modifications, simple projections, and open inclusions.
The pseudomeromorphic currents define an analytic sheaf PMZ on Z . This sheaf was
introduced in [
8
] and somewhat extended in [
6
]. If nothing else is explicitly stated,
proofs of the properties listed below can be found in, e.g., [
6
].
If τ is pseudomeromorphic and has support on an analytic subset V , and h is a
holomorphic function that vanishes on V , then h¯τ = 0 and dh¯ ∧ τ = 0.
Given a pseudomeromorphic current τ and a subvariety V of some open subset
U ⊂ Z , the natural restriction to the open set U \V of τ has a natural extension to a
pseudomeromorphic current on U that we denote by 1U\V τ . Throughout this paper
we let χ denote a smooth function on [0, ∞) that is 0 in a neighborhood of 0 and 1 in
a neighborhood of ∞. If h is a holomorphic tuple whose common zero set is V , then
(2.1)
(2.2)
(2.3)
(2.4)
1U\V τ = lim χ (|h|2/ )τ.
→0+
1V 1W τ = 1V ∩W τ.
1V (ξ ∧ τ ) = ξ ∧ 1V τ.
Notice that 1V τ := (1 − 1U\V )τ is also pseudomeromorphic and has support on V .
If W is another analytic set, then
This action of 1V on the sheaf of pseudomeromorphic currents is a basic tool. In fact
one can extend this calculus to all constructible sets so that (2.2) holds, see [
8
]. One
readily checks that if ξ is a smooth form, then
If f : Z → Z is a modification and τ is in PMZ then f∗τ is in PMZ . The same
holds if f is a simple projection and τ has compact support in the fiber direction. In
any case we have
1V f∗τ = f∗(1 f −1V τ ).
It is not hard to check that if τ is in PMZ and τ is in PMZ , then τ ⊗ τ is in
PMZ×Z , see, e.g., [4, Lemma 3.3]. If V ⊂ U ⊂ Z and V ⊂ U ⊂ Z , then
(1V τ ) ⊗ 1V τ = 1V ×V (τ ⊗ τ ).
(2.5)
Another basic tool is the dimension principle, that states that if τ is a pseudomero
morphic (∗, p)-current with support on an analytic set with codimension larger than
p, then τ must vanish.
A pseudomeromorphic current τ on Z has the standard extension property, SEP, if
1V τ = 0 for each germ V of an analytic set with positive codimension on Z . The set
WZ of all pseudomeromorphic currents on Z with the SEP is a subsheaf of PMZ .
By (2.3), WZ is closed under multiplication by smooth forms.
Let f be a holomorphic function (or a holomorphic section of a Hermitian line
bundle), not vanishing identically on any irreducible component of Z . Then 1/ f , a
priori defined outside of { f = 0}, has an extension as a pseudomeromorphic current,
the principal value current, still denoted by 1/ f , such that 1{ f =0}(1/ f ) = 0. The
current 1/ f has the SEP and
1 1
f = →lim0+ χ (| f |2/ ) f .
We say that a current a on Z is almost semi-meromorphic if there is a modification
π : Z → Z , a holomorphic section f of a line bundle L → Z and a smooth
form γ with values in L such that a = π∗(γ / f ), cf., [10, Section 4]. If a is almost
semi-meromorphic, then it is clearly pseudomeromorphic. Moreover, it is smooth
outside an analytic set V ⊂ Z of positive codimension, a is in WZ , and in particular,
a = lim →0+ χ (|h|/ )a if h is a holomorphic tuple that cuts out (an analytic set of
positive codimension that contains) V . The Zariski singular support of a is the Zariski
closure of the set where a is not smooth.
One can multiply pseudomeromorphic currents by almost semi-meromorphic
currents; and this fact will be crucial in defining WX0,∗, when X is non-reduced. Notice
that if a is almost semi-meromorphic in Z then it also is in any open U ⊂ Z .
Proposition 2.1 ([10, Theorem 4.8, Proposition 4.9]) Let Z be a reduced space,
assume that a is an almost semi-meromorphic current in Z , and let V be the Zariski
singular support of a.
(i) If τ is a pseudomeromorphic current in U ⊂ Z , then there is a unique
pseudomeromorphic current a ∧ τ in U that coincides with (the naturally defined
current) a ∧ τ in U \V and such that 1V (a ∧ τ ) = 0.
(ii) If W ⊂ U is any analytic subset, then
Notice that if h is a tuple that cuts out V , then in view of (2.1),
It follows that if ξ is a smooth form, then
1W (a ∧ τ ) = a ∧ 1W τ.
a ∧ τ = lim χ (|h|2/ )a ∧ τ.
→0+
ξ ∧ (a ∧ τ ) = (−1)deg ξ deg aa ∧ (ξ ∧ τ ).
(2.6)
(2.7)
(2.8)
For future reference we will need the following result.
Proposition 2.2 Let Z be a reduced space. Then PMZ = WZ + ∂¯ WZ .
Proof First assume that Z is smooth. Since WZ is closed under multiplication by
smooth forms, so is WZ + ∂¯ WZ . The statement that PMZ = WZ + ∂¯ WZ is local, and
since both sides are closed under multiplication by cutoff functions, we may consider
a pseudomeromorphic current μ with compact support in Cn. If μ has bidegree (∗, 0),
then it is in WZ in view of the dimension principle. Thus we assume that μ has bidegree
(∗, q) with q ≥ 1. Let
K μ(z) =
ζ
k(ζ, z) ∧ μ(ζ ),
(2.9)
where k is the Bochner–Martinelli kernel. Here (2.9) means that K μ = p∗(k ∧ μ ⊗ 1),
where p is the projection Cζn × Cnz → Cnz, (ζ, z) → z. Recall that we have the
Koppelman formula μ = ∂¯ K μ + K (∂¯ μ). It is thus enough to see that K μ is in WZ
if μ is pseudomeromorphic. Let χ = χ (|ζ − z|2/ ). It is easy to see, by a blowup of
Cn × Cn along the diagonal, that k is almost semi-meromorphic on Cn × Cn. Thus,
by (2.7), χ k ∧ (μ ⊗ 1) → k ∧ (μ ⊗ 1). In view of Proposition 2.1 it follows that
k ∧ (μ ⊗ 1) is pseudomeromorphic. Finally, if W is a germ of a subvariety of Cn of
positive codimension, then by (2.4) and (2.5),
1W p∗(k ∧ μ ⊗ 1) = →lim0+ p∗ (1Cn×W (χ k ∧ (μ ⊗ 1)))
= →lim0+ p∗ (χ k ∧ (1Cn×W μ ⊗ 1))
= →lim0+ p∗ (χ k ∧ (1Cn μ ⊗ 1W 1)) = 0,
since 1W 1 = 0. Thus K μ is in WZ .
If Z is not smooth, then we take a smooth modification π : Z → Z . For any μ in
PMZ there is some μ in PMZ such that π∗μ = μ, see [4, Proposition 1.2]. Since
μ = τ + ∂¯ u with τ, u in WZ , we have that μ = π∗τ + ∂¯ π∗u.
2.1 Pseudomeromorphic currents with support on a subvariety
Let be an open set in CN and let Z be a (reduced) subvariety of pure dimension
n. Let PMZ denote the sheaf of pseudomeromorphic currents τ on with support
on Z , and let W Z denote the subsheaf of PMZ of currents of bidegree (N , ∗) with
the SEP with respect to Z , i.e., such that 1W τ = 0 for all germs W of subvarieties
of Z of positive codimension. The sheaf CHZ of Coleff–Herrera currents on Z is the
subsheaf of W Z of ∂¯ -closed (N , p)-currents, where p = N − n.
Remark 2.3 In [
3,6
] CHZ denotes the sheaf of pseudomeromorphic (0, p)-currents
with support on Z and the SEP with respect to Z . If this sheaf is tensored by the
canonical bundle K we get the sheaf CHZ in this paper. Locally these sheaves are
thus isomorphic via the mapping μ → μ∧α, where α is a non-vanishing holomorphic
(N , 0)-form.
We have the following direct consequence of Proposition 2.1.
Proposition 2.4 Let Z ⊂ be a subvariety of pure dimension, let a be almost
semimeromorphic in , and assume that it is smooth generically on Z . If τ is in W Z , then
a ∧ τ is in W Z as well.
Assume that we have local coordinates (z, w) ∈ Cn × C p in
0}. We will use the short-hand notation
such that Z = {w =
for multiindices γ = (γ1, . . . , γ p) with γ j ≥ 0, and let γ ! := γ1! · · · γ p!. Notice that
for test forms ξ . If τ is in WZ , then it follows by (2.5) and the fact that
supp ∂¯ (1/wγ +1) = {w = 0} that τ ⊗ ∂¯ (1/wγ +1) is in W Z . We have the
following local structure result, see [11, Proposition 4.1 and (4.3)] and [10, Theorem 3.5].
Proposition 2.5 Assume that we have local coordinates (z, w) such that Z = {w =
0}. Then τ in W Z has a unique representation as a finite sum
τ =
γ
dw
τγ ∧ d z ⊗ ∂¯ wγ +1 , τγ ∈ WZ0,∗,
where d z := d z1 ∧ · · · ∧ d zn. If π is the projection (z, w) → z, then
τγ ∧ d z = (2π i )− pπ∗(wγ τ ).
If in addition ∂¯ τ is in W Z then its coefficients in the expansion (2.11) are ∂¯ τγ , cf.,
(2.12). In particular, ∂¯ τ = 0 if and only if ∂¯ τγ = 0 for all γ .
Let us now consider the pairing between W Z and germs φ at Z of smooth (0,
∗)forms. We assume that Z is smooth and that we have coordinates (z, w) as before, that
τ is in W Z , and that (2.11) holds. Moreover, we assume that φ is a smooth (0, ∗)-form
in a neighborhood of Z in . For any positive integer M we have the expansion
φ =
φα(z) ⊗ wα + O |w|M
+ O(w¯ , dw¯ ),
(2.13)
|α|<M
(2.10)
(2.11)
(2.12)
where
1 ∂φ
φα(z) = α! ∂wα (z, 0)
and O(w¯ , dw¯ ) denotes a sum of terms, each of which contains a factor w¯ j or dw¯ j for
some j . If M in (2.13) is chosen so that O(|w|M )τ = 0, then
i.e.,
Thus φ ∧ τ = 0 if and only if
of conditions!).
φ ∧ τ =
φ ∧ τ =
α≤γ
,
φγ ∧ τ +γ ∧ d z ⊗ ∂¯ w +1
.
dw
(2.14)
γ ≥0 φγ ∧ τ +γ = 0 for all (which is a finite number
2.2 Intrinsic pseudomeromorphic currents on a reduced subvariety
Currents on a reduced analytic space Z are defined as the dual of the sheaf of test
forms. If i : Z → Y is an embedding of a reduced space Z into a smooth manifold
Y , then the push-forward mapping τ → i∗τ gives an isomorphism between currents
τ on Z and currents μ on Y such that ξ ∧ μ = 0 for all ξ in EY such that i ∗ξ = 0.
When defining pseudomeromorphic currents in the non-reduced case it is desirable
that it coincides with the previous definition in case Z is reduced. From [4,
Theorem 1.1] we have the following description of pseudomeromophicity from the point
of view of an ambient smooth space.
Proposition 2.6 Assume that we have an embedding i : Z → Y of a reduced space
Z into a smooth manifold Y .
(i) If τ is in PMZ , then i∗τ is in PMY .
(ii) If τ is a current on Z such that i∗τ is in PMY and 1Zsing (i∗τ ) = 0, then τ is in
PMZ .
Since i∗(i ∗χ (|h|2/ )τ ) = χ (|h|2/ )i∗τ for any current τ on Z , we get by (2.1)
that for a subvariety V ⊂ U ⊂ Z ,
1V (i∗τ ) = i∗(1V τ ),
(2.15)
i.e., (2.4) holds also for an embedding i : Z → Y . The condition 1Zsing (i∗τ ) = 0 in
(ii) is fulfilled if i∗τ has the SEP with respect to Z .
Corollary 2.7 We have the isomorphism
i∗ : WZn,∗ → Hom(O /J , W Z ),
where J is the ideal defining Z in .
Notice that Hom(O /J , W Z ) is precisely the sheaf of μ in W Z such that J μ = 0.
Proof The map i∗ is injective, since it is injective on any currents, and it maps into
Hom(O /J , W Z ) by (2.15).
To see that i∗ is surjective, we take a μ in Hom(O /J , W Z ). We assume first that
we are on Zreg, with local coordinates such that Zreg = {w = 0}. If ξ is in E 0,∗ and
i ∗ξ = 0, then ξ is a sum of forms with a factor dw¯ j , w j or w¯ j . Since w j ∈ J , w j
annihilates μ by assumption, and since w j vanishes on the support of μ, w¯ j and dw¯ j
annihilate μ since μ is pseudomeromorphic. Thus, μ.ξ = 0, so μ = i∗τ for some
current τ on Z . By Proposition 2.6 (ii), τ is pseudomeromorphic, and by (2.15), has
the SEP, i.e., τ is in WZn,∗.
Remark 2.8 We do not know whether i∗τ ∈ PMZ implies that τ ∈ PMZ .
By [11, Proposition 3.12 and Theorem 3.14], we get
Proposition 2.9 Let ϕ and φ1, . . . , φm be currents in WZ . If ϕ = 0 on the set on Zreg
where φ1, . . . , φm are smooth, then ϕ = 0.
3 Local embeddings of a non-reduced analytic space
Let X be an analytic space of pure dimension n with structure sheaf OX and let
Z = Xred be the underlying reduced analytic space. For any point x ∈ X there is, by
definition, an open set ⊂ CN and an ideal sheaf J ⊂ O of pure dimension n with
zero set Z such that OX is isomorphic to O /J , and all associated primes of J at any
point have dimension n. We say that we have a local embedding i : X → ⊂ CN
at x . There is a minimal such N , called the Zariski embedding dimension Nˆ of X at
x , and the associated embedding is said to be minimal. Any two minimal embeddings
are identical up to a biholomorphism, and any embedding i : X → has locally at x
the form
j
X →
ι
→
:=
× U , i = ι ◦ j,
(3.1)
where j is minimal, U is an open subset of Cmw, m = N − Nˆ , and the ideal in is
J = J ⊗1+(w1, . . . , wm ). Notice that we then also have embeddings Z → → ;
however, the first one is in general not minimal.
Now consider a fixed local embedding i : X → ⊂ CN , assume that Z is smooth,
and let (z, w) be coordinates in such that Z = {w = 0}. We can identify OZ with
holomorphic functions of z, and we can define an injection
OZ → OX , φ (z) → φ˜ (z, w) = φ (z).
In this way OX becomes an OZ -module, which however depends on the choice of
coordinates.
Proposition 3.1 Assume that Z is smooth. Let OX have the OZ -module structure from
a choice of local coordinates as above. Then OX is a coherent OZ -module, and OX is
a free OZ -module at x if and only if OX is Cohen–Macaulay at x .
Recall that f1, . . . , fm ∈ R is a regular sequence on the R-module M if fi is a non
zero-divisor on M/( f1, . . . , fi−1) for i = 1, . . . , m, and ( f1, . . . , fm )M = M . If R is
a local ring, then depthR M is the maximal length d of a regular sequence f1, . . . , fd
such that f1, . . . , fd are contained in the maximal ideal m; furthermore, M is Cohen–
Macaulay if depthR M = dim R M , where dim R M = dim R (R/ann R M ). If R is
Cohen–Macaulay, and M has a finite free resolution over R, then the Auslander–
Buchsbaum formula, [14, Theorem 19.9], gives that
depthR M + pdR M = dim R R,
(3.2)
where pdR M is the length of a minimal free resolution of M over R. In this case, M
is Cohen–Macaulay as an R-module if and only if M has a free resolution over R of
length codim M .
Remark 3.2 Notice that if we have a local embedding i : X → as above, then the
depth and dimension of OX,x = O ,x /J as an O ,x -module coincide with the depth
and dimension of OX,x as an OX,x -module. Thus OX,x is Cohen–Macaulay as an
OX,x -module if and only if it is Cohen–Macaulay as an O ,x -module, and this holds
in turn if and only if O ,x /J has a free resolution of length N − n.
Proof of Proposition 3.1 By the Nullstellensatz there is an M such that wα is in J
in some neighborhood of x if |α| = M . Let M ⊂ O be the ideal generated by
{wα; |α| = M }. Then M = O /M is a free, finitely generated OZ -module. Thus,
O /J M /J M is a coherent OZ -module, which we note is generated by the
finite set of monomials wα such that |α| < M .
We shall now show that
depthOX,x OX,x = depthOZ,x OX,x
dimOX,x OX,x = dimOZ,x OX,x .
(3.3)
(3.4)
and
We claim that a sequence f1, . . . , fm in OX,x is regular (on OX,x ) if and only if
f˜1, . . . , f˜m ∈ OZ,x is regular on OX,x , where f˜j (z) = f j (z, 0). In fact, since OX,x
has pure dimension, a function g ∈ OX,x = O ,x /J is a non zero-divisor if and only
if g is generically non-vanishing on each irreducible component of Z (J ). Thus f1 is
a non zero-divisor if and only if f˜1 is. If it is, then OX,x /( f1) = O ,x /(J + ( f1))
again has pure dimension. Thus the claim follows by induction, and the fact that
Z (J + ( f1, . . . , fk )) = Z (J + ( f˜1, . . . , f˜k )). The claim immediately implies (3.3).
To see (3.4), we note first that dimOX,x OX,x is just the usual (geometric)
dimension of X or Z , i.e., in this case, n. Now, ann OZ,x OX,x = {0}, so dimOZ,x OX,x =
dimOZ,x OZ,x /(ann OZ,x OX,x ) = dimOZ,x OZ,x = n.
From (3.3) and (3.4) we conclude that OX,x is Cohen–Macaulay as an OZ,x -module if and only if it is Cohen–Macaulay (as an OX,x -module). Hence, by (3.2), with
R = OZ,x and M = OX,x ,
depthOZ,x OX,x + pdOZ,x OX,x = n,
so OX,x is Cohen–Macaulay as an OZ,x -module if and only if pdOZ,x OX,x = 0, that
is, if and only if OX,x is a free OZ,x -module.
In the proof above, we saw that OX is generated (locally) as an OZ -module by all
monomials wα with |α| ≤ M for some M .
Corollary 3.3 Assume that 1, wα1 , . . . , wαν−1 is a minimal set of generators at a
given point x (clearly 1 must be among the generators!). Then we have a unique
representation (1.4) for each φ ∈ OX,x if and only if OX,x is Cohen–Macaulay.
By coherence it follows that if OX,x is free as an OZ,x -module, then OZ,x is free
as an OZ,x -module for all x in a neighborhood of x , and 1, wα1 , . . . , wαν−1 is a basis
at each such x .
Example 3.4 Let J be the ideal in C4 generated by (w12, w22, w1w2, w1z2 − w2z1).
It is readily checked that OX is a free OZ -module at a point on Z = {w1 = w2 = 0}
where z1 or z2 is = 0. If, say, z1 = 0, then we can take 1, w1 as generators. At the
point z = (0, 0), e.g., 1, w1, w2 form a minimal set of generators, and then OX is not
a free OZ -module, since there is a non-trivial relation between w1 and w2.
We claim that OX has pure dimension. That is, we claim that there is no embedded
associated prime ideal at (0, 0); since Z is irreducible, this is the same as saying that J
is primary with respect to Z . To see the claim, let φ and ψ be functions such that φψ is
in J and ψ is not in √J . The latter assumption means, in view of the Nullstellensatz,
that ψ does not vanish identically on Z , i.e., ψ = a(z) + O(w), where a does not
vanish identically. Since in particular φψ must vanish on Z it follows that φ = O(w).
It is now easy to see that φ is in J . We conclude that J is primary.
The pure-dimensionality of OX can also be rephrased in the following way: If φ is
holomorphic and is 0 generically, then φ = 0. If we delete the generator w1w2 from
the definition of J in the example, then φ = w1w2 is 0 generically in O /J but is
not identically zero. Thus J then has an embedded primary ideal at (0, 0).
Example 3.5 Let = C2z,w and J = (w2) so that Z = {w = 0}. Then 1, w is
a basis for OX = OC2 /(w2) so each function φ in OX has a unique representation
a0(z) ⊗ 1 + a1(z) ⊗ w. Let us consider the new coordinates ζ = z − w, η = w. Then
J = (η2) and since
a0(z) + a1(z)w = a0(ζ + η) + a1(ζ + η)η = a0(ζ ) + (∂a0/∂ζ )(ζ )η + a1(ζ )η + J
we have the representation a0(ζ ) ⊗ 1 + (a1(ζ ) + ∂a0/∂ζ )(ζ ) ⊗ η with respect to
(ζ, η).
More generally, assume that, at a given point in Xreg ⊂ , we have two different
choices (z, w) and (ζ, η) of coordinates so that Z = {w = 0} = {η = 0}, and bases
1, . . . , wαν−1 and 1, . . . , ηβν−1 for OX as a free module over OZ . Then there is a ν ×
νmatrix L of holomorphic differential operators so that if (a j ) is any tuple in (OZ )ν and
(b j ) = L(a j ), then a0 ⊗ 1 + · · · + aν−1 ⊗ wαν−1 = b0 ⊗ 1 + · · · + bν−1 ⊗ ηβν−1 + J .
4 Smooth (0, ∗)-forms on a non-reduced space X
Let i : X → be a local embedding of X . In order to define the sheaf of smooth
(0, ∗)-forms on X , in analogy with the reduced case, we have to state which smooth
(0, ∗)-forms in “vanish” on X , or more formally, give a meaning to i ∗ = 0.
We will see, cf., Lemma 4.8 below, that the suitable requirement is that locally on
Xreg, belongs to E 0,∗J + E 0,∗J¯Z + E 0,∗dJ¯Z , where Jz is the ideal sheaf defining
Z . However, it turns out to be more convenient to represent the sheaf Ker i ∗ of such
forms as the annihilator of certain residue currents, and this is the path we will follow.
Moreover, these currents play a central role themselves later on.
The following classical duality result is fundamental for this paper; see, e.g., [
3
]
for a discussion.
Proposition 4.1 If J has pure dimension, then
J = ann O Hom(O /J , CHZ ).
That is, φ is in J if and only if φμ = 0 for all μ in Hom(O /J , CHZ ). It is also
well-known, see, e.g., [3, Theorem 1.5], that
so Hom(O /J , CHZ ) is a coherent analytic sheaf. Locally we thus have a finite
number of generators μ1, . . . , μm . In Example 6.9, we compute explicitly such generators
for the ideal J in Example 3.4.
Let ξ be a smooth (0, ∗)-form in . Without first giving meaning to i ∗, we define
the sheaf Ker i ∗ by saying that ξ is in Ker i ∗ if
ξ ∧ μ = 0, μ ∈ Hom(O /J , CHZ ).
Notice that if ξ is holomorphic, then, in view of the duality (4.1), ξ is in Ker i ∗ if and
only if ξ is in J .
Definition 4.2 We define the sheaf of smooth (0, ∗)-forms on X as
EX0,∗ := E 0,∗/Ker i ∗.
(4.1)
(4.2)
(4.3)
We will prove below that this sheaf is independent of the choice of embedding and
thus intrinsic on X .
Given φ in E 0,∗, let i ∗φ be its image in EX0,∗. In particular, i ∗ξ = 0 means that ξ
belongs to Ker i ∗, which then motivates this notation. Notice that Ker i ∗ is a
twosided ideal in E 0,∗, i.e., if φ is in E 0,∗ and ξ is in Ker i ∗, then φ ∧ ξ and ξ ∧ φ are in
Ker i ∗. It follows that we have an induced wedge product on EX0,∗ such that
i ∗(φ ∧ ξ ) = i ∗φ ∧ i ∗ξ.
Remark 4.3 It follows from Lemma 4.8 below that in case X = Z is reduced, then ξ
is in Ker i ∗ if and only its pullback to Xreg vanishes. Thus our definition of EX0,∗ is
consistent with the usual one in that case.
Lemma 4.4 Using the notation of (3.1),
ι∗ : HomO (O /J , W Z ) → HomO (O /J , W Z )
(4.4)
We can realize the mapping in (4.4) as the tensor product τ → τ ∧ [w = 0], where
[w = 0] is the Lelong current in associated with the submanifold {w = 0}.
Proof To begin with, ι∗ maps pseudomeromorphic (Nˆ , pˆ + )-currents with support
on Z ⊂ to pseudomeromorphic (N , p + )-currents with support on Z ⊂ . If, in
addition, τ has the SEP with respect to Z , then ι∗τ has, as well by (2.15). Moreover, if
τ is annihilated by J , then ι∗τ is annihilated by J = J ⊗ 1 + (w). Thus the mapping
(4.4) is well-defined, and it is injective since ι is injective.
Now assume that μ is in Hom(O /J , W Z ). Arguing as in the proof of
Corollary 2.7, we see that μ = ι∗μˆ for a current μˆ in W Z . Since J = ι∗J and J μ = 0, it
follows that J μˆ = 0. Thus (4.4) is surjective.
Since ι∗ is injective, ∂¯ τ = 0 if and only if ∂¯ ι∗τ = 0, and thus we get
Corollary 4.5 Using the notation of (3.1),
ι∗ : HomO (O /J , CHZ ) → HomO (O /J , CHZ )
is an isomorphism.
Corollary 4.6 Using the notation in (3.1),
ι∗ : E 0,∗/Ker i ∗ → E 0,∗/Ker j ∗,
Proof It follows immediately from (4.5) that the mapping (4.6) is well-defined and
injective. Given ξ in E 0,∗, let ξ = ξ ⊗ 1. Then ι∗ξ = ξ and so (4.6) is indeed surjective
as well.
It follows from (4.6) and (4.3) that the sheaf EX0,∗ is intrinsically defined on X .
Since ∂¯ maps Ker i ∗ to Ker i ∗, we have a well-defined operator ∂¯ : EX0,∗ → EX0,∗+1
such that ∂¯ 2 = 0. Unfortunately the sheaf complex so obtained is not exact in general,
see, e.g., [6, Example 1.1] for a counterexample already in the reduced case.
(4.5)
(4.6)
4.1 Local representation on X reg of smooth forms
Recall that Xreg is the open subset of X , where the underlying reduced space is
smooth and OX is Cohen–Macaulay. Let us fix some point in Xreg, and assume that
we have local coordinates (z, w) such that Z = {w = 0}. We also choose generators
1, wα1 , . . . , wαν−1 of OX as a free OZ -module, which exist by Corollary 3.3, and
generators μ1, . . . , μm of Hom(O /J , CHZ ).
Notice that for each smooth (0, ∗)-form in , → ∧ μ only depends on its
class φ in EX0,∗, and φ is in fact determined by these currents. By Proposition 2.5 each
of these currents can (locally) be represented by a tuple of currents in WZ0,∗. Putting
all these tuples together, we get a tuple in (WZ0,∗)M , where M = M1 + · · · + Mm and
M j is the number of indices in (2.11) in the representation of μ j .
Recall from Corollary 3.3 that φ in OX has a unique representative
φˆ = φˆ0 + φˆ1 ⊗ wα1 + · · · + φˆν−1 ⊗ wαν−1 ,
where φˆ j are in OZ . We thus have an OZ -linear morphism
T : (OZ )ν → (OZ )M .
The morphism is injective by Proposition 4.1, and the holomorphic matrix T is
therefore generically pointwise injective.
Lemma 4.7 Each φ in EX0,∗ has a unique representation (4.7) where φˆ j are in EZ0,∗.
Proof To begin with notice that a given smooth φ must have at least one such
representation. In fact, taking the finite Taylor expansion (2.13) we can forget about high
order terms, since they must annihilate all the μ j , and the terms w¯ and dw¯ annihilate
all the μ j as well since they are pseudomeromorphic with support on {w = 0}. On
the other hand, each wα not in the set of generators must be of the form
wα = a0 + a1 ⊗ wα1 + · · · + aν−1 ⊗ wαν−1 + J ,
and hence φα ⊗ wα is of the form (4.7). Thus the representation exists. To show
uniqueness of the representation, we assume that φˆ is in Ker i ∗. Then the tuple (φˆ j )
is mapped to 0 by the matrix T , and since T is generically pointwise injective we
conclude that each φˆ j vanishes.
By the above proof we get
Lemma 4.8 A smooth (0, ∗)-form ξ in
E 0,∗J¯Z + E 0,∗dJ¯Z on Xreg, where JZ is the radical sheaf of Z . is in Ker i ∗ if and only if ξ is in E 0,∗J +
Remark 4.9 This is not the same as saying that ξ is in E 0,∗J + E 0,∗J¯Z + E 0,∗dJ¯Z at
singular points. For a simple counterexample, consider φ = x y¯ on the reduced space
Z = {x y = 0} ⊂ C2.
(4.7)
(4.8)
However, this can happen also when Z is irreducible at a point. For example, the
variety Z = {x 2 y − z2 = 0} ⊂ C3 is irreducible at 0, but there exist points arbitrarily
close to 0 such that (Z , z) is not irreducible. In this case, the ideal of smooth functions
vanishing on (Z , 0) is strictly larger than E 0,0JZ,0 + E 0,0J¯Z,0 see [26, Proposition 9,
Chapter IV], and [25, Theorem 3.10, Chapter VI].
Remark 4.10 It is easy to check that if we have the setting as in the discussion at the end
of Sect. 3 but (a j ) is instead a tuple in EZ0,∗, then we can still define (b j ) = L(a j ) if we
consider the derivatives in L as Lie derivatives; in fact, since a j has no holomorphic
differentials, L only acts on the smooth coefficients, and it is easy to check that
a0 ⊗ 1 + · · · + aν−1 ⊗ wαν−1 and b0 ⊗ 1 + · · · + bν−1 ⊗ ηβν−1 are equal modulo
E 0,∗J + E 0,∗J¯Z + E 0,∗dJ¯Z , and thus define the same element in EX0,∗.
For future needs we prove in Sect. 6.1:
Lemma 4.11 The morphism T is pointwise injective.
We can thus choose a holomorphic matrix A such that
is pointwise exact, and we can also find holomorphic matrices S and B such that
0 → OZ → OZM →A OZM
ν T
I = T S + B A.
(4.9)
(4.10)
5 Intrinsic (n, ∗)-currents on X
In analogy with the reduced case we have the following definition when X is possibly
non-reduced.
test forms, i.e., forms in EX0,n−q with compact support.
Definition 5.1 The sheaf CnX,q of (n, q)-currents on X is the dual sheaf of (0, n −
q)
Here, just as in the case of reduced spaces, cf., for example [19, Section 4.2], the
space of smooth forms EX0,n−q is equipped with the quotient topology induced by a
local embedding.
More concretely, this means that given an embedding i : X → , currents ψ in
CnX,q precisely correspond to the (N , N − n +q)-currents τ on that vanish on Ker i ∗.
Since Ker i ∗ is a two-sided ideal in E 0,∗ this holds if and only if ξ ∧ τ = 0 for all ξ
in Ker i ∗. It is natural to write τ = i∗ψ so that
i∗ψ.ξ = ψ.i ∗ξ.
n,q
Clearly, we get a mapping ∂¯ : CX
n,q+1 such that ∂¯ 2 = 0.
→ CX
Proposition 5.2 If τ is in W Z and J τ = 0, then ξ ∧ τ = 0 for all smooth ξ such
that i ∗ξ = 0.
Proof Because of the SEP it is enough to prove that ξ ∧τ = 0 on Xreg. By assumption,
J annihilates τ , and by general properties of pseudomeromorphic currents, since τ has
support on Z , J¯Z and dJ¯Z annihilate τ . Thus the proposition follows by Lemma 4.8.
Definition 5.3 An (n, ∗)-current ψ on X is in WXn,∗ if i∗ψ is in Hom(O /J , W Z ).
By definition we thus have the isomorphism
n,∗
i∗ : WX
Hom(O /J , W Z ).
n,∗ is intrinsically defined.
It follows from Lemma 4.4 that WX
Remark 5.4 By Corollary 2.7, this definition is consistent with the previous definition
n,∗ when X is reduced. We cannot define PMnX,∗ in the analogous simple way,
of WX
cf., Remark 2.8.
n,∗ and a is an almost semi-meromorphic (0, ∗)-current
Definition 5.5 If ψ is in WX
on that is generically smooth on Z , then the product a ∧ ψ is a current in WXn,∗
defined as follows: By definition, i∗ψ is in Hom(O /J , W Z ) and by Proposition 2.4
and (2.8), one can define a ∧ i∗ψ in Hom(O /J , W Z ); now a ∧ ψ is the unique
n,∗ such that i∗(a ∧ ψ ) = a ∧ i∗ψ .
current in WX
By (2.7),
a ∧ ψ = lim χ (|h|2/ )a ∧ ψ
→0+
if h cuts out the Zariski singular support of a.
Definition 5.6 We let ωnX be the sheaf of ∂¯ -closed currents in WXn,0.
This sheaf corresponds via i∗ to ∂¯ -closed currents in Hom(O /J , W Z ) so we have
the isomorphism
i∗ : ωnX
Hom(O /J , CHZ ).
forms ξ in EX0,∗ and τ in PMnX,∗.
When X is reduced ωnX is the sheaf of (n, 0)-forms that are ∂¯ -closed in the Barlet–
Henkin–Passare sense. Let μ1, . . . , μm be a set of generators for Hom(O /J , CHZ ).
They correspond via (5.3) to a set of generators h1, . . . , hm for the OX -module ωn .
X
We will also need a definition of PMnX,∗. Let FX be the subsheaf of CnX,∗ of τ such
that i∗τ is in PMZ . If τ is a section of FX and W is a subvariety of some open subset
of Z , then 1W i∗τ is in PMZ , and by (2.3), 1W i∗τ is annihilated by Ker i ∗. Hence we
can define 1W τ as the unique current in FX such that i∗1W τ = 1W i∗τ . Clearly, 1W τ
has support on W and it is easily checked that the computational rule (2.3) holds also
in FX . Moreover, FX is closed under ∂¯ since PMZ is.
Definition 5.7 The sheaf PMnX,∗ is the smallest subsheaf of FX that contains WX
n,∗
and is closed under ∂¯ and multiplication by 1W for all germs W of subvarieties of Z .
In view of Proposition 2.2 this definition coincides with the usual definition in case
X is reduced. It is readily checked that the dimension principle holds for FX , and hence
it also holds for the (possibly smaller) sheaf PMnX,∗, and in addition, (2.3) holds for
(5.1)
(5.2)
(5.3)
6 Structure form on X
Let i : X → ⊂ CN be a local embedding as before, let p = N − n be the
codimension of X , and let J be the associated ideal sheaf on . In a slightly smaller
set, still denoted , there is a free resolution
fN0 f3 f2 f1
0 → O(E N0 ) −→ · · · −→ O(E2) −→ O(E1) −→ O(E0)
(6.1)
of O /J ; here Ek are trivial vector bundles over
This resolution induces a complex of vector bundles
and E0 is the trivial line bundle.
fN0 f3 f2 f1
0 → E N0 −→ · · · −→ E2 −→ E1 −→ E0
that is pointwise exact outside Z . Let Xk be the set where fk does not have optimal
rank. Then
· · · ⊂ Xk+1 ⊂ Xk ⊂ · · · ⊂ X p+1 ⊂ X p = · · · = X1 = Z ;
these sets are independent of the choice of resolution and thus invariants of O /J .
Since O /J has pure codimension p,
codim Xk ≥ k + 1, for k ≥ p + 1,
see [14, Corollary 20.14]. Thus there is a free resolution (6.1) if and only if Xk = ∅
for k > N0. Unless n = 0 (which is not interesting in relation to the ∂¯ -equation), we
can thus choose the resolution so that N0 ≤ N − 1. The variety X is Cohen–Macaulay
at a point x , i.e., the sheaf O /J is Cohen–Macaulay at x , if and only if x ∈/ X p+1.
Notice that Z \(Xreg)red = Zsing ∪ X p+1. The sets Xk are independent of the choice
of embedding, see [9, Lemma 4.2], and are thus intrinsic subvarieties of Z = Xred ,
and they reflect the complexity of the singularities of X .
Let us now choose Hermitian metrics on the bundles Ek . We then refer to (6.1) as a
Hermitian resolution of O /J in . In \Xk we have a well-defined vector bundle
morphism σk+1 : Ek → Ek+1, if we require that σk+1 vanishes on (Im fk+1)⊥, takes
values in (Ker fk+1)⊥, and that fk+1σk+1 is the identity on Im fk+1. Following [7,
Section 2] we define smooth Ek -valued forms
(6.2)
(6.3)
uk = (∂¯ σk ) · · · (∂¯ σ2)σ1 = σk (∂¯ σk−1) · · · (∂¯ σ1)
(6.4)
in
\X ; for the second equality, see [7, (2.3)]. We have that
f1u1 = 1, fk+1uk+1 − ∂¯ uk = 0, k ≥ 1,
in \X . If f := ⊕ fk and u := uk , then these relations can be written economically
as ∇ f u = 1, where ∇ f := f − ∂¯ . To make the algebraic machinery work properly one
has to introduce a superstructure on the bundle E =: ⊕Ek so that vectors in E2k are
even and vectors in E2k+1 are odd; hence f , σ := ⊕σk , and u := uk are odd. For
details, see [
7
]. It turns out that u has a (necessarily unique) almost semi-meromorphic
extension U to . The residue current R is defined by the relation
∇ f U = 1 − R.
(6.5)
Remark 6.1 In case J is generated by the single non-trivial function f , then we have
f
the free resolution 0 → O → O → O /( f ) → 0; thus U is just the principal
value current 1/ f and R = ∂¯ (1/ f ). More generally, if f = ( f1, . . . , f p) is a complete
intersection, then
R = ∂¯
1
f p
,
where the right hand side is the so-called Coleff–Herrera product of f , see for example
[1, Corollary 3.5].
There are almost semi-meromorphic αk in , cf., [7, Section 2] and the proof of [6, Proposition 3.3], that are smooth outside Xk , such that
Rk+1 = αk+1 Rk
outside Xk+1 for k ≥ p. In view of (6.3) and the dimension principle, 1Xk+1 Rk+1 = 0
and hence (6.6) holds across Xk+1, i.e., Rk+1 is indeed equal to the product αk+1 Rk in
the sense of Proposition 2.1. In particular, it follows that Rk has the SEP with respect
to Z .
In this section, we let (z1, . . . , z N ) denote coordinates on CN , and let d z := d z1 ∧
· · · ∧ d z N .
Lemma 6.2 There is a matrix of almost semi-meromorphic currents b such that
R ∧ d z = bμ,
where μ is a tuple of currents in Hom(O /J , CHZ ).
Proof As in [6, Section 3], see also [32, Proposition 3.2], one can prove that R p =
σF μ, where μ is a tuple of currents in Hom(O /J , CHZ ) and σF is an almost
semimeromorphic current that is smooth outside X p+1.
Let b p = σF and bk = αk · · · α p+1σF for k ≥ p + 1. Then each bk is almost
semi-meromorphic, cf., [10, Section 4.1]. In view of (6.6) we have that Rk = bk μ
outside X p+1 since bk is smooth there. It follows by the SEP that it holds across X p+1
as well since Rk has the SEP with respect to Z . We then take b = b p + b p+1 + · · · .
(6.6)
(6.7)
By Proposition 2.4 we get
Corollary 6.3 The current R ∧ d z is in Hom(O /J , W Z ).
From Lemma 6.2, Corollary 6.3, (5.1), and (5.3) we get the following analogue to
[6, Proposition 3.3]:
Proposition 6.4 Let (6.1) be a Hermitian resolution of O /J in , and let R be the
n,∗ such that
associated residue current. Then there exists a (unique) current ω in WX
There is a matrix b of almost semi-meromorphic (0, ∗)-currents in , smooth outside
of X p+1, and a tuple ϑ of currents in ωnX such that
i∗ω = R ∧ d z.
ω = bϑ.
(6.8)
(6.9)
More precisely, ω = ω0 + ω1 + · · · + ωn,1 where ωk ∈ Wn,k (X, E p+k ), and if
f j := f p+ j , then
f 0ω0 = 0, f j+1ω j+1 − ∂¯ ω j = 0, for j ≥ 0.
(6.10)
We will also use the short-hand notation ∇ f ω = 0. As in the reduced case, following
[
6
], we say that ω is a structure form for X . The products in (6.9) are defined according
to Definition 5.5.
Remark 6.5 Recall that X p+1 = ∅ if X is Cohen–Macaulay, so in that case ω = bϑ ,
where b is smooth. If we take a free resolution of length p, then ω = ω0, and ∂¯ ω0 =
f 1ω1 = 0, so ω is in ωn .
X
Remark 6.6 If X = { f = 0} is a reduced hypersurface in , then R = ∂¯ (1/ f ) and ω
is the classical Poincaré residue form on X associated with f , which is a meromorphic
form on X . More generally, if X is reduced, since forms in ωnX are then meromorphic,
by (6.9), ω can be represented by almost semi-meromorphic forms on X .
We now consider the case when X is non-reduced. We recall that a differential
operator is a Noetherian operator for an ideal J if Lϕ ∈ √J for all ϕ ∈ J . It is
proved by Björk, [
13
], see also [32, Theorem 2.2], that if μ ∈ Hom(O /J , CHZ ),
then there exists a Noetherian operator L for J with meromorphic coefficients such
that the action of μ on ξ equals the integral of Lξ over Z . By (5.3), the action of h in
ωnX on ξ in EX0,∗ can then be expressed as
h.ξ =
Z
Lξ.
1 In [6, Proposition 3.3], the sum ends with ωn−1 instead of ωn, which, as remarked above, one can indeed
assume when n ≥ 1 and the resolution is chosen to be of length ≤ N − 1.
One can then verify using this formula and (6.9) that the action of the structure form
ω on a test form ξ in EX0,∗ equals
ω.ξ =
Z
L˜ξ,
where L˜ is now a tuple of Noetherian operators for J with almost semi-meromorphic
coefficients, cf., [32, Section 4].
Notice that (6.1) gives rise to the dual Hermitian complex
f ∗ f p∗ f p∗+1
0 → O(E0∗) →1 · · · → O(E ∗p−1) → O(E ∗p) −→ · · · .
(6.11)
Let ξ = ξ0 ∧ d z be a holomorphic section of the sheaf
Hom(E p, K )
O(E ∗p) ⊗ O(K )
such that f p∗+1ξ0 = 0. Then ∂¯ (ξ0ω0) = ±ξ0∂¯ ω0 = ±ξ0 f p+1ω1 = ±( f p∗+1ξ0)ω1 = 0,
so that ξ0ω0 is in ωn . Moreover, if ξ0 = f p∗η for η in O(E ∗p−1), then ξ0ω0 = f p∗ηω0 =
X
±η f pω0 = 0. We thus have a sheaf mapping
H p(Hom(E•, K )) → ωnX , ξ0 ∧ d z → ξ0ω0.
(6.12)
Proposition 6.7 The mapping (6.12) is an isomorphism, which establishes an intrinsic
isomorphism
E x t p(O /J , K )
ωnX .
Proof If h is in ωnX , then i∗h is in Hom(O /J , CHZ ). We have mappings
H p(Hom(E•, K )) → ωnX → Hom(O /J , CHZ ),
where the first mapping is (6.12), and the second is h → i∗h. In view of (6.8), the
composed mapping is ξ = ξ0 ∧d z → ξ R p = ξ0 R p ∧d z.2 This mapping is an intrinsic
isomorphism
(6.13)
(6.14)
E x t p(O /J , K )
according to [3, Theorem 1.5]. It follows that (6.12) also establishes an intrinsic
isomorphism.
In particular it follows that ωnX is coherent, and we have:
If ξ 1, . . . , ξ m are generators of H p(Hom(E•∗, K ))), where ξ
h := ξ0 ω0, = 1, . . . , m, generate the OX -module ωnX , and μ
generate the O -module Hom(O /J , CHZ ).
= ξ0 ∧ d z, then
= i∗h = ξ R p
2 There is a superstructure involved, with respect to which R p has even degree, and therefore dz ∧ R p =
R p ∧ dz, explaining the lack of a sign in the last equality, see [6] or [7].
Now take φ in Hom(ωn , WXn,∗). Let us choose a basis μ1, . . . , μm for ωnX and let
X
φ˜ be the element in (WZ0,∗)M obtained from the coefficients of φμ j when expressed
as in (2.11), cf., Sect. 4.1. We claim that Aφ˜ = 0. Taking this for granted, by the
exactness of (7.9), φ˜ is the image of the tuple φˆ = Sφ˜ . Now φˆ ∧ μ j = φμ j since they
are represented by the same tuple in (WZ0,∗)M . Thus φˆ gives the desired representation
of φ.
In view of Proposition 2.9 it is enough to prove the claim where φ˜ is smooth. Let
us therefore fix such a point, say 0, and show that ( Aφ˜ )(0) = 0. From the proof
of Lemma 4.11, if we let I be the ideal generated by z, and let X0 be defined by
OX0 := O /(J + I), then μ1 ∧ μz , . . . , μm ∧ μz generate ω0X0 . If we let φ0 be the
morphism in Hom(ω0X0 , ω0X0 ) given by φ0(μi ∧ μz ) := φμi ∧ μz (which indeed
gives a well-defined such morphism), then, as in the proof of Lemma 4.11, φ˜0 = φ˜ (0).
In addition, the sequence (4.9) for X0 is
0 → Cν T (0) CM A(0) CM .
→ →
Since X0 is 0-dimensional, the morphism OX0 → Hom(ωX0 , ωX0 ) is an
isomorphism by Theorem 7.3, and thus φ0 is given as multiplication by a function in OX0 ,
which we also denote by φ0, i.e., φ˜0 = T (0)φˆ0. Hence, A(0)φ˜0 = A(0)T (0)φˆ0 = 0,
and thus ( Aφ˜ )(0) = 0.
Example 7.6 (Meromorphic functions) Assume that we have a local embedding X →
. Given meromorphic functions , in that are holomorphic generically on Z ,
we say that ∼ if and only if − is in J generically on Z . If = A/B
and = A /B , where B and B are generically non-vanishing on Z , the condition
is precisely that A B − A B is in J . We say that such an equivalence class is a
meromorphic function φ on X , i.e., φ is in MX . Clearly we have OX ⊂ MX . We
claim that
MX ⊂ WX0,∗.
To see this, first notice that if we take a representative in M of φ, then it can be
considered as an almost semi-meromorphic current on with Zariski-singular support
of positive codimension on Z , since it is generically holomorphic on Z . As in Definition
5.5 we therefore have a current ∧ h in WXn,0 for h in ωnX . Another representative
of φ will give rise to the same current generically and hence everywhere by the SEP.
Thus φ defines a section of Hom(ωnX , WXn,∗) = WX0,∗.
0,∗ can be multiplied by a current h in ωnX , and
By definition, a current φ in WX
the product φ ∧ h lies in WXn,∗. It will be crucial that we can extend to products by
somewhat more general currents. Notice that ωnX is a subsheaf of CnX,∗, which is an
EX0,∗-module. Thus, we can consider the subsheaf EX0,∗ωnX of CnX,∗ which consists of
finite sums
ξi ∧ hi , where ξi are in EX0,∗ and hi are in ωn .
X
morphism in HomEX0,∗ (EX0,∗ωnX , WXn,∗).
0,∗ = HomOX (ωnX , WXn,∗) has a unique extension to a
Lemma 7.7 Each φ in WX
Proof The uniqueness follows by EX0,∗-linearity, i.e., if b = ξ1 ∧ h1 + · · · + ξr ∧ hr
is in EX0,∗ωnX , then one must have
φb =
(−1)(deg ξi )(deg φ)ξi ∧ φhi .
i
(7.10)
We must check that this is well-defined, i.e., that the right hand side does not depend
on the representation ξ1 ∧ h1 + · · · + ξr ∧ hr of b. By the SEP, it is enough to
prove this locally on Xreg, and we can then assume that φ has a representation (7.1).
By Proposition 2.9, it is then enough to prove that it is well-defined assuming that
φ0, . . . , φν−1 in (7.1) are all smooth. In this case, the right hand side of (7.10) is
simply the product of ξ1 ∧ h1 + · · · + ξr ∧ hr = b by the smooth form φ in EX0,∗, and
this product only depends on b.
0,∗ and let α be a current in WX
Corollary 7.8 Let φ be a current in WX n,∗ of the form
α = ai ∧ hi , where ai are almost semi-meromorphic (0, ∗)-currents on which
are generically smooth on Z , and hi are in ωnX . Then one has a well-defined product
φ ∧ α =
(−1)(deg ai )(deg φ)ai ∧ (φ ∧ hi ).
(7.11)
Proof The right hand side of (7.11) exists as a current in WXn,∗, and we must prove is
that it only depends on the current α and not on the representation ai ∧ hi . Notice
that all the ai are smooth outside some subvariety V of Z and there the right hand side
of (7.11) is the product of φ and α in EX0,∗ωnX , cf., Lemma 7.7. It follows by the SEP
that the right hand side only depends on α.
Remark 7.9 Recall from (6.9) that ω = bϑ . If φ is in WX0,∗, then we can define the
product φ ∧ ω by Corollary 7.8.
Expressed extrinsically, if μ = i∗ϑ , and if we write R ∧ d z = bμ as in
Lemma 6.2, then we can define the product R ∧ d z ∧ φ := bμ ∧ φ as a current
in Hom(O /J , W Z ).
Lemma 7.10 Assume that φ is in WX0,∗, and that φ ∧ ω = 0 for some structure form
ω, where the product is defined by Remark 7.9. Then φ = 0.
Proof Considering the component with values in E p, we get that φ ∧ ω0 = 0. By
Proposition 6.7, any h in ωnX can be written as h = ξ ω0, where ξ is a holomorphic
section of E ∗p, so by O-linearity, φ ∧ h = 0, i.e., φ = 0.
We end this section with the following result, first part of [10, Theorem 3.7]. We
include here a different proof than the one in [
10
], since we believe the proof here is
instructive.
Proposition 7.11 If Z is smooth, then WZ is closed under holomorphic differential
operators.
Proof Let τ be any current in WZ . It suffices to prove that if ζ are local coordinates
on Z , then ∂τ /∂ζ1 is in WZ . Consider the current
on the manifold Y := Z × Cw. Clearly τ has support on Z , and it follows from (2.5)
that τ is in WYZ . Let
p : (z, w) → ζ = (z1 + w, z2, . . . , zn ),
which is just a change of variables on Y followed by a projection. It follows from (2.4)
that p∗τ is in WZ . Since
it is readily verified that p∗τ = ∂τ /∂ζ1, so we conclude that ∂τ /∂ζ1 is in WZ .
0,∗
8 The ∂¯ -operator on WX
We already know the meaning of ∂¯ on WXn,∗, and we now define ∂¯ on WX0,∗.
Definition 8.1 Assume that φ, v are in WX0,∗, We say that ∂¯ v = φ if
∂¯ (v ∧ h) = φ ∧ h, h ∈ ωnX .
If we have an embedding X →
, (8.1) means, cf., (7.8), that
∂¯ (v ∧ μ) = φ ∧ μ, μ ∈ Hom
O /J , CHZ .
In view of Remark 7.9 we can define the product φ ∧ ω for φ in WX0,∗.
Definition 8.2 We say that v belongs to Dom ∂¯X if v is in Dom ∂¯ , i.e., ∂¯ v = φ for
some φ and in addition ∂¯ (v ∧ ω), a priori only in PMnX,∗, is in WXn,∗, for each structure
form ω from any possible embedding.
If X is Cohen–Macaulay, then any such ω is of the form a1h1 + · · · + am hm , where
h j are in ωnX and a j are smooth, see Remark 6.5, and hence Dom ∂¯X coincides with
Dom ∂¯ in this case.
(8.1)
(8.2)
Example 8.3 Assume that v is in EX0,∗ and φ = ∂¯ v in the sense in Section 4. Then
clearly
∂¯ (v ∧ ω) = φ ∧ ω + (−1)deg v v ∧ ∂¯ ω.
n,∗ is closed under multiplication with forms in EX0,∗, we get
Since ∂¯ ω = f ω, and WX
n,∗, so v is in Dom ∂¯ X and ∂¯ X v = φ.
that ∂¯ (v ∧ ω) is in WX
If w is in Dom ∂¯ X and v is in EX0,∗, then
∂¯ (v ∧ w ∧ ω) = ∂¯ v ∧ w ∧ ω + (−1)deg v v ∧ ∂¯ (w ∧ ω).
Thus v ∧ w is in Dom ∂¯ X , and the Leibniz rule ∂¯ (v ∧ w) = ∂¯ v ∧ w + (−1)deg v v ∧ ∂¯ w
holds.
Let χδ = χ (|h|2/δ) where h is a tuple of holomorphic functions that cuts out Xsing .
Lemma 8.4 If v is in W 0,∗( X ), and it is in Dom ∂¯ X on Xreg, then v is in Dom ∂¯ X on
all of X if and only if
for all structure forms ω. In this case,
∂¯ χδ ∧ v ∧ ω → 0, δ → 0,
− ∇ f (v ∧ ω) = ∂¯ v ∧ ω.
n,∗ is closed under multiplication by f , v is in Dom ∂¯ X if and only if
Proof Since WX
n,∗ for all structure forms ω. Since v is in Dom ∂¯ X on Xreg, thus
∇ f (v ∧ ω) is in WnX,∗ on Xreg. By (2.2), ∇ f (v ∧ ω) is then in WX
∇ f (v ∧ ω) is in WX n,∗ on all of X if and
only if
1Xreg ∇ f (v ∧ ω) = ∇ f (v ∧ ω).
By the Leibniz rule,
∇ f (χδ v ∧ ω) = −∂¯ χδ ∧ v ∧ ω + χδ ∇ f (v ∧ ω).
Since v is in W X0,∗, v∧ω is in W Xn,∗, so the left hand side of (8.6) tends to ∇ f (v∧ω) when
δ → 0, whereas the second term on the right hand side of (8.6) tends to 1Xreg ∇ f (v ∧ω).
Thus (8.5) holds if and only if (8.3) does. Thus the first statement in the lemma is
proved.
Recall, cf., (6.9), that ω = bϑ where b is smooth on Xreg and ϑ is in ωn . By the
X
Leibniz rule thus −∇ f (v ∧ ω) = ∂¯ v ∧ ω on Xreg, since ∇ f ω = 0. Therefore, (8.6) is
equivalent to −∇ f (χδ v ∧ ω) = ∂¯ χδ ∧ v ∧ ω + χδ ∂¯ v ∧ ω. If (8.3) holds, we therefore
get (8.4) when δ → 0.
Remark 8.5 In case X is reduced the definition of ∂¯ X is precisely the same as in [6].
However, the definition of ∂¯ v = φ given here, for v, φ in W X0,∗, does not coincide
with the definition in, e.g., [
6
]. In fact, that definition means that ∂¯ (v ∧ h) = φ ∧ h for
all smooth h in ωn , which in general is a strictly weaker condition. For example, for
X
(8.3)
(8.4)
(8.5)
(8.6)
any weakly holomorphic function v, we have ∂¯ (v ∧ h) = 0 for all smooth h in ωn ,
X
while if X is a reduced complete intersection, or more generally Cohen–Macaulay,
then ∂¯ (v ∧ h) = 0 for all h in ωnX is equivalent to v being strongly holomorphic, see
[33, p. 124] and [
2
].
We conclude this section with a lemma that shows that ∂¯ means what one should
expect when φ, v are expressed with respect to a local basis wα j for OX over OZ as
in Lemma 7.5.
Lemma 8.6 Assume that we have a local embedding Xreg →
represented as in (7.1). Then ∂¯ v = φ if and only if
0,∗
and φ, v in WX
∂¯ vˆ j = φˆ j , j = 0, . . . , ν − 1.
(8.7)
Proof Let us use the notation from the proof of Lemma 7.5. Recall that vˆ = Sv˜. In
view of (8.2) and (2.12), ∂¯ v = ∂¯ v˜. Since S is holomorphic therefore ∂¯ v = S∂¯ v =
S∂¯ v˜ = ∂¯ (Sv˜) = ∂¯ vˆ.
9 Solving ∂¯ u = φ on X
We will find local solutions to the ∂¯ -equation on X by means of integral formulas.
We use the notation and machinery from [6, Section 5]. Let i : X → ⊂ CN be a
local embedding such that is pseudoconvex, let ⊂⊂ be a relatively compact
subdomain of , and let X = X ∩ .
Theorem 9.1 There are integral operators
K : E 0,∗+1(X ) → W0,∗(X ) ∩ Dom ∂¯X , P : E 0,∗(X ) → E 0,∗(X )
such that, for φ ∈ E 0,k (X ),
φ = ∂¯ K φ + K (∂¯ φ) + Pφ.
(9.1)
The operators K and P are described below; they depend on a choice of weight g.
Since is Stein one can find such a weight g that is holomorphic in z, by which we
mean that it depends holomorphically on z ∈ and has no components containing
any d z¯i , cf., Example 5.1 in [
6
]. In this case, Pφ is holomorphic when k = 0, and
vanishes when k ≥ 1, i.e.,
φ = ∂¯ K φ + K (∂¯ φ), φ ∈ E 0,k (X ), k ≥ 1.
(9.2)
If ∂¯ φ = 0 in , and k ≥ 1, then K φ is a solution to ∂¯ v = φ. If k = 0, then φ = Pφ is
holomorphic. It follows that a smooth ∂¯ -closed function is holomorphic. In the reduced
case this is a classical theorem of Malgrange [24]. In Sect. 10 we prove that K φ is
smooth on Xreg.
We now turn to the definition of K and P. For future need, in Sect. 11, we define them
acting on currents in W0,∗(X ) and not only on smooth forms. Let π : ζ × z → z
be the natural projection. Let us choose a holomorphic Hefer form3 H , a smooth
weight g with compact support in with respect to z ∈ ⊂⊂ , and let B be the
Bochner–Martinelli form. Since we are only are concerned with (0, ∗)-forms, we will
here assume that H and B only have holomorphic differentials in ζ , i.e., the factors
dηi = dζi − d zi in H and B in [
6
] should be replaced by just dζi .
If γ is a current in ζ × z we let (γ )N be the component of bidegree (N , ∗) in ζ
and (0, ∗) in z, and let ϑ (γ ) be the current such that
Consider now μ in Hom(O /J , W Z ) and φ in WX0,∗. We can give meaning to
ϑ (γ ) ∧ dζ = (γ )N .
(g ∧ H R(ζ ))N ∧ φ (ζ ) ∧ μ(z)
×
as a tensor product of currents in the following way: first of all, by Remark 7.9, we can
form the product R(ζ ) ∧ dζ ∧ φ (ζ ) as a current in W Z . In view of [11, Corollary 4.7]
the tensor product R(ζ ) ∧ dζ ∧ φ (ζ ) ∧ μ(z) is in W Zζ××Z z , where Z = Z ∩ .
Finally, we multiply this with the smooth form ϑ (g ∧ H ) to obtain (9.4). Similarly,
outside of , the diagonal in , where B is smooth, we can define
(B ∧ g ∧ H R(ζ ))N ∧ φ (ζ ) ∧ μ(z)
as a tensor product of currents.
Lemma 9.2 For μ in Hom(O /J , W Z ) and φ ∈ W0,∗(X ), the current (9.5), a
priori defined as a current in W Zζ××Z \z\ has an extension across . The current (9.4)
and the extension of (9.5) depend O /J -bilinearly on μ and φ, and are such that
and
are in Hom(O /J , W Z ).
K φ ∧ μ := π∗ (B ∧ g ∧ H R(ζ ))N ∧ φ (ζ ) ∧ μ(z)
Pφ ∧ μ := π∗ (g ∧ H R(ζ ))N ∧ φ (ζ ) ∧ μ(z)
It follows that K φ ∧ μ and Pφ ∧ μ are C-linear in φ and O /J -linear in μ. In view
of (7.8), by considering μ in Hom(O /J , CHZ ), we have defined linear operators
K : W0,∗+1(X ) → W0,∗(X ),
P : W0,∗(X ) → W0,∗(X ).
(9.8)
Proof of Lemma 9.2 In order to define the extension of (9.5) across , we note first
that since B is almost semi-meromorphic with Zariski singular support , ϑ (B∧g∧ H )
3 We are only concerned with the component H 0 of this form, so for simplicity we write just H .
(9.3)
(9.4)
(9.5)
(9.6)
(9.7)
is an almost semi-meromorphic (0, ∗)-current on ζ × z , which is smooth outside
the diagonal. We can thus form the current ϑ (B ∧ g ∧ H ) ∧ R(ζ ) ∧ dζ ∧ φ (ζ ) ∧ μ(z)
Z×Z , cf., Proposition 2.4, and this is the extension of (9.5) across .
in W ζ × z
From the definitions above, it is clear that (9.4) and the extension of (9.5) are O
bilinear in φ and μ. Both these currents are annihilated by Jz and Jζ , cf., (2.8), so
they depend O /J -bilinearly. In view of (2.4) we conclude that (9.6) and (9.7) are
in Hom(O /J , W Z ).
Proposition 9.3 If φ ∈ W0,k (X ), then Pφ ∈ E 0,k (X ), and if in addition g is
holomorphic in z, then Pφ ∈ O(X ) if k = 0 and vanishes if k ≥ 1.
Proof Since ϑ (g ∧ H ) is smooth, we get that
π∗ ϑ (g ∧ H ) ∧ R(ζ ) ∧ dζ ∧ φ ∧ μ(z)
= π∗ ϑ (g ∧ H ) ∧ R(ζ ) ∧ dζ ∧ φ ∧ μ(z) = π∗ (g ∧ H R)N ∧ φ ∧ μ(z),
cf., for example [20, (5.1.2)]. Thus Pφ (z) = π∗ (g ∧ H R(ζ ))N ∧ φ which is smooth
on . If g depends holomorphically on z, then Pφ is holomorphic in if φ is a
(0, 0)-current, and vanishes for degree reasons if φ has positive degree.
We shall now approximate K φ by smooth forms. Let B
= χ (|ζ − z|2/ )B.
Proposition 9.4 For any φ ∈ W0,k (X ), k ≥ 1,
K φ := π∗ (B ∧ g ∧ H R(ζ ))N ∧ φ
= π∗ ϑ (B ∧ g ∧ H ) ∧ R(ζ ) ∧ dζ ∧ φ
is in E 0,k−1(X ) and K φ → K φ when
→ 0.
The last statement means that
K φ ∧ μ → K φ ∧ μ, μ ∈ Hom(O /J , CHZ ).
Proof Since B is smooth, the current we push forward is R(ζ ) ∧ φ (ζ ) times a smooth
form of ζ and z. Therefore K φ is smooth. As in the proof of Proposition 9.3, we
obtain since B is smooth that
K φ ∧ μ = π∗ (B ∧ g ∧ H R(ζ ))N ∧ φ ∧ μ(z) .
(9.9)
(9.10)
By (5.2) applied to a = B we have that
which implies (9.9).
(9.11)
9.1 Proof of Theorem 9.1
By definition K φ and Pφ are currents in W0,∗(X ) such that (9.6) and (9.7) hold for
μ in Hom(O /J , CHZ ). We claim that
and
K φ ∧ R ∧ d z = π∗ (B ∧ g ∧ H R(ζ ))N ∧ φ ∧ R(z) ∧ d z
Pφ ∧ R ∧ d z = π∗ (g ∧ H R(ζ ))N ∧ φ ∧ R(z) ∧ d z ;
(9.12)
(9.13)
here the left hand sides are defined in view of Remark 7.9, whereas the right hand sides
have meaning by Lemma 9.2 and the fact that R(z) ∧ d z is in Hom(O /J , W Z ) by
Corollary 6.3.
Recall from Lemma 6.2 that R ∧ d z = bμ, where μ is a tuple of currents in
Hom(O /J , CHZ ) and b is an almost semi-meromorphic matrix that is smooth
generically on Z . Therefore (9.12) and (9.13) hold where b is smooth, in view of
Lemma 7.7, and since both sides are in Hom(O /J , W Z ), the equalities hold
everywhere by the SEP.
As in [
6
] we let Rλ = ∂¯ | f |2λ ∧ U for Re λ 0. It has an analytic continuation
to λ = 0 and R = Rλ|λ=0. Notice that R(z) ∧ B is well-defined since it is a tensor
product with respect to the coordinates z, η = ζ − z. Also R(z) ∧ Rλ(ζ ) ∧ B admits
such an analytic continuation and defines a pseudomeromorphic current4 when λ = 0.
Let Bk,k−1 be the component of B of bidegree (k, k − 1).
Lemma 9.5 For all k,
Bk,k−1 ∧ H Rλ(ζ ) ∧ R(z)|λ=0 = Bk,k−1 ∧ H R(ζ ) ∧ R(z).
(9.14)
Proof of Lemma 9.5 Notice that the equality holds outside . Let T be the left hand
side of (9.14). In view of Proposition 2.1 it is therefore enough to check that 1 T = 0.
Fix j, k and let
T = Bk,k−1 ∧ H R λj(ζ ) ∧ R (z)|λ=0.
Clearly T = 0 if < p so first assume that = p. Since H R j has bidegree ( j, j )
in ζ , the current vanishes unless j + k ≤ N . Thus the total antiholomorphic degree is
≤ N − n + N − 1. On the other hand, the current has support on ∩ Z × Z Z × { pt }
which has codimension N + N − n. Thus it vanishes by the dimension principle.
We now prove by induction over ≥ p that 1 T = 0. Note that by (6.6), outside
of Z , R (z) = α (z)R −1(z), where α (z) is smooth. Thus, outside of Z × , T is
a smooth form times T −1, and thus, by induction and (2.3), 1 T has its support in
∩ (Z × Z ) Z × { pt }, which has codimension ≥ N + + 1, see (6.3). On the
other hand, the total antiholomorphic degree is ≤ + j + k − 1 ≤ + N − 1, so the
current vanishes by the dimension principle. We conclude that (9.14) holds.
4 One can consider this current as R(z) ∧ B multiplied by the residue of the almost semi-meromorphic
current U in (6.5), cf., [10, Section 4.4].
By the same argument5 as for [6, (5.2)] we have the equality
∇ f (z) (B ∧g ∧ H Rλ(ζ ))N ∧ R(z)∧d z = [ ] ∧ R(z)∧d z −(g ∧ H Rλ)N ∧ R(z)∧d z,
(9.15)
also for our R, where [ ] denotes the part of [ ] where dηi = dζi − d zi has been
replaced6 by dζi . In view of (9.14) we can put λ = 0 in (9.15), and then we get
∇ f (z) (B ∧g∧ H R(ζ ))N ∧ R(z)∧d z = [ ] ∧ R(z)∧d z −(H R(ζ )∧g)N ∧ R(z)∧d z.
(9.16)
Multiplying (9.16) by the smooth form φ, and using (9.12) and (9.13), we get
φ ∧ R ∧ d z = −∇ f (K φ ∧ R ∧ d z) + K (∂¯ φ) ∧ R ∧ d z + Pφ ∧ R ∧ d z,
or equivalently,
φ ∧ ω = −∇ f (K φ ∧ ω) + K (∂¯ φ) ∧ ω + Pφ ∧ ω.
(9.17)
Multiplying by suitable holomorphic ξ0 in E ∗p such that f p∗+1ξ0 = 0, cf.,
Proposition 6.7, we see that φ ∧ h = ∂¯ (K φ ∧ h) + K (∂¯ φ) ∧ h + Pφ ∧ h for all h in ωX .
Thus by definition (9.1) holds.
Since WX 0,∗ is in
0,∗ is closed under multiplication by OX , we get that ψ in WX
Dom ∂¯X if and only if −∇ f (ψ ∧ ω) is in WXn,∗. Thus, we conclude from (9.17) that
K φ is in Dom ∂¯X since all the other terms but −∇ f (K φ ∧ ω) are in WX0,∗.
9.2 Intrinsic interpretation of K and P
So far we have defined K and P by means of currents in ambient space. We used
this approach in order to avoid introducing push-forwards on a non-reduced space.
However, we will sketch here how this can be done. We must first define the product
space X × X . Given a local embedding i : X → as before, we have an embedding
(i × i ) : X × X → × such that the structure sheaf is O × /(JX + JX ).
One can check that this sheaf is independent of the chosen embedding, i.e., OX×X
is intrinsically defined. Thus we also have definitions of all the various sheaves on
X × X like EX0,×∗X . The projection p : X × X → X is determined by p∗φ : OX →
OX×X , which in turn is defined so that p∗i ∗ = (i × i )∗π ∗ for in O , where
π : × → as before. Again one can check that this definition is independent
of the embedding, and also extends to smooth (0, ∗)-forms φ. Therefore, we have the
well-defined mapping p∗ : C2Xn×,∗X+n → CnX,∗, and clearly
i∗ p∗ = π∗(i × i )∗.
(9.18)
5 There is a sign error in [6, (5.2)] due to R(z) ∧ dz being odd with respect to the super structure. Since
we here move R(z) ∧ dz to the right, we get the correct sign.
6 This change is due to the fact that we do the same change of the differentials in the definition of H and
B above.
As before we have the isomorphism
2n,∗
(i × i )∗ : WX×X
Hom
O ×
/(JX + JX ), W Z××Z
.
As in the proof of Lemma 9.2 we see that π∗ maps a current in W Z××Z annihilated by
JX to a current in Hom(O /J , W Z ). It follows by (9.18) that
p∗ : WX2n×,∗X+n → WXn,∗.
Now, take h in ωnX and let μ = i∗h. Then, cf., the proof of Lemma 9.2,
(B ∧ g ∧ H R(ζ ))N ∧ φ (ζ ) ∧ μ(z) = (i × i )∗ ϑ (B ∧ g ∧ H ) ∧ ω(ζ ) ∧ φ (ζ ) ∧ h .
Thus we can define K φ intrinsically by
K φ ∧ h = p∗ (ϑ (B ∧ g ∧ H ) ∧ ω(ζ ) ∧ φ (ζ ) ∧ h(z)) .
From above it follows that K φ ∧ h is in WXn,∗. In the same way we can define Pφ by
Pφ ∧ h = p∗ (ϑ (g ∧ H ) ∧ ω(ζ ) ∧ φ (ζ ) ∧ h(z)) .
(9.19)
(9.20)
It is natural to write
ζ
K φ (z) =
ϑ (B ∧ g ∧ H ) ∧ ω(ζ ) ∧ φ (ζ ),
Pφ (z) =
ϑ (g ∧ H ) ∧ ω(ζ ) ∧ φ (ζ ),
ζ
although the formal meaning is given by (9.19) and (9.20).
10 Regularity of solutions on X r eg
We have already seen, cf., Proposition 9.3, that Pφ is always a smooth form. We shall
now prove that K preserves regularity on Xreg. More precisely,
Theorem 10.1 If φ in WX0,∗ is smooth near a point x ∈ Xreg, then K φ in Theorem 9.1
is smooth near x .
Throughout this section, let us choose local coordinates (ζ, τ ) and (z, w) at x
corresponding to the variables ζ and z in the integral formulas, so that Z = {(ζ, τ ); τ =
0}.
Lemma 10.2 Let B := χ (|ζ − z|2/ )B, and assume that φ has compact support in
our coordinate neighborhood. Then K φ can be approximated by the smooth forms
K φ := π∗ (B ∧ g ∧ H R)N ∧ φ .
Notice that here we cut away the diagonal in Z × Z times Cτ × Cw in contrast
to Proposition 9.4, where we only cut away the diagonal in .
×
Proof Clearly B is smooth so that each K φ is smooth in a full neighborhood in
of x . Let T = μ(z, w) ∧ (H R(ζ, τ ) ∧ B ∧ g)N ∧ φ, and let W = × Cτ × Cw.
Since μ(z, w) ⊗ R(ζ, τ ) has support on {w = τ = 0}, T = 1{w=τ =0}T . Therefore,
1W T = 1W 1{w=τ =0}T = 0 since W ∩ {w = τ = 0} ⊂ and 1 T = 0 by definition,
cf., Proposition 2.1 (i). Now notice that 1W T = 0 implies (9.11) and in turn (9.9) with
our present choice of B .
We first consider a simple but nontrivial example of Theorem 10.1.
Example 10.3 Let X = Cζ ⊂ Cζ2,τ and J = (τ m+1). Then R = ∂¯ (1/τ m+1). For an
arbitrary point (z, w) we can choose the Hefer form
From the Bochner–Martinelli form B we only get a contribution from the term
1
H = 2π i
1 (ζ¯ − z¯)dζ + (τ¯ − w¯ )dτ
B1 = 2π i |ζ − z|2 + |τ − w|2
.
It follows that
Let ⊂⊂ be open balls with center at the origin, and let ϕ = ϕ(|ζ |2 + |τ |2) be a
smooth cutoff function with support in that is ≡ 1 in a neighborhood of . Then
we can choose a holomorphic weight g = ϕ + · · · , see, [6, Example 5.1] with respect
to , and with support in . Now 1, τ, . . . , τ m is a set of generators for OX over OZ .
Assume that
φ = (φˆ0(ζ ) ⊗ 1 + · · · + φˆm (ζ ) ⊗ τ m )dζ¯
is a smooth (0, 1)-form. We want to compute K φ. We know that
K φ = a0(z) ⊗ 1 + · · · + am (z) ⊗ wm
with ak (z) in WZ0,0. By Lemma 10.2 and its proof, we have smooth K φ (z, w) in
such that
1 1
K φ ∧ d z ∧ dw ∧ ∂¯ wm+1 → K φ ∧ d z ∧ dw ∧ ∂¯ wm+1
.
(10.1)
(10.2)
ak (z) = lim
→0 k! ∂wk
1 ∂k
K φ (z, w) w=0.
Notice that
1
(B ∧ g ∧ H R(τ ))2 = B1 ∧ g0,0 ∧ H ∧ ∂¯ τ m+1
=0
m
τ m− w dτ ∧
(ζ¯ − z¯)dζ + (τ¯ − w¯ )dτ
|ζ − z|2 + |τ − w|2
τ m− w ∧ |ζ − (zζ¯|2−+z¯|)τdζ− w|2 .
For each fixed
> 0, |ζ − z| > 0 on supp χ , cf., Lemma 10.2, so we have
dτ (ζ¯ − z¯)dζ¯ ∧ dζ
∂¯ τ +1 ∧ w χ |ζ − z|2 + |τ − w|2 ∧
k=0
φˆk (ζ ) ⊗ τ k .
= ζ,τ ϕ (2π i )2
1
A simple computation yields that
where
Letting tend to 0 we get K φ as in (10.1), where
K φ (z, w) =
ak (z) ⊗ wk + O(w¯ ),
1
ak (z) = 2π i ζ ϕ(|ζ |2)χ
φˆk (ζ )dζ¯ ∧ dζ
ζ − z
.
1
ak (z) = 2π i ζ
ϕ(|ζ |2) φˆk (ζ )dζ¯ ∧ dζ .
ζ − z
It is well-known that these Cauchy integrals ak (z) are smooth solutions to ∂¯ v = φˆk d z¯
in Z = Z ∩ . Thus K φ is smooth.
Remark 10.4 The terms O(w¯ ) in the expansion (10.4) of K φ (z, w) do not converge
to smooth functions in general when → 0. For a simple example, take φ = ζ dζ¯ ⊗τ m .
Then K φ (0, w) tends to
wm
ϕ(|ζ |2)
1 |ζ |2dζ¯ ∧ dζ
2π i |ζ |2 + |w|2
which is a smooth function of w plus (a constant times) wm |w|2 log |w|2, and
thus not smooth. However, it is certainly in C m . One can check that K φ (z, w) =
(10.3)
(10.4)
k=0 ζ
m
k=0 ζ
j
lim →0+ K φ (z, w) exists pointwise and defines a function in at least C m and that
our solution can be computed from this limit. In fact, by a more precise computation
we get from (10.3) that
ϕ(|ζ |2)χ
1 (ζ¯ − z¯)φˆk (ζ )dζ¯ ∧ dζ k m−k
2π i w
|ζ − z|2 + |w|2
It is now clear that we can let
→ 0. By a simple computation we then get
C φˆk (z) ⊗ wk
wk
Let ψ = ϕφˆk . Then the kth term in the second sum is equal to
wk
If we integrate outside the unit disk, then we certainly get a smooth function. Thus it
is enough to consider the integral over the disk. Moreover, if ψ (z + ζ ) = O(|ζ |M ) for
a large M , then the integral is at least C m . By a Taylor expansion of ψ (z + ζ ) at the
point z, we are thus reduced to consider
|ζ |<1
ζ αζ¯ β
For symmetry reasons, they vanish, except when α = β + 1. Thus we are left with
|ζ |<1
|ζ |2β
wk = C wk |w|2(m−k+1)
1
sβ ds
0 (s + |w|2)m−k+1
for non-negative integers β. The right hand side is a smooth function of w if β ≤
m − k − 1 and a smooth function plus
C wk |w|2(β+1) log |w|2
if β ≥ m − k. The worst case therefore is when k = m and β = 0; then we have
wm |w|2 log |w|2 that we encountered above.
Proposition 10.5 Let z, w be coordinates at a point x ∈ Xreg such that Z = {w = 0}
and x = (0, 0). If φ is smooth, and has support where the local coordinates are
defined, then
v (z, w) =
χ (|ζ − z|2/ )(H R ∧ B ∧ g)N ∧ φ,
is smooth for
> 0, and for each multiindex there is a smooth form v such that
∂wv |w=0 → v
as currents on Z .
Taking this proposition for granted we can conclude the proof of Theorem 10.1.
Proof of Theorem 10.1 If φ ≡ 0 in a neighborhood of x ∈ Xreg, then K φ is smooth
near x , cf., the proof of Proposition 9.4. Thus, it is sufficient to prove Theorem 10.1
assuming that φ is smooth and has support near x .
Recall that given a minimal generating set 1, wα1 , . . . , wαν−1 , one gets the
coefficients vˆ j in the representation
v = vˆ0 ⊗ 1 + · · · + vˆν−1 ⊗ wαν−1
from ∂wv |w=0, | | ≤ M by a holomorphic matrix, cf., the proof of Lemma 4.7. It thus
follows from Proposition 10.5 that there are smooth vˆ j such that vˆ j → vˆ j as currents
on Z . Let v = vˆ0 ⊗ 1 + · · · + vˆν−1 ⊗ wαν−1 . In view of (2.14), v ∧ μ → v ∧ μ for
all μ in Hom(O /J , CHZ ). From Lemma 10.2 we conclude that v ∧ μ = K φ ∧ μ
0,∗ and hence K φ is smooth.
for all such μ. Thus K φ = v in WX
Proof of Proposition 10.5 Assume that X is embedded in ⊂ CζN,τ . After a suitable
rotation we can assume that Z is the graph τ = ψ (ζ ). The Bochner–Martinelli kernel
in is rotation invariant, so it is
where
B = σ + σ ∧ ∂¯ σ + σ ∧ (∂¯ σ )2 + · · · ,
σ =
(ζ¯ − z¯ ) · dζ + (τ¯ − w¯ ) · dτ
|ζ − z |2 + |τ − w |2
.
We now choose the new coordinates ζ = ζ , τ = τ − ψ (ζ ) in , so that Z =
{(ζ, τ ); τ = 0}.
Recall that on Xreg we have that R ∧ d z is a smooth form times μ = (μ1, . . . , μm ),
where μ j is a generating set for Hom(O /J , CHZ ). Thus we are to compute ∂w|w=0
of integrals like
where k ≤ n and φ is smooth with compact support near x . It is clear that the symbols
τ¯, w¯ , dτ¯ can be omitted in the expression for
dτ
ζ,τ ¯ τ α+1 ∧ Bk ∧ φ (ζ, z, w, τ ),
∂
B = χ B = χ (|ζ − z|2/ )B,
(10.5)
since τ¯ and dτ¯ annihilate ∂¯ (1/τ α+1), and since we only take holomorphic derivatives
with respect to w and set w = 0.
Let us write ψ (ζ ) − ψ (z) = A(ζ, z)η, where η := ζ − z is considered as a column
matrix and A is a holomorphic (N − n) × n-matrix. Then
where ν is the (1, 0)-form valued column matrix
Since η∗ν is a (1, 0)-form we have that
σ =
η∗ν
|ζ − z|2 + |τ − w + ψ (ζ ) − ψ (z)|2
,
ν = dζ + A∗d(τ + ψ (ζ )).
Bk = χ
η∗ν ∧ ((dη∗)ν + η∗∂¯ ν)k−1
(|ζ − z|2 + |τ − w + ψ (ζ ) − ψ (z)|2)k
.
Lemma 10.6 Let
i i ∂ i ∂
ξ = ξ1 ∂ζ1 + · · · + ξn ∂ζn
be smooth (1, 0)-vector fields, and let Li = Lξi be the associated Lie derivatives for
i = 1, . . . , ρ. Let
γk := η∗ν ∧ ((dη∗)ν + η∗∂¯ ν)k−1.
If we have a modification π : W˜ →
a holomorphic function, then
such that locally π ∗η = η0η , where η0 is
π ∗(L1 · · · Lρ γk ) = η¯0kβ,
Recall that if a is a form, then Lξ a = d(ξ ¬a) + ξ ¬(da), and that Lξ (β¬a) =
[ξ, β]¬a + β¬(Lξ a) if β is another vector field.
Proof Introduce a nonsense basis e and its dual e∗ and consider the exterior algebra
spanned by e j , e∗, and the cotangent bundle. Let
Notice that γk is a sum of terms like
Since Li c = 0 and Li (η∗b) = η∗ Li b it follows after a finite number of applications
of Li ’s that we get
where ν j and b j are smooth. Since
(ν1e∗)¬ · · · (ν e∗)¬c (η∗b1) · · · (η∗bk− ),
π ∗c = η¯0(η )∗e ∧ (d(η )∗e) −1,
the lemma now follows.
We note that η∗(I + A∗ A)η = |ζ − z|2 + |ψ (ζ ) − ψ (z)|2. Thus, differentiating
(10.5) with respect to w, setting w = 0, and evaluating the residue with respect to τ
using (2.10), we obtain a sum of integrals like
ζ
χ (η∗a1) · · · (η∗at+1) ∧ γk ∧ φ
(η∗(I + A∗ A)η)k+t+1
,
where a1, . . . , at+1 are column vectors of smooth functions. We must prove that the
limit of such integrals when → 0 are smooth in z.
Lemma 10.7 Let
I r,s
,
where a1, . . . , ar are tuples of smooth functions, γ˜k = L1 · · · Lρ γk , where Li = Lξi
are Lie derivatives with respect to smooth (1, 0)-vector fields ξ i as above for i =
1, . . . , ρ, φ is a test form with support close to z, and := η∗(I + A∗ A)η. If r ≥ 1
and r + s ≥ + 1, then we have the relation
I r+,s1 = I r−1,s + I r+−11,s+1 + I r,s−1 + o(1)
(10.6)
(10.7)
when
→ 0.
Proof If
Thus
and L = Lξ , then using that
= ηt (I + A∗ A)t η¯, one obtains that
ξ = art (I + A∗ A)−t ∂ ,
∂ζ
L
= η∗ar + O(|η|2).
I r,s
+1 =
χ (η∗a1) · · · (η∗ar−1)O(|η|2s )γ˜k ∧ φ L
1
k+
+ I r−1,s+1
+1
in view of (10.7). We now integrate by parts by L in the integral. If a derivative with
respect to ζ j falls on some η∗ai , we get a term I r−1,s . If it falls on O(|η|2s ) we get
either O(|η|2(s−1)) times η∗b, for some tuple b of smooth functions, and this gives rise
to the term I r,s−1 or O(|η|2s ), and this gives rise to another term I r−1,s . If it falls on φ
or γ˜k we get an additional term I r−1,s . The only possibility left is when the derivative
falls on χ = χ (|η|2/ ). It remains to show that an integral of the form
χ (|η|2/ ) (η∗a1) · · · (η∗ar−1)(η∗b) O(|η|2s )γk ∧ φ
k+
tends to 0, where the factor η∗b comes from the derivative of |η|2. We now choose
a resolution V → × such that η = η0η where η is non-vanishing and η0 is
(locally) a monomial. Notice that π ∗ = |η0|2 where is smooth and strictly
positive. In view of Lemma 10.6 we thus obtain integrals of the form
(10.8)
V
χ (|η0|2v/ )
β
f m
where v is smooth and strictly positive and α is smooth.
In order to see that the limit of (10.8) tends to 0, we note first that if we let
then just as χ , χ˜ is also a smooth function on [0, ∞) that is 0 in a neighborhood of 0
and 1 in a neighborhood of ∞. By assumption, r + s − − 1 ≥ 0. Since the principal
value current 1/ f m acting on a test form β can be defined as
for any cut-off function as above, the principal value current 1/η0k+ −s acting on
η¯0r+s− −1α equals
lim
→0+ V
χ |η0|2v/
η¯0r+s− −1
ηk+ −s α = lim
0 →0+ V
χ˜ |η0|2v/
η¯0r+s− −1
ηk+ −s α.
0
Taking the difference between the left and right hand side, we conclude that (10.8)
tends to 0 when → 0.
Now we can conclude the proof of Proposition 10.5. From the beginning we have
I ,0. After repeated applications of (10.6) we end up with
I 0, + I 0−,1−1 + · · · + I00,0 + o(1).
However, any of these integrals has an integrable kernel even when = 0. This means
that we are back to the case in [6, Lemma 6.2], and so the limit integral is smooth in
z.
11 A fine resolution of OX
We first consider a generalization of Theorem 9.1.
Lemma 11.1 Assume that φ ∈ W0,k (X ) ∩ EX0,k (Xreg) ∩ Dom ∂¯X and that K φ is in
Dom ∂¯X (or just in Dom ∂¯ ). Then (9.1) holds on X .
Proof Let χδ be functions as before that cut away Xsing. From Koppelman’s formula
(9.1) for smooth forms we have
χδφ ∧ h = ∂¯ (K (χδφ)) ∧ h + K (χδ∂¯ φ) ∧ h + P(χδφ) ∧ h + K (∂¯ χδ ∧ φ) ∧ h, h ∈ ωnX ,
(11.1)
for z ∈ Xreg. Clearly the left hand side tends to φ ∧ h when δ → 0. From Lemma 9.2
it follows that K (χδφ) ∧ h → K φ ∧ h. Thus the first term on the right hand side of
(11.1) tends to ∂¯ (K φ) ∧ h. In the same way the second and third terms on the right
hand side tend to K (∂¯ φ) ∧ h and Pφ ∧ h, respectively. It remains to show that the last
term tends to 0. If z belongs to a fixed compact subset of Xreg, then B is smooth in
(9.5) when ζ is in supp ∂¯ χδ for small δ. Hence it suffices to see that
and since this is a tensor product of currents, it suffices to see that
R(ζ ) ∧ dζ ∧ ∂¯ χδ ∧ φ (ζ ) ∧ i∗h → 0,
R(ζ ) ∧ dζ ∧ ∂¯ χδ ∧ φ (ζ ) → 0,
or equivalently, ω(ζ ) ∧ ∂¯ χδ ∧ φ (ζ ) → 0, which follows by Lemma 8.4 since φ is in
Dom ∂¯X . We have thus proved that
χδφ ∧ h = χδ∂¯ (K φ) ∧ h + χδ K (∂¯ φ) ∧ h + χδ Pφ ∧ h.
The first term on the right hand side is equal to ∂¯ (χδ K φ ∧ h) − ∂¯ χδ ∧ K φ ∧ h, where
the latter term tends to 0 if K φ is in Dom ∂¯X or just in Dom ∂¯ , cf., Lemma 8.4. Thus
we get
φ ∧ h = ∂¯ (K φ) ∧ h + K (∂¯ φ) ∧ h + Pφ ∧ h, h ∈ ωnX ,
which precisely means that (9.1) holds.
Definition 11.2 We say that a (0, q)-current φ on an open set U ⊂ X is a section of
AXq over U , φ ∈ A q (U ), if, for every x ∈ U , the germ φx can be written as a finite
sum of terms
ξν ∧ Kν (· · · ξ2 ∧ K2(ξ1 ∧ K1(ξ0))),
where ξ j are smooth (0, ∗)-forms and K j are integral operators with kernels k j (ζ, z) at
x , defined as above, and such that ξ j has compact support in the set where z → k j (ζ, z)
is defined.
Clearly AX∗ is closed under multiplication by ξ in EX0,∗. It follows from (9.8) that
0,∗ and from Theorem 10.1 that AXk = EX0,∗ on Xreg. Clearly
AX∗ is a subsheaf of WX
any operator K as above maps AX∗+1 → AX∗.
Lemma 11.3 If φ is in AX , then φ and K φ are in Dom ∂¯X .
Proof Notice that [6, Lemma 6.4] holds in our case by verbatim the same proof, since
we have access to the dimension principle for (tensor products of) pseudomeromorphic
(n, ∗)-currents, and the computation rule (2.3), cf., the comment after Definition 5.7.
Since AX∗ = EX0,∗ on Xreg it is enough by Lemma 8.4 to check that ∂¯ χδ ∧ ω ∧ φ → 0,
and this precisely follows from [6, Lemma 6.4].
In view of Lemmas 11.1 and 11.3 we have
Proposition 11.4 Let K , P be integral operators as in Theorem 9.1. Then
K : A k+1(X ) → A k (X ), P : A k (X ) → E 0,k (X ),
and the Koppelman formula (9.1) holds.
Proof of Theorem 1.1 By definition, it is clear that AXk are modules over EX0,k , and by
Theorem 10.1, AXk coincides with EX0,k on Xreg. Since we have access to Koppelman
formulas, precisely as in the proof of [6, Theorem 1.2] we can verify that ∂¯ : AXk →
AXk+1.
It remains to prove that (1.2) is exact. We choose locally a weight g that is
holomorphic in z, so the term Pφ vanishes if φ is in AXk , k ≥ 1, and is holomorphic in z
when k = 0. Assume that φ is in AXk and ∂¯ φ = 0. If k ≥ 1, then ∂¯ K φ = φ, and if
k = 0, then φ = Pφ.
Assume that E → X is a holomorphic vector bundle; this means that the transition
matrices have entries in OX . For instance if we have a global embedding i : X →
and a holomorphic vector bundle F → , then F defines a vector bundle i ∗ F → X .
The sheaves AX∗(E ) give rise to a fine resolution of the sheaf OX (E ), and by standard
homological algebra we have the isomorphisms
H q (X, O(E )) =
Ker (A q (X, E ) →∂¯ A q+1(X, E ))
Im (A q−1(X, E ) →∂¯ A q (X, E ))
, q ≥ 1.
Thus, if φ ∈ A q+1(X, E ), ∂¯ φ = 0, and its canonical cohomology class vanishes, then
the equation ∂¯ ψ = φ has a global solution in A q (X, E ). In particular, the equation
is always solvable if X is Stein. If for instance X is a pure-dimensional projective
variety i : X → PN , then the ∂¯ -equation is solvable, e.g., if E is a sufficiently ample
line bundle.
12 Locally complete intersections
Let us consider the special case when X locally is a complete intersection, i.e., given a
local embedding i : X → ⊂ CN there are global sections f j of O(d j ) → PN such
that J = ( f1, . . . , f p), where p = N − n. In particular, Z = { f1 = · · · = f p = 0}.
In this case Hom(O /J , CH ) is generated by the single current
μ = ∂¯
1
∧ · · · ∧ ∂¯ f ∧ d z1 ∧ · · · ∧ d z N ,
1
see, e.g., [
3
]. Each smooth (0, q)-form φ in EX0,q is thus represented by a current ∧μ,
where is smooth in a neighborhood of Z and i ∗ = φ. Moreover, X is Cohen–
Macaulay so Xreg coincides with the part of X where Z is regular, and ∂¯ φ = ψ if and
only if ∂¯ (φ ∧ μ) = ψ ∧ μ.
Henkin and Polyakov introduced, see [17, Definition 1.3], the notion of residual
currents φ of bidegree (0, q) on a locally complete intersection X ⊂ PN , and the
∂¯ -equation ∂¯ ψ = φ. Their currents φ correspond to our φ in EX0,q and their ∂¯ -operator
on such currents coincides with ours.
Remark 12.1 In [
18
] Henkin and Polyakov consider a global reduced complete
intersection X ⊂ PN . They prove, by a global explicit formula, that if φ is a global ∂¯ -closed
smooth (0, q)-form with values in O( ), = d1 + · · · d p − N − 1, then there is a
smooth solution to ∂¯ ψ = φ at least on Xreg, if 1 ≤ q ≤ n −1. When q = n a necessary
obstruction term occurs. However, their meaning of “∂¯ -closed” is that locally there is
a representative of φ and smooth g j such that ∂¯ = g1 f1 + · · · + g p f p. If this
holds, then clearly ∂¯ φ = 0. The converse implication is not true, see Example 12.2
below. It is not clear to us whether their formula gives a solution under the weaker
assumption that ∂¯ φ = 0, neither do we know whether their solution admits some
intrinsic extension across Xsing as a current on X .
Example 12.2 Let X = { f = 0} ⊂ ⊂ Cn+1 be a reduced hypersurface, and
assume that d f = 0 on Xreg, so that J = ( f ). Let φ be a smooth (0, q)-form in a
neighborhood of some point x on X such that ∂¯ φ = 0. We claim that ∂¯ u = φ has a
smooth solution u if and only if φ has a smooth representative in ambient space
such that ∂¯ = f g for some smooth form g. In fact, if such a exists then 0 = f ∂¯ g
and thus ∂¯ g = 0. Therefore, g = ∂¯ γ for some smooth γ (in a Stein neighborhood of
x in ambient space) and hence ∂¯ ( − f γ ) = 0. Thus there is a smooth U such that
∂¯ U = − f γ ; this means that u = i ∗U is a smooth solution to ∂¯ u = φ. Conversely,
if u is a smooth solution, then u = i ∗U for some smooth U in ambient space, and thus
= ∂¯ U is a representative of φ in ambient space. Thus ∂¯ = f g (with g = 0).
There are examples of hypersurfaces X where there exist smooth φ with ∂¯ φ = 0
that do not admit smooth solutions to ∂¯ u = φ, see, e.g., [6, Example 1.1]. It follows
that such a φ cannot have a representative in ambient space as above.
Acknowledgements We thank the referee for very careful reading and many valuable remarks.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
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It follows directly that R is ∇ f -closed. In addition, R has support on Z and is a sum Rk , where Rk is a pseudomeromorphic Ek -valued current of bidegree (0, k) . It follows from the dimension principle that R = R p + R p+ 1 + · · · + RN . If we choose a free resolution that ends at level N − 1, then RN = 0. If X is Cohen-Macaulay and N0 = p in (6.1), then R = R p, and the ∇ f -closedness implies that R is ∂¯ -closed. If φ is in J then φ R = 0 and in fact, J = ann R, see [7, Theorem 1 .1].
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