Global Koppelman Formulas on (Singular) Projective Varieties

Let i:X→PN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i:X\rightarrow {\mathbb {P}}^N$$\end{document} be a projective manifold of dimension n embedded in projective space PN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {P}}^N$$\end{document}, and let L be the pullback to X of the line bundle OPN(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}_{{\mathbb {P}}^N}(1)$$\end{document}. We construct global explicit Koppelman formulas on X for smooth (0,∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,*)$$\end{document}-forms with values in Ls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^s$$\end{document} for any s. The same construction works for singular, even non-reduced, X of pure dimension, if the sheaves of smooth forms are replaced by suitable sheaves AX∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}_X^*$$\end{document} of (0,∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,*)$$\end{document}-currents with mild singularities at Xsing\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{sing}$$\end{document}. In particular, if s≥regX-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\ge \mathrm{reg\,}X -1$$\end{document}, where regX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{reg\,}X$$\end{document} is the Castelnuovo–Mumford regularity, we get an explicit representation of the well-known vanishing of H0,q(X,Ls-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{0,q}(X, L^{s-q})$$\end{document}, q≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 1$$\end{document}. Also some other applications are indicated.


Introduction
During the last decade, global Koppelman formulas for∂ on various special projective varieties have been constructed, see, e.g., [17][18][19][22][23][24]. The aim of this paper is to present a quite general explicit construction of intrinsic 1 Koppelman formulas on any projective, possibly non-reduced, subvariety i : X → P N of pure dimension n.
Let us first assume that X is smooth; even in this case, such global formulas are previously known only in case X is (locally) a complete intersection in P N . Let L → X be the restriction of the ample line bundleO P N (1) to X , and let E 0,q X (L r ) denote the 1 Here "intrinsic" means that the operators in the formula only depend on intrinsic forms on X .
The author was partially supported by the Swedish Research Council. sheaf of smooth (0, q)-forms on X with values in L r . We introduce integral operators K : E 0,q+1 (X , L s ) → E 0,q (X reg , L s ) and P : E 0,q (X , L s ) → E 0,q (X , L s ) such that the Koppelman formula φ(z) =∂Kφ + K(∂φ) + Pφ (1.1) holds on X . In some situations, see below, we can choose the operators so that Pφ = 0; if∂φ = 0, then ψ = Kφ is a smooth solution to∂ψ = φ on X . In certain cases, we can choose P such that P : E 0,0 (X , L s ) → O(P N , O(s)). Then Pφ is a holomorphic extension to P N of a holomorphic section φ of L r on X . We get no new existence results; the novelty is that we have explicit formulas for the global solutions and for the holomorphic extensions. The operators K are given by kernels k(ζ, z) that are defined on X × X and integrable in ζ for any z ∈ X . Simply speaking the operators locally behave like standard integral operators for∂ in C n ; in particular, they extend to L p -spaces, etc., and all classical local norm estimates hold.
Let us now turn our attention to the case when i : X → P N is a subvariety of pure dimension n. In case X is reduced, there are well-known definitions of smooth forms and currents on X , cf. Sect. 3. In [9] are introduced reasonable definitions of sheaves E 0, * X of smooth (0, * )-forms and suitable sheaves of currents even in the non-reduced case, see Sect. 6 below. In [6] and [9] are introduced, by means of local Koppelman formulas, fine sheaves A k X (in fact, modules over E 0, * X ) of (0, q)-currents on X , or any pure-dimensional analytic space, with the following properties: There are sheaf inclusions E 0,q X ⊂ A q X with equality on X reg , whereas A q X have "mild" singularities at X sing , and is a (fine) resolution of the structure sheaf O X of holomorphic functions on X . By the abstract de Rham theorem, we therefore have canonical isomorphisms Remark 1 It was proved already in [21] that if X is a local reduced complete intersection and φ is a smooth∂-closed form, then there is locally a smooth solution tō ∂ψ = φ on X reg . The case with a general X was proved only in [5]. The analogous result for a local non-reduced space is a special case of the main result in [9]. It is known that in general there is no solution that is smooth across X sing , see, e.g., [6,Example 1.1].
Let J X be the homogeneous ideal in the graded ring S = C[z 0 , . . . , z N ] associated with X . Let S(−r ) be the module S but with the grading shifted by r . There is a free graded resolution (1.4) is exact, and the cokernel of the right-most mapping is precisely S/J X . Since 0 is not an associated prime ideal of J X it follows from [11,Corollary 20.14] that one can choose (1.4) such that N 0 ≤ N . Our integral formulas are explicitly constructed out of a resolution (1.4).
Recall that the (Castelnuovo-Mumford) regularity of X is defined as the regularity of the ideal J X which turns out to be 1 plus the regularity of the module S/J X , so that if (1.4) is a minimal free resolution of S/J X , cf. [12,Ch. 4]. It is well known, see, e.g., [12,Proposition 4.16], that and that the natural mapping is surjective for s ≥ reg X − 1.
In [4,Example 3.4] is described an extension operator that provides an explicit proof of surjectivity of (1.7), in case X is reduced. The non-reduced case is obtained in precisely the same way following the ideas in [9]. By appropriate choices of operators K we can give an explicit proof of the vanishing of (1.6), provided that X is irreducible. That is, we have Theorem 1.1 Let X be a, possibly non-reduced, irreducible, subvariety of P N of pure dimension n and assume that s ≥ reg X − 1. For each q ≥ 1 there is an integral oper- In fact, for fixed q it is enough that s ≥ max ≤N −q d i − (N − q); this follows from the proof below. When X is not irreducible a slightly less sharp version of the theorem still holds, see Proposition 7.2.
Koppelman formulas on P n were found by Götmark, [17], and on more general symmetric spaces in [18]. In [20], explicit formulas for the∂-equation are used on a smooth Riemann surface embedded in P 2 , for L 1 -estimates. Similar formulas were also introduced in [19], cf. Sect. 8.2 below. Koppelman formulas for global, even nonreduced, complete intersections are constructed in the recent papers [23,24], cf. Sect. 8 below.
As already mentioned, the main novelty in this paper is Koppelman formulas for an arbitrary embedded projective variety X . We think that these formulas will be of interest even when X is smooth. We prove Theorem 1.1 as an illustration of the utility and indicate some other applications in Sect. 7.1. We hope that our Koppelman formulas will be useful for other purposes as well.
In Sect. 2 we describe, based on [3,4,17], how one can obtain weighted integral formulas on P N . We need some elements from residue theory that we have collected in Sect. 3. In Sects. 4 and 5, we then describe the construction of our Koppelman formulas on a pure-dimensional subvariety. In order to keep the technicalities on a reasonable level, we restrict here to the case where X is reduced. The reader who is mainly interested in the smooth case can just think that X sing is empty, in that way avoiding a lot of technicalities. In Sect. 6, we discuss the non-reduced case.
Throughout this paper we will only consider forms and currents that only contain holomorphic differentials with respect to ζ , whereas anti-holomorphic differentials with respect to both z and ζ may occur.
Given a vector bundle F → P N , let F z denote the pullback of F to P N ζ × P N z under the natural projection P N ζ × P N z → P N z and define F ζ analogously. A weight with respect to F is a smooth section g of L 0 (Hom (F ζ , F z )) such that ∇ η g = 0 and g 0 = I F on the diagonal in P N ζ × P N z , where g 0 denotes the term in g with bidegree (0, 0). In general, we let lower index on a form denote degree with respect to holomorphic differentials of ζ . Notice that if g is a weight with respect to F, then from (2.1) we get Identifying terms of full degree in dζ thus By Stokes' theorem we get the following Koppelman formula, cf. [17] and [18].

Example 1
It is easy to check that Since α 0 is equal to I O(1) on the diagonal, α is a weight with respect to F = O(1). For each natural number ρ therefore g = α ρ is a weight with respect to F = O(ρ).
For ≥ −N , we thus have the Koppelman formula Since α is holomorphic in z and has no differentials dz, the last term in (2.5) vanishes if q ≥ 1, and for degree reasons also if q = 0 and ≤ −1.
We thus have an explicit proof of the well-known vanishing H 0,q (P N , O( )) = 0 for ≥ −N if 1 ≤ q ≤ N , and for q = 0 if ≤ −1.

Remark 3
Using the transposed integral operators in Example 1, we get an explicit proof of the vanishing H N ,q Let us now consider a weight with respect to O(−1). Let is a projective form. More explicitly, so each term has the same degree in dζ as in dz, and is holomorphic in ζ .

Some Preliminaries
Let X be any reduced analytic space of pure dimension n. By definition there is, locally, some embedding i : X → ⊂ C N . Let J X ⊂ O be the ideal sheaf of holomorphic functions in that vanish on X . Then the sheaf of holomorphic functions on X , the structure sheaf O X , is represented as of smooth forms on X , and have a natural mapping i * : One can prove that E p, * X so defined is independent of the choice of embedding and is thus an intrinsic sheaf on X . We define the sheaf C p, * X of currents as the dual of E n− p,n− * X . More concretely this means that currents τ in C . Clearly∂ is defined on smooth forms and extends to currents by duality. Also the wedge product φ∧τ is well defined as long as at least one of the factors is smooth. Thus the currents form a module over the smooth forms.
We say that a current in C M s of the form where γ is a test form, is elementary. A current τ on X is pseudomeromorphic if locally it is a finite sum of direct images under holomorphic mappings of elementary currents; see, e.g., [10] for a precise definition and basic properties. The pseudomeromorphic currents form a sheaf PM X that is closed under multiplication by E p, * X and the action of∂. Given a pseudomeromorphic current τ in an open set U and a subvariety V ⊂ U, the natural restriction of τ to U \ V has a canonical extension to a pseudomeromorphic current 1 U \V τ such that Let χ be a smooth function on [0, ∞) that is 0 in a neighborhood of 0 and 1 in a neighborhood of ∞ and let h be a tuple of holomorphic functions, or a section of some holomorphic Hermitian vector bundle such that the zero set of h is precisely V . Then We say that a current a in X is almost semi-meromorphic, a ∈ AS M(X ), if there is a smooth modification π : X → X , a generically non-vanishing holomorphic section σ of a line bundle L → X and a smooth L-valued form γ such that a = π * (γ /σ ).
Let Z SS(a) be the smallest analytic subset of X such that a is smooth in X \ Z SS(a). It follows that Z SS(a) has positive codimension. Clearly an almost semi-meromorphic a is pseudomeromorphic. We refer to (ii) as the dimension principle.

A Structure Form Associated to X
Let i : X → P N be a reduced subvariety of pure dimension n, and let (1.4) be a free graded resolution of the S-module M X = S/J X . In particular, then a 1 = (a 11 1 , . . . , a 1r 1 1 ) is a tuple of homogeneous forms that define the homogeneous ideal J X in the graded ring S = C[z 0 , . . . , z N ]. Let E j k be disjoint trivial line bundles over P N with basis elements e k, j and let is a complex of vector bundles over P N that is pointwise exact outside Z , and the corresponding complex of locally free sheaves See, e.g., [7,Sect. 6]. We equip E k with the natural Hermitian metric . . , ξ r k ), so that (4.1) becomes a Hermitian complex. In [7] were introduced pseudomeromorphic currents on P N associated to (4.1) with the following properties: The currents U k are almost semi-meromorphic (0, k − 1)-currents, smooth outside X , that take values in Hom (E 0 , E k ) E k , and R k are (0, k)-currents with support on X , taking values in Hom (E 0 , E k ) E k . Moreover, we have the relations which can be compactly written as Let X k be the analytic subset of P N where a k does not have optimal rank. Then Since J X has pure dimension see [11,Corollary 20.14]. By the dimension principle R k = 0 for k < N −n. Moreover, there are almost semimeromorphic Hom (E k , E k+1 )-valued (0, 1)-currents α k+1 , smooth outside X k+1 , such that R k+1 = α k+1 R k there. By (4.6) and the dimension principle it follows that this equality must hold across X k+1 if the right-hand side is interpreted in the sense of Proposition 3.1. By a simple induction argument, using (4.6) and the dimension principle, it follows that 1 X sing R = 0. It follows that Rφ is well-defined for φ in E 0, * X .
Proof Locally at a point x ∈ X reg we can choose coordinates (z, w) such that X = {w = 0}. By a Taylor expansion of in w, using that w j R =w j R = dw j ∧R = 0, cf. Proposition 3.2 (i), we find that that R = 0 if and only if i * = 0. If R = 0 on X reg it follows from (3.1) and (4.7) that R = 0 identically.
Let Ω = δ z (dz 0 ∧ · · · ∧dz N ) be the unique, up to a multiplicative constant, nonvanishing global (N , 0)-form with values in O(N + 1). From [6, Proposition 3.3] we have Proposition 4.2 There is a unique almost semi-meromorphic current ω = ω 0 +· · ·+ω n on X that is smooth on X reg , ω have bidegree (n, ) and take values in E := i * E N −n+ , and We say that ω is a structure form on X .
For any smooth form ξ on P N there is a unique form ϑ(ξ ) such that 9) where ξ N denotes the components of ξ of bidegree (N , * ). From (4.8) and (4.9) we have that where we in the last term, for simplicity, write ϑ(ξ )∧ω rather then i * ϑ(ξ )∧ω. Proof Let W be the zero set of h. Notice that i * 1 X sing ω = 1 X sing i * ω = 1 X sing Ω∧R = Ω∧1 X sing R = 0 in view of Lemma 4.1. Thus 1 X sing ω = 0, and hence 1 W 1 X sing ω = 0.

Koppelman Formulas on a Projective Variety
Let U λ = |a 1 | 2λ U and R λ = 1 − |a 1 | 2λ +∂|a 1 | 2λ ∧U . Then R λ and U λ are as smooth as we may wish if Re λ is sufficiently large. In particular, R λ and U λ are well-defined currents. Moreover, they admit analytic continuations to Re λ > − , and the values at λ = 0 are precisely R and U , respectively, see [7].
H k = 0 for k < , the term (H ) 0 of bidegree (0, 0) is the identity I E on , and is a weight with respect to O(ρ). Here

Proposition 5.2 Let F be holomorphic vector bundle over P N and let g be a weight with respect to F. Moreover, assume that H is a Hefer morphism for E
By blowing up P N × P N along the diagonal one can verify that B is almost semimeromorphic. In view of Proposition 3.1 thus (H R∧g∧B) N ∧φ is a well-defined pseudomeromorphic current on P N × P N . Formally (5.3) means that where π is the projection (ζ, z) → z.
Proof Since g λ ∧g is a weight with respect to O(ρ) ⊗ F, and a 1 (z) = 0 when z ∈ X , from Proposition 2.2 we get (5.3) with R λ instead of R. In view of (the proof of) [6, Lemma 5.2], see also [9, Lemma 9.5], we can take λ = 0 and so we get the proposition, keeping in mind that the product B∧τ can be defined as the value of |η| 2λ B∧τ at λ = 0 in view of [6, (2.2) and (2. for z ∈ X reg . In view of (4.10) we have that It is apparent from (5.6) that K and P are intrinsic integral operators on X . Locally they are precisely of the type in [6], so it follows that Kφ is smooth on X reg if φ is smooth. Moreover, from [6, Theorem 1.4] we get:

Theorem 5.3 Let F be holomorphic vector bundle over P N and let g be a weight with respect to F on X . Moreover, assume that H is a Hefer morphism for E • ⊗ O(ρ) and
K and P are defined by (5.6). Then and the global Koppelman formula (5.5) holds on X for φ ∈ A q (X , F ⊗ L ρ−N ).

The Non-reduced Case
Now assume that i : X → P N has pure dimension n but is non-reduced. Then we still have an ideal sheaf J X ⊂ O P N that has pure dimension n but J X is no longer radical, i.e., there are nilpotent elements. Still the structure sheaf of X has the representation O X = O P N /J X . The underlying reduced space X red is associated with the radical ideal √ J X = J X red . Let X reg be the subset of X where X red is smooth and in addition J X is Cohen-Macaulay. In a neighborhood U of a point x in X reg , which is an open dense subset of X , we can choose local coordinates (z, w) such that X red ∩ U = {w = 0}. It turns out, see, e.g., [9], that there are monomials 1, w α 1 , . . . , w ν−1 such that each φ in O X has a unique representation φ =φ 0 (z) ⊗ 1 + · · · +φ ν−1 (z) ⊗ w ν−1 . (6.1) Thus O X has the structure of a free O X red -module in U. We say that in E 0, * P N is in Ker i * if in a neighborhood of each point in X reg , is in the subsheaf of E 0, * P N generated by J X ,J X red , and dJ X red . As in the reduced case we define E 0, * X = E P N /Ker i * , and again it is independent of the choice of local embedding of X . It turns out that at each point in X reg and coordinates (z, w) as above we have a unique representation (6.1) of φ in E 0, * X whereφ j are in E 0, * X red .
We define the sheaf of (n, * )-currents on X as the dual of E 0,n− * X , so that such a current τ is represented by a (N , N − n + * )-current i * τ in P N that is annihilated by Ker i * .
Basically all facts in Sect. 4 now hold verbatim, except for that one has to replace X by X red in (4.5) and slightly modify the proof of Lemma 4.1. It follows from Lemma 4.1 that there is a current ω such that (4.8) holds. However, in the non-reduced case we give no meaning to that ω is almost semi-meromorphic and smooth on X reg . The first part of (4.11) just means that 1 W R = 0 and this follows from [9, Corollary 6.3]. The second part of (4.11) follows from the first part precisely as before.
Following [9] we can also make the construction of Koppelman formulas in Sect. 5 and define sheaves A * on X so that Theorem 5.3 holds.

Remark 4
Recall that a holomorphic differential operator L is Noetherian with respect to the ideal J if Lφ vanishes on Z if φ ∈ J . As in the local case, cf.

Global Solutions
To begin with we consider a Hefer morphism, introduced in [3], for the complex E • ⊗ O(ρ), a for large ρ. Let E denote the complex of trivial bundles over C N +1 that we get from E, and let A denote the corresponding mappings (which then formally are just the original matrices a). Let δ w−z denote interior multiplication by Notice that is a projective form and that Given h k in Proposition 7.1 we let τ * h k be the projective form we obtain by replacing w by αζ and dw j by γ j . We then have (7.4) in light of (7.3) and (2.4). It is proved in [3] that if is a Hefer morphism for (4.1) with ρ = κ 0 . Clearly, H is holomorphic in z.
Recall that g = α ν is a holomorphic weight with respect to F = O(ν) for ν ≥ 0. From Proposition 5.2 we thus obtain explicit solutions to the∂-equation in L for ≥ κ 0 (X ) − N .

Proposition 7.2
Assume that X is a possibly singular projective subvariety of P N of pure dimension n, and s ≥ κ 0 − N . If φ is a∂-closed section in A q (X , L s ), q ≥ 1, then Thus the proposition gives a weaker form of Theorem 1.1.

Remark 5
The associated operator P is precisely the operator in [4,Example 3.4] that realizes the surjectivity of (1.7). That is, if φ ∈ O(X , L s ), then is a global section of O(s) → P N that coincides with φ on X .

Example 2
Assume that X is a complete intersection, i.e., J is generated by homogeneous forms a 1 1 , . . . , a p 1 , of degrees d 1 , . . . , d p , where p = N − n. Then the Koszul complex generated by a j 1 provides a minimal free resolution, and it is then easy to see that κ 0 = d 1 + · · · + d p , cf. Sect. 8 below. Moreover, by the adjunction formula Here K X is the Grothendieck dualizing sheaf, which in this case is a line bundle that is generated by ω = ω 0 . When X is smooth K X is just the usual canonical bundle. If we define (n, q)-forms as (0, q)-forms with values in K X , then Proposition 7.2 gives an explicit realization of the vanishing If X is smooth this follows precisely from Kodaira's theorem, since L is strictly positive on X .
For the proof of Theorem 1.1 we must make a more careful analysis of the kernels. Let us introduce the notation Notice that Proof of Theorem 1.1 Notice that the section h(ζ, is non-vanishing on . Let χ(t) be a cutoff function as before and let χ δ := χ(|h| 2 /δ) = χ |ζ · z| 2 /|z| 2 |ζ | 2 δ .
Here |h| denote the natural norm of the section h, whereas in the last term | | denotes norm of points in C N +1 . For small δ, χ δ is identically 1 in a neighborhood of and thus is a smooth weight (with respect to the trivial line bundle). For fixed z, g δ vanishes in a neighborhood of the hyperplane h = 0, and therefore α −r ∧g δ is a smooth weight with respect to O(−r ) for any r , though not holomorphic in z. Now fix q ≥ 1 and let t = s − q. In particular, for t ≥ κ q − N and φ ∈ A q (X , L t ), q ≥ 1, with∂φ = 0 we have the formula We claim that K δ φ tends to a current Kφ in A q−1 (X , L t ) and that P δ φ → 0. Taking this for granted, the theorem follows in view of (7.7).
To settle the claim we first consider the expression for P δ φ in (7.9). Since φ has bidegree (0, q) only components H N − R N − with ≥ q can occur in the integral. Thus the total power of α is in view of (7.6). Thus where ξ are smooth and holomorphic in z. Since q ≥ 1 we need some antiholomorphic differentials with respect to z and they must come from∂χ δ ∧B; hence we can forget about χ δ . Since χ δ = 1 in a neighborhood of the diagonal, we can consider B as smooth. Thus we have to verify that Since X is irreducible, h = 0 has positive codimension on X , and if φ is smooth thus (7.10) holds in view of Lemma 4.3. If φ is in A q , then it is in Dom X , cf. [6,9], and then (7.10) follows from (the proofs of) [6, Lemma 4.1] and [9,Lemma 8.4]. In fact, (7.10) can be reformulated as 1 h=0∂ (ω∧φ) = 0. We conclude that P δ φ → 0. Notice now that B∧B = 0 so that It is proved in [6,9] that (H R∧α t−κ 0 +N ∧B) N ∧φ is in the space W X ×P N , and this implies that It follows that

Examples with Negative Curvature
We now turn our attention to the case of negative curvature. We define a Hefer morphism from h k in Proposition 7.1 by replacing w by ζ , z by βz, and dw j by the γ j from Proposition 2.4. The morphism so obtained is a Hefer morphism for (4.1) (i.e., for E • ⊗ O(ρ), a with ρ = 0). This is verified in the same way as [3,Proposition 4.4]. This time H is not holomorphic in z but in ζ instead. Let δ be the depth of the ring S/J . This is a number, 0 ≤ δ ≤ n, and choosing (1.4) minimal, (4.1) will end up at k = N − δ, which means that R = R N −n + · · · + R N −δ . The variety X is Cohen-Macaulay precisely when δ = n.
From the Koppelman formula we get solutions to∂ (representation of the cohomology in the smooth case) for (0, q)-forms φ with values in O( ) for ≤ −N and thus solutions as soon as the obstruction term vanishes. Notice that H R has degree at most N − δ in dζ since β and γ j only contain holomorphic differentials with respect to ζ . Therefore (7.11) must vanish if N −δ+q < N , i.e., 0 ≤ q ≤ δ − 1.

Theorem 7.3
Assume that X is a subvariety of P N of pure dimension n and ≤ −N .

Global Complete Intersections
Let us compute the resulting formulas in case i : X → P N is a global complete intersection as in Example 2. Let f 1 , . . . , f p , f j = a j 1 , be our given homogeneous forms of degrees d j from Example 2, and recall that p = N − n. Assume that E 0 is the trivial line bundle and let where E j are trivial line bundles. Let e j be basis elements for E j and let e * j be the dual basis elements. We take O(−(d I 1 + · · · + d I k ))E I 1 ⊗ · · · ⊗ E I k and a k : E k → E k−1 as interior multiplication by f = f j e * j . Now Moreover, cf. Sect. 5 and [2]; here | λ=0 means evaluation at λ = 0 after analytic continuation.
Since codim Z = p the resulting residue current R just consists of the term R p ; it coincides with the classical Coleff-Herrera product We now compute Hefer morphisms for the Koszul complex. Leth j (w, z) be (1, 0)forms in C n+1 × C n+1 of polynomial degrees d j − 1 such that and let h j = τ * h j . We only have to care about k ≤ p so κ 0 = d 1 + · · · + d p . Then is a Hefer morphism. The components of most interest for us are H 0 k and H 1 k . Since it can be more compactly written formally as where δ h denotes formal interior multiplication with In the same way Our description of U , H 1 k , etc., is just to illustrate what these currents look like in the complete intersection case since they play a rule in the proofs above. As we have seen, however, in the final Koppelman formula only the term of H R occurs. It follows that the operator K in Proposition 7.2, with has the more explicit form and the operator P in Remark 5 is Proposition 8.1 Assume that the projective space i : X → P N of codimension p is defined by the homogeneous forms f j on C N +1 of degree d j , j = 1, . . . , p, and assume that s ≥ d 1 + · · · + d p − N . For φ ∈ E 0,k (X , L s ), or φ ∈ A k (X , L s ), we have the Koppelman formula (5.5) with K and P defined by (8.3) and (8.4), respectively. Moreover, P vanishes if k ≥ 1.

Remark 6
In [23] similar Koppelman formulas are obtained on a, not necessarily reduced, global complete intersection X for (0, * )-forms with values in L d 1 +···+d p −N −1 in the notation from Example 2. The authors construct Koppelman formulas on homogeneous subvarieties of C N +1 , keep track of homogeneities and so obtain Koppelman formulas on X . They use the same definition of∂ as we do. However, they only consider solutions to∂u = φ on X reg when φ is smooth on X and satisfies a condition ( * ) that in general is stronger than∂φ = 0. There is no discussion whether their solution has some meaning as a current across X sing . The condition ( * ) on φ means that (locally) there is a smooth extension to ambient space such that∂ is in EJ . Clearly this implies that∂φ = 0 on X but in general the converse does not hold. In fact, consider a reduced hypersurface X = {a = 0} ⊂ C n+1 so that J = (a). Then ( * ) means that there is an extension such that∂ = ξ a for some smooth form ξ . Then 0 = a∂ξ and thus∂ξ = 0; hence ξ =∂η for some smooth η. Now − aη is∂-closed and therefore there is a smooth solution to∂ = − aη. It follows that ψ = i * is a smooth solution to∂ψ = φ. However, it is well known that there are smooth φ with∂φ = 0 such that∂ψ = φ has no smooth solution, see, e.g., [6, Example 1.1].

The Reduced Case
Let us consider a more intrinsic-looking representation of K and P as in (5.6) and (5.7). In order to avoid a Noetherian operator, cf. Remark 4, let us in addition assume that X is reduced. Let A 1 , . . . , A p be holomorphic vector fields on C N +1 such that or equivalently, Notice that since δ ζ anti-commutes with δ A j , is a projective form. Following the proof of [4, Proposition 6.3] we see that ω is a representative for the structure form on X , that is,

Curves in P 2
Assume now that N = 2 and Notice that for degree reasons, In what follows, for simplicity, let us right α rather than α 0 . If ∂ f /∂ζ 2 is generically non-vanishing on X and then δ A d f = 2πi (generically) on X . We get Notice furthermore that The second term in the brackets actually cancels out the singularity if X is smooth.
Proof Differentiating (8.14) with respect to w j gives We conclude that Without loss of generality we may assume that ζ 0 = z 0 = 1 and that ζ 1 is a local coordinate on X so that ζ = g(ζ 1 ). Since α = 1 on the diagonal we then have that B = B (ζ 1 − z 1 ) where B is holomorphic in z. After homogenization we get that B = B (z 0 ζ 1 − z 1 ζ 0 ), where B is holomorphic in z. Thus the proposition follows.
Even if X is not smooth, by the same argument, one can identify the principal term of the kernel k.
By formula (8.17) we can now express k completely in terms of the parameters [t 0 , t 1 ] and [τ 0 , τ 1 ]. However, we restrict to considering the principal term. Since we are primarily interested in the singularity, we consider the standard affinization where z 0 = ζ 0 = 1. By the recipe above, then k = 1 2πi dζ 1 ζ 1 − z 1 α κ z 2 + αζ 2 2ζ 2 + · · · (8.20) In this case · · · is not smooth but at least the singularity is smaller than in the leading term. Since α − 1 = O(ζ 1 − z 1 ) + O(ζ 2 − z 2 ), we can delete α as well in the leading term in (8.20). We then have Letting t = t 1 /t 0 and τ = τ 1 /τ 0 we are then left with where the leading term is precisely the kernel in the last example in [6,Sect. 8].
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