A Note on the Singularities of Residue Currents of Integrally Closed Ideals

Abstract

Given a free resolution of an ideal \(\mathfrak a\) of holomorphic functions, there is an associated residue current R that coincides with the classical Coleff-Herrera product if \(\mathfrak a\) is a complete intersection ideal and whose annihilator ideal equals \(\mathfrak a\). In the case when \(\mathfrak a\) is an Artinian monomial ideal, we show that the singularities of R are small in a certain sense if and only if \(\mathfrak a\) is integrally closed.

Introduction

Given (a germ of) a holomorphic function f at 0 ∈Cn, Herrera and Lieberman [11] proved that one can define the principal value current

$$ \frac{1}{f}.\xi:=\lim\limits_{\epsilon\to 0}{\int}_{|f|^{2}>\epsilon}\frac{\xi}{f}, $$
(1.1)

for test forms ξ. It follows that \(\bar {\partial }(1/f)\) is a current with support on the variety of f; such a current is called a residue current. The duality principle asserts that a holomorphic germ g is in the ideal generated by f if and only if \(g\bar {\partial } (1/f)=0\).

Given a (locally) free resolution

$$ 0 \to E_{N} \stackrel{\varphi_{N}}{\longrightarrow} E_{N-1} \to {\cdots} \to E_{1} \stackrel{\varphi_{1}}{\longrightarrow} E_{0}\to 0 $$
(1.2)

of a general ideal (sheaf) \(\mathfrak a\), in [3] with Andersson, we defined a vector (bundle) valued residue current R with support on the variety of \(\mathfrak a\) that satisfies the duality principle for \(\mathfrak a\), cf. Section 2.2 below. If \(\mathfrak a\) is Cohen-Macaulay, then R is essentially independent of the resolution. In particular, if (E,φ) is the Koszul complex of a minimal set of generators f1,…,fp of a complete intersection ideal, then R coincides with the classical Coleff-Herrera product [9],

$$ \bar{\partial}\frac{1}{f_{p}}\wedge\cdots\wedge\bar{\partial}\frac{1}{f_{1}}. $$
(1.3)

By means of these residue currents, we were able to extend several results previously known for complete intersections. These currents have also turned out to be particularly useful for analysis on singular spaces; for example, they have been used to obtain new results on the \(\bar {\partial }\)-equation [2] and new global versions of the classical Briançon-Skoda theorem [5] on singular spaces.

In view of the duality principle, the residue current R can be thought of as a current representing the ideal \(\mathfrak a\); this idea is central to many applications of residue currents, including the ones mentioned above. Various properties of the ideal \(\mathfrak a\) are reflected in the residue current R. For example, R has a natural geometric decomposition corresponding to a primary decomposition of \(\mathfrak a\), see [4], and the fundamental cycle of \(\mathfrak a\) admits a natural representation in terms of R that generalizes the classical Poincaré-Lelong formula, see [14].

In this note we study the singularities of R and show that, for a monomial ideal \(\mathfrak a\), they are small in a certain sense if and only if \(\mathfrak a\) is integrally closed. For simplicity, we will work in a local setting; let \(\mathcal {O}_{0}^{n}\) be the ring of germs of holomorphic functions at 0 ∈Cn and let \(\mathfrak a\) be an ideal in \(\mathcal {O}_{0}^{n}\). Recall that \(g\in \mathcal {O}_{0}^{n}\) is in the integral closure\(\overline {\mathfrak {a}}\) of \(\mathfrak a\) if |g|≤ C|f|, where C is a constant and f is a set of generators \(f_{1},\ldots , f_{m}\in \mathcal {O}_{0}^{n}\) of \(\mathfrak a\), or equivalently if g satisfies a monic equation gq + h1gq− 1 + ⋯ + hq = 0, where \(h_{k}\in \mathfrak a^{k}\). If \(\overline {\mathfrak {a}}=\mathfrak a\), then \(\mathfrak a\) is said to be integrally closed. Assume that \(\pi :\widetilde X\to (\mathbf {C}^{n},0)\) is a log-resolution of \(\mathfrak a\), i.e., \(\widetilde X\) is a complex manifold, π is a biholomorphism outside the variety of \(\mathfrak a\), and \(\mathfrak a\cdot \mathcal {O}_{\widetilde X}=\mathcal {O}_{\widetilde X}(-D)\), where \(D={\sum }_{i=1}^{N}r_{i} D_{i}\) is an effective divisor with simple normal crossings support. Then \(\overline {\mathfrak {a}}=\pi _{*} (\mathcal {O}_{\widetilde X}(-D))\), which means that \(g\in \mathcal {O}_{0}^{n}\) is in \(\overline {\mathfrak {a}}\) if and only if \(\text {ord}_{D_{i}}(g)\geq r_{i}\) for each i, where \(\text {ord}_{D_{i}}\) denotes the divisorial valuation defined by the prime divisor Di.

If \(\pi : \widetilde X\to (\mathbf {C}^{n},0)\) is a common log-resolution of \(\mathfrak a\) and the Fitting ideals of \(\mathfrak a\), i.e., the ideals generated by the minors of optimal rank of the φk in (1.2), then there is a section σ of a line bundle \(L=\mathcal {O}_{\widetilde X}(-F)\) over \(\widetilde X\) and a current \(\widetilde R\) on \(\widetilde X\) such that

$$ \widetilde R\wedge \pi^{*} dz= \eta\wedge \bar{\partial} \frac{1}{\sigma}, $$
(1.4)

where dz = dz1 ∧⋯ ∧ dzn and η is a vector (bundle) valued smooth form with values in L, such that \(\pi _{*}\widetilde R=R\), see [3, Section 2] and Section 2.2 below. The observation that residue currents in this way can be seen as pushforwards of residue currents of principal ideal sheaves is crucial for many applications of residue currents, cf. Section 2.1 below.

Assume that

$$ \sigma=\sigma_{1}^{a_{1}}{\cdots} \sigma_{N}^{a_{N}}, $$
(1.5)

where σi are holomorphic sections of line bundles \(\mathcal {O}(-D_{i})\) defining the prime divisors Di of \(F={\sum }_{i=1}^{N} a_{i} D_{i}\). We are interested in the exponents ai. Naively, one could hope that one could choose ai as \(r_{i}=\text {ord}_{D_{i}}(\mathfrak a)\). However, this can only be true if \(\mathfrak a\) is integrally closed. Indeed, assume that \(R=\pi _{*} \widetilde R\), where \(\widetilde R\) satisfies (1.4) with σ given by (1.5) with airi. Take \(g\in \overline {\mathfrak {a}}\); then \(\text {ord}_{D_{i}}(g)\geq r_{i}\) for each i and thus \(\pi ^{*} g = \sigma g^{\prime }\), where \(g^{\prime }\) is a holomorphic section of L− 1. Therefore, by the duality principle, \(\pi ^{*} g\bar {\partial } (1/\sigma )=0\) and so \(\pi ^{*} g\widetilde R=0\), which implies that gR = 0, and hence \(g\in \mathfrak a\). To conclude, if we can choose airi for each i, then \(\mathfrak a\) is integrally closed.

We are interested in whether the converse holds, i.e., if \(\mathfrak a\) is integrally closed, is it then always possible to find an \(\widetilde R\) as above with σ given by (1.5) with airi? In this note, we answer this question affirmatively when R is the residue current associated with a cellular resolution, introduced by Bayer-Sturmfels [8], see Section 2.3 below, of an Artinian, i.e., 0-dimensional, monomial ideal, and when we moreover allow η to be semi-meromorphic, i.e., of the form (1/f)ω, where ω is smooth and f is holomorphic. For the definition of the product ημ, where η is a semi-meromorphic form and μ is a residue current, or more generally a so-called pseudomeromorphic current, see Section 2.1 below. Multiplication from the left by η does not increase the singularities in the sense that if g is a holomorphic function such that gμ = 0, then gημ = 0.

Theorem 1.1

Let\(M\subset \mathcal {O}_{0}^{n}\)bean integrally closed Artinian monomial ideal and let R be the residuecurrent associated with a cellular resolution of M. Then there is alog-resolution\(\pi :\widetilde X\to (\mathbf {C}^{n},0)\)of M, such that\(M\cdot \mathcal {O}_{\widetilde X}=\mathcal {O}_{\widetilde X} (-D)\),where\(D={\sum }_{i=1}^{N} r_{i} D_{i}\), and acurrent\(\widetilde R\)on\(\widetilde X\)suchthat\(\pi _{*}\widetilde R=R\)and

$$ \widetilde R\wedge \pi^{*} dz= \eta\wedge \bar{\partial} \frac{1}{\sigma_{1}^{r_{1}}{\cdots} \sigma_{N}^{r_{N}}}, $$
(1.6)

where σi is a holomorphic section defining Di and η is a semi-meromorphic form.

The proof uses explicit descriptions of residue currents of monomial ideals, [19], as well as so-called Bochner-Martinelli residue currents, [13], cf. Sections 2.3 and 2.6 below, and it should be possible to extend to general, not necessarily Artinian, monomial ideals. There is a brief discussion of this and other aspects of our result at the end of Section 3.

Preliminaries

Pseudomeromorphic Currents

To get a coherent approach to principal value and residue currents, in [4] with Andersson, we introduced the sheaf of pseudomeromorphic currents which essentially are pushforwards of tensor products of principal value and residue currents times smooth forms, like

$$ \frac{1}{s_{2}^{b_{2}}{\cdots} s_{m}^{b_{m}}} \omega \wedge \bar{\partial} \frac{1}{s_{1}^{b_{1}}}, $$

where s1,…,sm are (local) coordinates in some Cm and ω is a smooth form. Principal value currents and the residue currents mentioned in this paper are typical examples of pseudomeromorphic currents.

Pseudomeromorphic currents have a geometric nature similar to positive closed currents. For example, the dimension principle states that if the pseudomeromorphic current μ has bidegree (∗,p) and support on a variety of codimension larger than p, then μ vanishes. Moreover if μ is a pseudomeromorphic current and 1V is the characteristic function of an analytic variety V, then the product 1Vμ, defined through a suitable regularization, is a well-defined pseudomeromorphic current with support on V, see [4, Proposition 2.2].

A current of the form (1/f)ω where f is a holomorphic section of a line bundle LX and ω is a smooth form with values in L is said to be semi-meromorphic. If η is a semi-meromorphic form, or more generally the pushforward under a modification of a semi-meromorphic form, and μ is a pseudomeromorphic current, there is a unique pseudomeromorphic current ημ that coincides with the usual product where η is smooth and such that 1ZSS(η)ημ = 0, where ZSS(η) is the smallest analytic set containing the set where η is not smooth, see, e.g., [6, Section 4.2]. If h is a holomorphic tuple such that {h = 0} = ZSS(η) and χ(t) is (a smooth approximand of) the characteristic function of the interval \([1,\infty )\), then

$$ \eta\wedge \mu=\lim\limits_{\epsilon\to 0}\chi(|h|^{2}/\epsilon) \eta\wedge\mu. $$

It follows that for c > 0

$$ \frac{1}{{s_{i}^{c}}}~ \frac{1}{{s_{i}^{b}}}= \frac{1}{s_{i}^{b+c}}, \quad \frac{1}{{s_{i}^{c}}}~ \bar{\partial}\frac{1}{{s_{i}^{b}}}= 0. $$
(2.1)

For further reference, in view of (1.1), note that

$$ {s_{i}^{c}}~ \frac{1}{{s_{i}^{b}}}= \frac{1}{s_{i}^{b-c}}, \quad {s_{i}^{c}}~ \bar{\partial}\frac{1}{{s_{i}^{b}}}= \bar{\partial}\frac{1}{s_{i}^{b-c}}. $$
(2.2)

Example 2.1

Assume that s1,…,sn are (local) coordinates in Cn. If D = {s1 = 0}, by the dimension principle,

$$ \mathbf{1}_{D} ~\bar{\partial} \frac{1}{s_{1}^{b_{1}}{\cdots} s_{n}^{b_{n}}}= \frac{1}{s_{2}^{b_{2}}{\cdots} s_{n}^{b_{n}}} ~ \bar{\partial} \frac{1}{s_{1}^{b_{1}}}. $$

In view of (2.1) and (2.2), it follows that for c1b1 and any choices of c2,…,cn,

$$ \mathbf{1}_{D} ~\bar{\partial} \frac{1}{s_{1}^{b_{1}}{\cdots} s_{n}^{b_{n}}}= \mathbf{1}_{D} ~ s_{1}^{c_{1}-b_{1}}{\cdots} s_{n}^{c_{n}-b_{n}} ~\bar{\partial} \frac{1}{s_{1}^{c_{1}}{\cdots} s_{n}^{c_{n}}}, $$

where the factor \(s_{i}^{c_{i}-b_{i}}\) should be understood as a principal value if ci < bi.

Residue Currents from Complexes of Vector Bundles

Let

$$ 0 \to E_{N} \stackrel{\varphi_{N}}{\longrightarrow} E_{N-1} \to {\cdots} \to E_{1} \stackrel{\varphi_{1}}{\longrightarrow} E_{0}\to 0 $$
(2.3)

be a complex of Hermitian vector bundles over a complex manifold X of dimension n that is exact outside a variety ZX. In [3] with Andersson, we constructed an End(⊕Ek)-valued residue current R with support on Z that in some sense measures the exactness of the associated sheaf complex

$$ 0 \to \mathcal{O}(E_{N}) \stackrel{\varphi_{N}}{\longrightarrow} \mathcal{O}(E_{N-1}) \to {\cdots} \to \mathcal{O}(E_{1}) \stackrel{\varphi_{1}}{\longrightarrow} \mathcal{O}(E_{0})\to 0 $$
(2.4)

of holomorphic sections. If (2.4) is exact, then R satisfies the duality principle, which means that if ξ is a section of E0 that is generically in the image of φ1, then Rξ = 0 if and only if ξ ∈Imφ1; in particular, if (2.4) is a free resolution of an ideal \(\mathfrak a\subset \mathcal {O}_{0}^{n}\), then the annihilator ideal of R, i.e., the ideal of germs of holomorphic functions g such that gR = 0, equals \(\mathfrak a\). Moreover, then R is of the form \(R=\sum R_{k}\), where Rk is a Hom(E0,Ek)-valued pseudomeromorphic current of bidegree (0,k). Note that Rk vanishes for k < codimZ by the dimension principle, and for k > n for degree reasons. In particular, if (2.4) is a free resolution of an Artinian monomial ideal in \(\mathcal {O}_{0}^{n}\), then R = Rn.

Let ρk be the optimal rank of φk, and let \(\pi :\widetilde X\to X\) be a common log-resolution of the ideal sheaves generated by the ρk-minors of the φk, i.e., such that the pullback of the section \(\det ^{\rho _{k}} \varphi _{k}\) of \({\Lambda }^{\rho _{k}} E_{k}^{*} \otimes {\Lambda }^{\rho _{k}} E_{k-1}\) is of the form \(t_{k}\rho _{k}^{\prime }\), where tk is a section of some line bundle Lk and \(\varphi _{k}^{\prime }\) is a nonvanishing section of \(L_{k}^{-1}\otimes {\Lambda }^{\rho _{k}}\pi ^{*} E_{k}^{*} \otimes {\Lambda }^{\rho _{k}} \pi ^{*} E_{k-1}\). It was proved in [3, Section 2] that there is a current \(\widetilde R\) on \(\widetilde X\) such that \(\pi _{*} \widetilde R = R\) and \(\widetilde R = \omega \wedge \bar {\partial } (1/\sigma )\), where ω is smooth and \(\sigma =t_{1}{\cdots } t_{{\min \nolimits } (n,N)}\). The form ω may vanish along the divisor F of σ, and thus in general it may be possible to find a σ that vanishes to lower order along F than \(t_{1}{\cdots } t_{{\min \nolimits } (n,N)}\), cf. Example 2.3 below.

Monomial Ideals and Cellular Resolutions

Let us briefly recall the construction of cellular resolutions due to Bayer-Sturmfels, [8]. Let M be a monomial ideal in the polynomial ring S := C[z1,…,zn], i.e., M is generated by monomials m1,…,mr in S. Moreover, let K be an oriented polyhedral cell complex, where the vertices are labeled by the generators mi and the face τ of K is labeled by the least common multiple mτ of the labels mi of the vertices of τ. Then with K there is an associated graded complex of S-modules. For \(k=0,\ldots , \dim K+1\), let Ak be the free S-module with one generator eτ in degree mτ for each τKk, where Kk denotes the faces of K of dimension k − 1 (K0 should be interpreted as {}) and let φk : AkAk− 1 be defined by

$$ \varphi_{k}: e_{\tau} \mapsto \sum\limits_{\tau^{\prime}\subset \tau} \text{sgn}(\tau^{\prime},\tau)~\frac{m_{\tau}}{m_{\tau^{\prime}}}~ e_{\tau^{\prime}}, $$
(2.5)

where \(\text {sgn}(\tau ^{\prime },\tau )\) is a sign coming from the orientation of K. Now the complex (A,φ) is exact precisely if the labeled polyhedral cell complex K satisfies a certain acyclicity condition, see [8, Proposition 1.2]. We then say that the complex (A,φ) is a cellular resolution of M. Any monomial ideal admits a cellular resolution, cf. [8, Proposition 1.5].

Let M denote also the monomial ideal of germs of holomorphic functions at \(0\in \mathbf {C}^{n}_{z_{1},\ldots , z_{n}}\) generated by the monomials m1,…,mr. Since \(\mathcal {O}_{0}^{n}\) is flat over S, (A,φ) induces a free resolution of \(M\subset \mathcal {O}_{0}^{n}\). More precisely, for \(k=0,\ldots , N=\dim K+1\), let Ek be a trivial bundle over (a neighborhood of 0 in) Cn with a global frame \(\{e_{\tau }\}_{\tau \in K_{k}}\), endowed with the trivial metric, and where the differential φk is given by (2.5). Then (2.4) is exact if (A,φ) is. We will think of monomial ideals sometimes as ideals in S, sometimes as ideals in \(\mathcal {O}_{0}^{n}\), and sometimes as ideals in the ring of entire functions in Cn.

In [19], we computed the residue current R of a cellular resolution of a monomial ideal M. Note that if M is Artinian, then R = Rn is of the form \(R=\sum R_{\tau } e_{\tau }\otimes e_{\emptyset }^{*}\), i.e., with one component for each τKn. Proposition 3.1 in that paper asserts that if \(z^{\alpha }:=z_{1}^{\alpha _{1}}\cdots z_{n}^{\alpha _{n}}\) is the label of τ, then Rτ = cτRα, where cτC and

$$ R_{\alpha}= \bar{\partial}\frac{1}{z_{n}^{\alpha_{n}}}\wedge{\cdots} \wedge \bar{\partial} \frac{1}{z_{1}^{\alpha_{1}}}. $$
(2.6)

Toric Log-Resolutions

For an (Artinian) monomial ideal M in Cn, it is possible to find a log-resolution \(\pi :\widetilde X\to \mathbf {C}^{n}\) where \(\widetilde X\) is a toric variety. Let us briefly recall this construction, which can be found, e.g., in [7, p. 82]. For a general reference on toric varieties, see, e.g., [10]. A (rational strongly convex) cone in Rn is a set of the form \(\mathcal C=\sum \mathbf {R}_{+}v_{i}\), where vi are in the lattice Zn, that contains no line; here R+ denotes the non-negative real numbers. A cone is regular if the vi can be chosen as part of a basis for the lattice Zn. A fan Δ is a finite collection of cones such that all faces and intersections of cones in Δ are in Δ; Δ is regular if all cones are regular. A regular fan Δ determines a smooth toric variety X(Δ), obtained by patching together affine toric varieties corresponding to the cones in Δ.

Assume that M is an Artinian monomial ideal in \( \mathbf {C}^{n}_{z_{1},\ldots , z_{n}}\). Recall that the Newton polyhedron NP(M) of M is defined as the convex hull in Rn of the exponents of monomials in M. Let \( \mathcal S(M)\) be the collection of cones of the form \( \mathcal C=\mathbf {R}_{+}\rho \subset \mathbf {R}^{n}_{+}\), where ρ is a normal vector of a compact facet (face of maximal dimension) of NP(M). Let Δ be a regular fan that contains \(\mathcal S(M)\) and such that the support, i.e., the union of all cones in Δ, equals \(\mathbf {R}^{n}_{+}\). The cones in \(\mathcal S(M)\) determine a fan with support \(\mathbf {R}^{n}_{+}\) and by refining this is always possible to find such a Δ. Then π : X(Δ) →Cn is a log-resolution of M. The prime divisors Di of the exceptional divisor correspond to one-dimensional cones \(\mathcal C_{i}=\mathbf {R}_{+} \rho ^{i}\) in Δ and \(\text {ord}_{D_{i}}\) are monomial valuations (i.e., determined by its values on z1,…,zn). More precisely, if ρ = (ρ1,…,ρn) is the first non-zero lattice point met along \(\mathcal C_{i}\), then \(\text {ord}_{D_{i}}\) is the monomial valuation \(\text {ord}_{\rho }(z_{1}^{a_{1}}{\cdots } z_{n}^{a_{n}}):=\rho _{1} a_{1}+{\cdots } +\rho _{n} a_{n}\).

Rees Valuations

Given a non-zero ideal \(\mathfrak a\subset \mathcal {O}_{0}^{n}\), let ν : X+ → (Cn,0) be the normalized blow-up of \(\mathfrak a\) and let \(D=\sum r_{i} D_{i}\) be the exceptional divisor, such that \(\mathfrak a\cdot \mathcal {O}_{X^{+}}=\mathcal {O}_{X^{+}}(-D)\). The divisorial valuations \(\text {ord}_{D_{i}}\) are called the Rees valuations of \(\mathfrak a\), see, e.g., [15, Section 9.6.A]. Then \(\overline {\mathfrak {a}}=\nu _{*} (\mathcal {O}_{X^{+}}(-D))\), i.e., \(g\in \overline {\mathfrak {a}}\) if and only if \(\text {ord}_{D_{i}}(g)\geq \text {ord}_{D_{i}}(\mathfrak a)\) for each i. If \(\pi :\widetilde X\to (\mathbf {C}^{n}, 0)\) is any log-resolution of \(\mathfrak a\) (and thus factors through the normalized blow-up) with exceptional divisor \(D=\sum r_{i} D_{i}\), following [13] we say that the prime divisor Di is a Rees divisor if \(\text {ord}_{D_{i}}\) is a Rees valuation.

Let M be an Artinian monomial ideal (at 0) in \(\mathbf {C}^{n}_{z_{1},\ldots , z_{n}}\). Then the Rees valuations are monomial and in one-to-one correspondence with the compact facets of NP(M). We say that the normal vector ρ of a facet τ is primitive if it has minimal non-negative entries, i.e., if ρ is the first lattice point met along the cone \(\mathbf {R}_{+}\rho \subset \mathbf {R}^{n}_{+}\). If ρ is a primitive normal vector of a compact facet τ, then the Rees valuation corresponding to τ is the monomial valuation ordρ, see, e.g., [12, Theorem 10.3.5] and [7, p. 82]. It follows that in the toric log-resolution π : X(Δ) →Cn in the previous section, Di is a Rees divisor of M if and only if the corresponding cone \(\mathcal C_{i}\) is in \(\mathcal S(M)\).

Example 2.2

Given \(\beta = (\beta _{1},\ldots , \beta _{n})\in \mathbf {N}^{n}\), we let \(\mathfrak {m}^{\beta }\) denote the Artinian monomial complete intersection ideal generated by \(z_{1}^{\beta _{1}}, \ldots , z_{n}^{\beta _{n}}\). Then \(\text {NP}(\mathfrak {m}^{\beta })\) has a unique compact facet, namely the simplex τ with vertices (β1,0…,0),(0,β2,0,…,0), …,(0,…,0,βn). Let ρj = β1βj− 1βj+ 1βn; then ρ = (ρ1,…,ρn) is a normal vector of τ and thus the unique Rees valuation of \(\mathfrak {m}^{\beta }\) is of the form r ordρ for some rQ. Note that \(\text {ord}_{\rho }(z_{i}^{\beta _{i}})=\text {ord}_{\rho }(\mathfrak {m}^{\beta })\) for all i.

Bochner-Martinelli Residue Currents

Let f = (f1,…,fp) be a tuple (of germs) of holomorphic functions at 0 ∈Cn and let (2.3) be the Koszul complex of f, i.e., consider f as a section \(f=\sum f_{j} e_{j}^{*}\) of a trivial rank p bundle E over (a neighborhood of 0 in) Cn with a frame \(e_{1}^{*},\ldots , e_{p}^{*}\), let \(E_{j}=\bigwedge ^{j} E\), where E is the dual bundle of E, and let φk = δf be contraction with f. Assume that the complex is equipped with the trivial metric. Then the coefficients of the associated residue current are the so-called Bochner-Martinelli residue currents introduced by Passare et al. [17]. In particular, if f1,…,fp are minimal generators of a complete intersection ideal, then the only nonvanishing coefficient of R = Rp equals the Coleff-Herrera product (1.3), see [17, Theorem 4.1] and [1, Theorem 1.7].

In [13], together with Jonsson, we gave a geometric description of the residue current R in this case in terms of the Rees valuations of the ideal \(\mathfrak a=\mathfrak a(f)\) generated by f. It is proved in Section 4 in that paper that if \(\pi :\widetilde X\to (\mathbf {C}^{n},0)\) is a log-resolution of \(\mathfrak a\), then there is a current \(\widetilde R\) such that \(\pi _{*}\widetilde R=R\) and \(\widetilde R\) has support on the Rees divisors of \(\mathfrak a\). Moreover if \(D={\sum }_{i=1}^{N} r_{i} D_{i}\) is the exceptional divisor of π, then

$$ \widetilde R = \omega \wedge \bar{\partial} \frac{1}{\sigma_{1}^{n r_{1}}{\cdots} \sigma_{N}^{n r_{N}}}, $$
(2.7)

where σi is a holomorphic section defining Di and ω is a smooth form.

Example 2.3

Let \(\mathfrak a_{\ell }=(z_{1}^{\ell },\ldots , z_{n}^{\ell })\subset \mathcal {O}_{0}^{n}\), and let (2.3) be the Koszul complex of \((z_{1}^{\ell },\ldots , z_{n}^{\ell })\). Then the ρk-minors of the φk are monomials of degree ρk. It follows that the blow-up of Cn at 0 is a common log-resolution of \(\mathfrak a_{\ell }\) and the ideals generated by the ρk-minors of the φk. Let D = {σ1 = 0} denote the exceptional (prime) divisor. Then ordD(zi) = 1 for each i and ordD(dz) = n − 1. It follows that the section tk from Section 2.2 is of the form \(t_{k}=\sigma _{1}^{\rho _{k} \ell }\), so that according to Section 2.2 there is an \(\widetilde R\) that satisfies (1.4) with \(\sigma =\sigma _{1}^{(\rho _{1}+{\cdots } + \rho _{n})\ell - n+1}\) and where η is smooth. However, noting that \(\text {ord}_{D}(\mathfrak a_{\ell })=\ell \), in view of (2.7), we can, in fact, choose \(\widetilde R\) with \(\sigma =\sigma _{1}^{(n-1)\ell +1}\).

Proof of Theorem 1.1

Theorem 1.1 is a direct consequence of the following slightly more precisely formulated result.

Theorem 3.1

Let\(M\subset \mathcal {O}_{0}^{n}\)be anintegrally closed Artinian monomial ideal and let R be the residue current associated with acellular resolution of M corresponding to the labeled polyhedral cell complex K. Then there is alog-resolution\(\pi :\widetilde X\to (\mathbf {C}^{n},0)\)of M and acurrent\(\widetilde R\)on\(\widetilde X\)withsupport on the Rees divisors of M suchthat\(\pi _{*}\widetilde R= R\)and\(\widetilde R\wedge \pi ^{*} dz\)isof the form (1.6), where\(D=\sum r_{i} D_{i}\),σi, andηareas in Theorem 1.1. More precisely, for eachτKn, there is acurrent\(\widetilde R_{\tau }\)on\(\widetilde X\)anda Rees divisorDτsuchthat\(\widetilde R_{\tau }\)has support onDτ,\(\pi _{*}\widetilde R_{\tau }= R_{\tau }\),and

$$ \widetilde R_{\tau}\wedge \pi^{*} dz= \eta_{\tau}\wedge \bar{\partial} \frac{1}{\sigma_{1}^{r_{1}}{\cdots} \sigma_{N}^{r_{N}}}, $$

where ητ is a semi-meromorphic form.

Proof

Let \(\pi :\widetilde X\to (\mathbf {C}^{n},0)\) be a toric log-resolution of M in the sense of Section 2.4. Consider an entry Rτ = cτRα of R, where cτ≠ 0, cf. Section 2.3. Note that zα1Rα≠ 0, where 1 = (1,…,1). It follows that zα1Rτ≠ 0, and thus zα1R≠ 0, which by the duality principle implies that zα1M. Since M is integrally closed, there is a Rees divisor Dτ, that we may assume equals D1, of \(M=\overline M\) such that

$$ \text{ord}_{D_{1}}(z^{\alpha-\mathbf{1}})<\text{ord}_{D_{1}}(M), $$
(3.1)

see Section 2.5.

Since M is monomial, \(\text {ord}_{D_{1}}\) is a monomial valuation of the form ordρ, where ρ = (ρ1,…,ρn) is the primitive normal vector of one of the compact facets of NP(M); in particular, ρjN, see Sections 2.4 and 2.5. Let γj = ρ1ρj− 1ρj+ 1ρn and choose kN such that βj := kγjαj for all j. Then ρ is the primitive normal vector of the unique compact facet of the Newton polyhedron of \(\mathfrak {m}^{\beta }=(z_{1}^{\beta _{1}},\ldots , z_{n}^{\beta _{n}})\), so that D1 is the unique Rees divisor of \(\mathfrak {m}^{\beta }\), see Example 2.2. It follows that \(\pi :\widetilde X\to (\mathbf {C}^{n}, 0)\) is a log-resolution of \(\mathfrak {m}^{\beta }\), see Section 2.4. Recall from Section 2.6 that (the coefficient of) the Bochner-Martinelli residue current of \((z_{1}^{\beta _{1}},\ldots , z_{n}^{\beta _{n}})\) equals Rβ, defined as in (2.6). Thus in view of Section 2.6, on \(\widetilde X\) there is an \(\widetilde R_{\beta }\) with support on D1 such that \(\pi _{*}\widetilde R_{\beta } = R_{\beta }\) and

$$ \widetilde R_{\beta} = \omega_{\beta} \wedge \bar{\partial} \frac{1}{\sigma_{1}^{n \text{ord}_{D_{1}}(\mathfrak{m}^{\beta})}{\cdots} \sigma_{N}^{n \text{ord}_{D_{N}}(\mathfrak{m}^{\beta})}}, $$

where ωβ is smooth.

Let \(\widetilde R_{\alpha }=\pi ^{*} (z^{\beta -\alpha }) \widetilde R_{\beta }\). Then \(\widetilde R_{\alpha }\) has support on D1 and by (2.2), \(\pi _{*}\widetilde R_{\alpha }=R_{\alpha }\). Moreover

$$ \widetilde R_{\alpha} \wedge \pi^{*}dz = \omega \wedge \bar{\partial} \frac{1}{\sigma_{1}^{a_{1}}{\cdots} \sigma_{N}^{a_{N}}}, $$
(3.2)

where \(a_{i}=n\text {ord}_{D_{i}}(\mathfrak {m}^{\beta }) -\text {ord}_{D_{i}}(z^{\beta -\alpha })-\text {ord}_{D_{i}}(dz)\) and ω is smooth. A direct computation gives that \(\text {ord}_{D_{i}}(dz) \geq \text {ord}_{D_{i}}(z^{\mathbf {1}})-1\). Since \(n\text {ord}_{D_{1}} (\mathfrak {m}^{\beta })=\text {ord}_{D_{1}}(z^{\beta })\), see Example 2.2, it follows that

$$ a_{1}=\text{ord}_{D_{1}}(z^{\beta})-\text{ord}_{D_{1}}(z^{\beta-\alpha})-\text{ord}_{D_{1}}(dz)\leq \text{ord}_{D_{1}}(z^{\alpha-\mathbf{1}})+1\leq \text{ord}_{D_{1}}(M), $$

cf. (3.1).

That \(D={\sum }_{i=1}^{N} r_{i} D_{i}\) has simple normal crossings support means that at \(x\in \widetilde X\), we can choose coordinates s1,…,sn such that for some p, \(\pi ^{-1}(0)=\{s_{1}{\cdots } s_{p}=0\}\) and for each i either xDi or Di = {sj = 0} for some j. Thus, we may assume that at x, for i = 1,…,p, \(\sigma _{i}=s_{i}\sigma _{i}^{\prime }\), where \(\sigma _{i}^{\prime }\) does not vanish at x, and moreover σp+ 1,…,σN do not vanish at x. Since \(a_{1}\leq r_{1}=\text {ord}_{D_{1}}(M)\), in view of Example 2.1,

$$ \mathbf{1}_{D_{1}} \bar{\partial} \frac{1}{\sigma_{1}^{a_{1}}{\cdots} \sigma_{N}^{a_{N}}}= \mathbf{1}_{D_{1}} \sigma_{1}^{r_{1}-a_{1}}{\cdots} \sigma_{N}^{r_{N}-a_{N}} \bar{\partial} \frac{1}{\sigma_{1}^{r_{1}}{\cdots} \sigma_{N}^{r_{N}}}. $$
(3.3)

Let ηα be the semi-meromorphic form \(\eta _{\alpha }=\sigma _{1}^{r_{1}-a_{1}}{\cdots } \sigma _{N}^{r_{N}-a_{N}} \omega \). Since \(\widetilde R_{\alpha }\) has support on D1, it follows from (3.2) and (3.3) that

$$ \widetilde R_{\alpha} \wedge \pi^{*} dz = \eta_{\alpha}\wedge \bar{\partial} \frac{1}{\sigma_{1}^{r_{1}}{\cdots} \sigma_{N}^{r_{N}}}. $$

Now, let ητ = cτηα and \(\eta ={\sum }_{\tau \in X_{n}} \eta _{\tau } ~e_{\tau }\otimes e_{\emptyset }^{*}\). Then \(\widetilde R_{\tau }\) and \(\widetilde R\) are of the desired form. □

By using the description of residue currents of general, not necessarily Artinian, monomial ideals in [19, Section 5], it should be possible to extend Theorems 1.1 and 3.1 to this setting, although the formulations would become slightly more complicated. However, the arguments above rely heavily on the explicit descriptions of the log-resolution of a monomial ideal M and the residue current R of a cellular resolution of M, as well as the explicit description of Bochner-Martinelli residue currents, and it does not seem obvious how to extend them to non-monomial ideals.

In [16], Lazarsfeld and Lee proved that multiplier ideals are very special among integrally closed ideals by proving that the maps φj in a free resolution do not vanish to high order in a certain sense. It might happen that in a similar way, R has small singularities, in the sense that it is the pushforward of a current \(\widetilde R\) that satisfies (1.6), only for a restricted class of integrally closed ideals.

We finally remark that if the residue current R, associated with a general ideal \(\mathfrak a\subset \mathcal {O}_{0}^{n}\), is the pushforward of a current \(\widetilde R\) of the form (1.4), then, in general, the exponents ai in (1.5) have to be (at least) like nri, where ri is as in the introduction. Indeed, assume that for some νN, aiνri for each i, and take \(g\in \overline {\mathfrak a}^{\nu }\). Then πg is divisible by σ and thus gR = 0, cf. the arguments after (1.5). It follows that \(\overline {\mathfrak a}^{\nu }\subset \mathfrak a\). The classical Briançon-Skoda theorem [18] asserts that this inclusion holds for \(\nu =\min \nolimits (n,m)\), where m is the minimum number of generators. This theorem is sharp and therefore in general the ai need to be at least like nri, cf. Example 2.3.

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Acknowledgments

I would like to thank Mats Andersson for valuable discussions on the topic of this paper. I am also very grateful to the referee for the careful reading, for pointing out some obscurities and mistakes in a previous version of this paper, and for several useful comments and suggestions.

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The author was partially supported by the Swedish Research Council.

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Wulcan, E. A Note on the Singularities of Residue Currents of Integrally Closed Ideals. Acta Math Vietnam 45, 281–289 (2020). https://doi.org/10.1007/s40306-019-00342-5

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Keywords

  • Residue currents
  • Integrally closed ideals
  • Monomial ideals

Mathematics Subject Classification (2010)

  • 13B22
  • 13D02
  • 32A27
  • 32S45