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.

1 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,…,f_p 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 g^q + h₁g^q− 1 + ⋯ + h_q = 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 D_i.

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 φ_k 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 ∧⋯ ∧ dz_n 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 σ_i are holomorphic sections of line bundles $\mathcal {O}(-D_{i})$ defining the prime divisors D_i of $F={\sum }_{i=1}^{N} a_{i} D_{i}$. We are interested in the exponents a_i. Naively, one could hope that one could choose a_i 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 a_i ≤ r_i. 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 a_i ≤ r_i 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 a_i ≤ r_i? 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 η ∧ μ, 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 gη ∧ μ = 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 σ_i is a holomorphic section defining D_i 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.

2 Preliminaries

2.1 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,…,s_m are (local) coordinates in some C^m 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 1_V is the characteristic function of an analytic variety V, then the product 1_Vμ, 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 L → X 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 η ∧ μ that coincides with the usual product where η is smooth and such that 1_ZSS(η)η ∧ μ = 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,…,s_n 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 c₁ ≥ b₁ and any choices of c₂,…,c_n,

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

where the factor $s_{i}^{c_{i}-b_{i}}$ should be understood as a principal value if c_i < b_i.

2.2 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 Z ⊂ X. In [3] with Andersson, we constructed an End(⊕E_k)-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 R_k is a Hom(E₀,E_k)-valued pseudomeromorphic current of bidegree (0,k). Note that R_k 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 = R_n.

Let ρ_k be the optimal rank of φ_k, and let $\pi :\widetilde X\to X$ be a common log-resolution of the ideal sheaves generated by the ρ_k-minors of the φ_k, 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 t_k is a section of some line bundle L_k 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.

2.3 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,…,z_n], i.e., M is generated by monomials m₁,…,m_r in S. Moreover, let K be an oriented polyhedral cell complex, where the vertices are labeled by the generators m_i and the face τ of K is labeled by the least common multiple m_τ of the labels m_i 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 A_k be the free S-module with one generator e_τ in degree m_τ for each τ ∈ K_k, where K_k denotes the faces of K of dimension k − 1 (K₀ should be interpreted as {∅}) and let φ_k : A_k → A_k− 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,…,m_r. 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 E_k 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 φ_k 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 = R_n is of the form $R=\sum R_{\tau } e_{\tau }\otimes e_{\emptyset }^{*}$, i.e., with one component for each τ ∈ K_n. 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 R_τ = c_τR_α, where c_τ ∈C and

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

(2.6)

2.4 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 v_i are in the lattice Zⁿ, that contains no line; here R₊ denotes the non-negative real numbers. A cone is regular if the v_i 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 D_i 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,…,z_n). More precisely, if ρ = (ρ₁,…,ρ_n) is the first non-zero lattice point met along $\mathcal C_{i}$, then $\text {ord}_{D_{i}}$ is the monomial valuation $\text {ord}_{\rho }(z_{1}^{a_{1}}{\cdots } z_{n}^{a_{n}}):=\rho _{1} a_{1}+{\cdots } +\rho _{n} a_{n}$.

2.5 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 D_i 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 ord_ρ, 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, D_i 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,β_n). Let ρ_j = β₁⋯β_j− 1β_j+ 1⋯β_n; then ρ = (ρ₁,…,ρ_n) is a normal vector of τ and thus the unique Rees valuation of $\mathfrak {m}^{\beta }$ is of the form r ord_ρ for some r ∈Q. Note that $\text {ord}_{\rho }(z_{i}^{\beta _{i}})=\text {ord}_{\rho }(\mathfrak {m}^{\beta })$ for all i.

2.6 Bochner-Martinelli Residue Currents

Let f = (f₁,…,f_p) 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 φ_k = δ_f 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,…,f_p are minimal generators of a complete intersection ideal, then the only nonvanishing coefficient of R = R_p 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 σ_i is a holomorphic section defining D_i 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 ρ_k-minors of the φ_k are monomials of degree ρ_kℓ. 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 ρ_k-minors of the φ_k. Let D = {σ₁ = 0} denote the exceptional (prime) divisor. Then ord_D(z_i) = 1 for each i and ord_D(dz) = n − 1. It follows that the section t_k 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 $\sigma =\sigma _{1}^{(n-1)\ell +1}$.

3 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$D=\sum r_{i} D_{i}$,σ_i, andηareas in Theorem 1.1. More precisely, for eachτ ∈ K_n, there is acurrent$\widetilde R_{\tau }$on$\widetilde X$anda Rees divisorD_τsuchthat$\widetilde R_{\tau }$has support onD_τ,$\pi _{*}\widetilde R_{\tau }= R_{\tau }$,and

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

where η_τ 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 R_τ = c_τR_α of R, where c_τ≠ 0, cf. Section 2.3. Note that z^α−1R_α≠ 0, where 1 = (1,…,1). It follows that z^α−1R_τ≠ 0, and thus z^α−1R≠ 0, which by the duality principle implies that z^α−1∉M. Since M is integrally closed, there is a Rees divisor D_τ, 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 ord_ρ, where ρ = (ρ₁,…,ρ_n) is the primitive normal vector of one of the compact facets of NP(M); in particular, ρ_j ∈N, see Sections 2.4 and 2.5. Let γ_j = ρ₁⋯ρ_j− 1ρ_j+ 1⋯ρ_n and choose k ∈N such that β_j := kγ_j ≥ α_j 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 R_β, 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 ω_β 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,…,s_n such that for some p, $\pi ^{-1}(0)=\{s_{1}{\cdots } s_{p}=0\}$ and for each i either x∉D_i or D_i = {s_j = 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 σ_p+ 1,…,σ_N 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 η_α 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 η_τ = c_τη_α 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 φ_j 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 a_i in (1.5) have to be (at least) like nr_i, where r_i is as in the introduction. Indeed, assume that for some ν ∈N, a_i ≤ νr_i for each i, and take $g\in \overline {\mathfrak a}^{\nu }$. Then π^∗g 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 a_i need to be at least like nr_i, cf. Example 2.3.