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

Given a free resolution of an ideal a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak a$\end{document} of holomorphic functions, there is an associated residue current R that coincides with the classical Coleff-Herrera product if a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak a$\end{document} is a complete intersection ideal and whose annihilator ideal equals a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak a$\end{document}. In the case when a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak a$\end{document} is an Artinian monomial ideal, we show that the singularities of R are small in a certain sense if and only if a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak a$\end{document} is integrally closed.


Introduction
Given (a germ of) a holomorphic function f at 0 ∈ C n , Herrera and Lieberman [11] proved that one can define the principal value current 1 f .ξ := lim →0 |f | 2 > ξ f , (1.1) for test forms ξ . It follows that∂(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∂(1/f ) = 0. Given a (locally) free resolution of a general ideal (sheaf) a, in [3] with Andersson, we defined a vector (bundle) valued residue current R with support on the variety of a that satisfies the duality principle for a, cf. Section 2.2 below. If a is Cohen-Macaulay, then R is essentially independent of the Elizabeth Wulcan wulcan@chalmers.se resolution. In particular, if (E, ϕ) is the Koszul complex of a minimal set of generators f 1 , . . . , f p of a complete intersection ideal, then R coincides with the classical Coleff-Herrera product [9],∂ 1 f p ∧ · · · ∧∂ 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∂-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 a; this idea is central to many applications of residue currents, including the ones mentioned above. Various properties of the ideal a are reflected in the residue current R. For example, R has a natural geometric decomposition corresponding to a primary decomposition of a, see [4], and the fundamental cycle of 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 a, they are small in a certain sense if and only if a is integrally closed. For simplicity, we will work in a local setting; let O n 0 be the ring of germs of holomorphic functions at 0 ∈ C n and let a be an ideal in O n 0 . Recall that g ∈ O n 0 is in the integral closure a of a if |g| ≤ C|f |, where C is a constant and f is a set of generators f 1 , . . . , f m ∈ O n 0 of a, or equivalently if g satisfies a monic equation g q + h 1 g q−1 + · · · + h q = 0, where h k ∈ a k . If a = a, then a is said to be integrally closed. Assume that π : X → (C n , 0) is a log-resolution of a, i.e., X is a complex manifold, π is a biholomorphism outside the variety of a, and a · O X = O X (−D), where D = N i=1 r i D i is an effective divisor with simple normal crossings support. Then a = π * (O X (−D)), which means that g ∈ O n 0 is in a if and only if ord D i (g) ≥ r i for each i, where ord D i denotes the divisorial valuation defined by the prime divisor D i .
If π : X → (C n , 0) is a common log-resolution of a and the Fitting ideals of 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 = O X (−F ) over X and a current R on X such that where dz = dz 1 ∧ · · · ∧ dz n and η is a vector (bundle) valued smooth form with values in L, such that π * 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 σ = σ a 1 1 · · · σ a N N , We are interested in the exponents a i . Naively, one could hope that one could choose a i as r i = ord D i (a). However, this can only be true if a is integrally closed. Indeed, assume that R = π * R, where R satisfies (1.4) with σ given by (1.5) with a i ≤ r i . Take g ∈ a; then ord D i (g) ≥ r i for each i and thus π * g = σg , where g is a holomorphic section of L −1 . Therefore, by the duality principle, π * g∂(1/σ ) = 0 and so π * g R = 0, which implies that gR = 0, and hence g ∈ a. To conclude, if we can choose a i ≤ r i for each i, then a is integrally closed.
We are interested in whether the converse holds, i.e., if a is integrally closed, is it then always possible to find an 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.
0 be an integrally closed Artinian monomial ideal and let R be the residue current associated with a cellular resolution of M. Then there is a log-resolution π : 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.

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 where s 1 , . . . , 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 V μ, 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, ∞), then For further reference, in view of (1.1), note that In view of (2.1) and (2.2), it follows that for c 1 ≥ b 1 and any choices of c 2 , . . . , c n , where the factor s

Residue Currents from Complexes of Vector Bundles
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 of holomorphic sections. If (2.4) is exact, then R satisfies the duality principle, which means that if ξ is a section of E 0 that is generically in the image of ϕ 1 , then Rξ = 0 if and only if ξ ∈ Imϕ 1 ; in particular, if (2.4) is a free resolution of an ideal a ⊂ O n 0 , then the annihilator ideal of R, i.e., the ideal of germs of holomorphic functions g such that gR = 0, equals a. Moreover, then R is of the form R = R k , where R k is a Hom(E 0 , 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 O n 0 , then R = R n . Let ρ k be the optimal rank of ϕ k , and let π : X → 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 ρ k ϕ k of ρ k E * k ⊗ ρ k E k−1 is of the form t k ρ k , where t k is a section of some line bundle L k and ϕ k is a nonvanishing section of L −1 k ⊗ ρ k π * E * k ⊗ ρ k π * E k−1 . It was proved in [3, Section 2] that there is a current R on X such that π * R = R and R = ω ∧∂(1/σ ), where ω is smooth and σ = t 1 · · · t min(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 · · · t min(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[z 1 , . . . , z n ], i.e., M is generated by monomials m 1 , . . . , 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, . . . , dim K + 1, let A k be the free Smodule 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 (K 0 should be interpreted as {∅}) and let ϕ k : A k → A k−1 be defined by More precisely, for k = 0, . . . , N = dim K + 1, let E k be a trivial bundle over (a neighborhood of 0 in) C n with a global frame {e τ } τ ∈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 O n 0 , and sometimes as ideals in the ring of entire functions in C n .
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 = R τ e τ ⊗ e * ∅ , i.e., with one component for each τ ∈ K n . Proposition 3.1 in that paper asserts that if z α := z α 1 1 · · · z α n n is the label of τ , then R τ = c τ R α , where c τ ∈ C and R α =∂ 1 z α n n ∧ · · · ∧∂ 1 z α 1 1 . (2.6)

Toric Log-Resolutions
For an (Artinian) monomial ideal M in C n , it is possible to find a log-resolution π : X → C n where 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 R n is a set of the form C = R + v i , where v i are in the lattice Z n , 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 Z n . A fan is a finite collection of cones such that all faces and intersections of cones in are in ; is regular if all cones are regular. A regular fan determines a smooth toric variety X( ), obtained by patching together affine toric varieties corresponding to the cones in .
Assume that M is an Artinian monomial ideal in C n z 1 ,...,z n . Recall that the Newton polyhedron NP(M) of M is defined as the convex hull in R n of the exponents of monomials in M. Let S(M) be the collection of cones of the form C = R + ρ ⊂ R n + , where ρ is a normal vector of a compact facet (face of maximal dimension) of NP(M). Let be a regular fan that contains S(M) and such that the support, i.e., the union of all cones in , equals R n + . The cones in S(M) determine a fan with support R n + and by refining this is always possible to find such a . Then π : X( ) → C n is a log-resolution of M. The prime divisors D i of the exceptional divisor correspond to one-dimensional cones C i = R + ρ i in and ord D i are monomial valuations (i.e., determined by its values on z 1 , . . . , z n ). More precisely, if ρ = (ρ 1 , . . . , ρ n ) is the first non-zero lattice point met along C i , then ord D i is the monomial valuation ord ρ (z a 1 1 · · · z a n n ) := ρ 1 a 1 + · · · + ρ n a n .

Rees Valuations
Given a non-zero ideal a ⊂ O n 0 , let ν : X + → (C n , 0) be the normalized blow-up of a and let D = r i D i be the exceptional divisor, such that a · O X + = O X + (−D). The divisorial valuations ord D i are called the Rees valuations of a, see, e.g., [15,Section 9.6.A]. Then a = ν * (O X + (−D)), i.e., g ∈ a if and only if ord D i (g) ≥ ord D i (a) for each i. If π : X → (C n , 0) is any log-resolution of a (and thus factors through the normalized blowup) with exceptional divisor D = r i D i , following [13] we say that the prime divisor D i is a Rees divisor if ord D i is a Rees valuation.

Bochner-Martinelli Residue Currents
Let f = (f 1 , . . . , f p ) be a tuple (of germs) of holomorphic functions at 0 ∈ C n and let (2.3) be the Koszul complex of f , i.e., consider f as a section f = f j e * j of a trivial rank p bundle E * over (a neighborhood of 0 in) C n with a frame e * 1 , . . . , e * p , let E j = 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 f 1 , . . . , 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 a = a(f ) generated by f . It is proved in Section 4 in that paper that if π : X → (C n , 0) is a log-resolution of a, then there is a current R such that π * R = R and R has support on the Rees divisors of a. Moreover if D = N i=1 r i D i is the exceptional divisor of π , then R = ω ∧∂ 1 σ nr 1 1 · · · σ nr N N , where σ i is a holomorphic section defining D i and ω is a smooth form. a = (z 1 , . . . , z n ) ⊂ O n 0 , and let (2.3) be the Koszul complex of (z 1 , . . . , z n ). Then the ρ k -minors of the ϕ k are monomials of degree ρ k . It follows that the blow-up of C n at 0 is a common log-resolution of a and the ideals generated by the ρ k -minors of the ϕ k . Let D = {σ 1 = 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. More precisely, for each τ ∈ K n , there is a current R τ on X and a Rees divisor D τ such that R τ has support on D τ , π * R τ = R τ , and

Example 2.3 Let
where η τ is a semi-meromorphic form.
Proof Let π : X → (C n , 0) be a toric log-resolution of M in the sense of Section 2.4. Consider an entry R τ = c τ R α of R, where c τ = 0, cf. Section 2.3. Note that z α−1 R α = 0, where 1 = (1, . . . , 1). It follows that z α−1 R τ = 0, and thus z α−1 R = 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 D 1 , of M = M such that see Section 2.5.
Since M is monomial, ord D 1 is a monomial valuation of the form ord ρ , where ρ = (ρ 1 , . . . , 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 = ρ 1 · · · ρ j −1 ρ 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 m β = (z β 1 1 , . . . , z β n n ), so that D 1 is the unique Rees divisor of m β , see Example 2.2. It follows that π : X → (C n , 0) is a log-resolution of m β , see Section 2.4. Recall from Section 2.6 that (the coefficient of) the Bochner-Martinelli residue current of (z β 1 1 , . . . , z β n n ) equals R β , defined as in (2.6). Thus in view of Section 2.6, on X there is an R β with support on D 1 such that π * R β = R β and where ω β is smooth.
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 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 a ⊂ O n 0 , is the pushforward of a current 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 ∈ a ν . Then π * g is divisible by σ and thus gR = 0, cf. the arguments after (1.5). It follows that a ν ⊂ a. The classical Briançon-Skoda theorem [18] asserts that this inclusion holds for ν = min(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.