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Residue currents of monomial ideals. (English) Zbl 1120.32004

Let \(f\) be a tuple of holomorphic function \(f_1,\dots, f_m\) in \(\mathbb C^n\) and \(Y = \{f_1 =\dots = f_m = 0\}\). If \(f\) is a complete intersection, that is, the codimension of \(Y\) is \(m\), the duality theorem, due to Dickenstein-Sessa and Passare asserts that a holomorphic function \(h\) locally belongs to the ideal \((f) = (f_1, \dots,f_m)\) iff \(hR_{CH}^f = 0\), where \(R_{CH}^f\) is the Coleff-Herrera residue current of \(f\). Passare, Tsikh and Yger introduced residue currents for arbitrary \(f\) by means of the Bochner-Martinelli kernel. For each ordered index set \(\mathcal J\subseteq\{1,\dots,m\}\) of cardinality \(k\), let \(R_\mathcal J^f\) be the analytic continuation to \(\lambda = 0\) of \[ {\overline \partial}| f| ^{2\lambda}\land\sum_{\ell=1}^k(-1)^{\ell-1}\frac{\overline {f_{i_\ell}}\land_{q\neq\ell}\overline{df_{i_q}}}{| f| ^{2k}}, \] where \(| f| ^2 = | f_1| ^2 + \dots + | f_m| ^2\). Then \(R_\mathcal J^f\) is a well-defined \((0,k)\)-current with support on \(Y\), that vanishes whenever \(k <\text{codim}\,Y\) or \(k > \min(m,n)\). In case \(f\) defines a complete intersection, the only nonvanishing current, \(R^f_{\{1,\dots,m\}}\), is shown to coincide with the Coleff-Herrera current. The concept of Bochner-Martinelli residue currents was further developed by Andersson. From his construction, based on the Koszul complex, follows that \(hR_{CH}^f = 0\) for all \(\mathcal J\) implies that the holomorphic function \(h\) belongs to the ideal \((f)\) locally. Thus, letting Ann\(\,R^f\) denote the annihilator ideal, \(\{h\,\, \text{holomorphic},\,\, hR_{CH}^f = 0\}\), we have that
\[ \text{Ann}\,R^f\subseteq(f).\tag{1} \]
The inclusion is strict in general, and thus the currents \(R_\mathcal J^f\) do not fully characterize \((f)\) as in the complete intersection case. Still the ideal \( \text{Ann}\,R^f\) is big enough to catch in some sense the size of \((f)\). Recall that a holomorphic function \(h\) belongs locally to the integral closure of \((f)\), denoted by \(\overline{(f)}\), if \(| h| \leq C| f| \) for some constant \(C\), or equivalently if \(h\) fulfills a monic equation \(h^r + g_1h^{r-1} + \dots + g_r = 0\) with \(g_i\in(f)^i\) for \(1 \leq i\leq r\). It is known that \(hR_\mathcal J^f = 0\) for any \(h\), that is, locally in the integral closure of \((f)^k\), where \(k =| \mathcal J| \), and thus we get
\[ \overline{(f)^\mu}\subseteq\text{Ann}R^f,\tag{2} \]
where \(\mu = \min(m,n)\). Now, combining (1) and (2) yields a proof of the Briançon-Skoda theorem: \(\overline{(f)^\mu}\subseteq (f)\). This motivates to study the ideal Ann\(\,R^f\).
In this paper, the author computes the Bochner-Martinelli currents \(R_{\mathcal J}^f\) in case the generators \(f_i\) are all monomials. The author gives a complete description of \(R_{\mathcal J}^f\) in terms of the Newton diagram associated with the generators, when \((f)\) is a monomial ideal of dimension \(0\). In particular it turns out that Ann\(\,R^f\) depends only on \((f)\), not on the particular choice of generators. Also, it follows that we have equality in (1) iff \((f)\) is a complete intersection and moreover that the inclusion (2) is always strict. The proof of this result amounts to computing residue currents in a certain toric variety constructed from the generators, using ideas of Varchenko and Khovanskii.
The author also provides partial results for the case of general monomial ideals.

MSC:

32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
32A27 Residues for several complex variables
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