Coleff-Herrera currents, duality, and noetherian operators
Let \(f=(f_1, \dots, f_p)\) be a tuple of holomorphic functions in a neighborhood of the origin in \(\mathbb{C}^n\) that defines a complete intersection. Then the Coleff-Herrera product NEWLINE\[NEWLINE\mu^f = \overline{\partial} \frac{1}{f_1} \wedge \cdots \wedge \overline{\partial} \frac{1}{f_p},NEWLINE\]NEWLINE introduced in [\textit{N. R. Coleff} and \textit{M. E. Herrera}. Les courants residuels associes à une forme méromorphe, Lect. Notes Math. 633. Berlin etc.: Springer (1978; Zbl 0371.32007)] is a \(\overline{\partial}\)-closed \((0,p)\)-current with support on \(\{f=0\}\), and it is independent (up to a non-vanishing holomorphic factor) of the choice of generators of the ideal sheaf \(\mathcal{I}\) generated by \(f\). It was proved in [\textit{A. Dickenstein} and \textit{C. Sessa}, Invent. Math. 80, 417--434 (1985; Zbl 0556.32005)] and [\textit{U. Oberst}, J. Algebra 222, No. 2, 595--620 (1999; Zbl 0948.13015)] that \(\mathcal{I}\) coincides with the ideal sheaf \(\text{ann\,} \mu^f\) of holomorphic functions \(\phi\) such that the current \(\mu^f\phi\) vanishes. This is often referred to as the duality principle. NEWLINENEWLINENEWLINE The Coleff-Herrera current described above is the model for the general notion of Coleff-Herrera currents introduced by Björk: given a variety \(Z\) of pure codimension \(p\), we say that a (possibly vector valued) \((0,p)\)-current \(\mu\) with support on \(Z\) is a Coleff-Herrera current on \(Z\) if it is \(\overline{\partial}\)-closed, annihilated by \(\overline{\mathcal{I}_Z}\) (i.e. \(\overline{\xi}\mu=0\) for each holomorphic function \(\xi\) vanishing on \(Z\)), and has the standard extension property SEP (see [\textit{J.-E. Björk}, Residues and \(\mathcal{D}\)-modules. in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, Oslo, Norway, 2002. Berlin: Springer. 605--651 (2004; Zbl 1069.32001)] or [the author, Ann. Fac. Sci. Toulouse, Math. (6) 18, No. 4, 651--661 (2009; Zbl 1187.32026)]). NEWLINENEWLINENEWLINEIn the paper under review, the author uses the residue theory developed in [the author and \textit{E. Wulcan}, Ann. Sci. Éc. Norm. Supér. (4) 40, No. 6, 985--1007 (2007; Zbl 1143.32003); J. Reine Angew. Math. 638, 103--118 (2010; Zbl 1190.32006)] to extend the duality for a complete intersection to a general pure-dimensional ideal (or submodule of a locally free) sheaf. NEWLINENEWLINENEWLINE To be more precise, let \(\mathcal{I}\) be a coherent subsheaf of a locally free sheaf \(\mathcal{O}(E_0)\) and suppose that \(\mathcal{J}=\mathcal{O}(E_0)/\mathcal{I}\) has pure codimension. Starting with a residue current \(R\) obtained from a locally free resolution of \(\mathcal{J}\), the author constructs a vector-valued Coleff-Herrera current \(\mu\) with support on the variety associated to \(\mathcal{J}\) such that \(\phi\) is in \(\mathcal{I}\) if and only if \(\mu\phi=0\).NEWLINENEWLINENEWLINENEWLINE Following an idea of Björk, such a current \(\mu\) is also derived algebraically from a fundamental theorem of Roos about the bidualizing functor [\textit{V. P. Palamodov}, Linear differential operators with constant coefficients. Berlin etc.: Springer (1970; Zbl 0191.43401)], and the relation between the two approaches is discussed.NEWLINENEWLINENEWLINENEWLINE By a construction due to Björk one gets Noetherian operators for \(\mathcal{I}\) from the current \(\mu\). The current \(R\) also provides an explicit realization of the Dickenstein-Sessa decomposition and another related canonical isomorphism.
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