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Toric residues. (English) Zbl 0904.14029

The Grothendieck local residue symbol \[ \text{Res}_0 \left({gdx_0 \wedge \cdots \wedge dx_n \over f_0\dots f_n} \right)= {1\over (2\pi i)^{n+1}} \int_{| f_i| = \varepsilon} {gdx_0 \wedge\dots \wedge dx_n \over f_0\dots f_n} \] [see P. Griffiths and J. Harris, “Principles of algebraic geometry” (1978; Zbl 0408.14001 or 1994; Zbl 0836.14001); chapter 5] is defined whenever \(g,f_0,\dots,f_n\) are holomorphic in a neighborhood of \(0\in \mathbb{C}^{n+1}\) and \(f_0, \dots, f_n\) do not vanish simultaneously except at 0. C. Peters and J. Steenbrink [in: Classification of algebraic and analytic varieties, Proc. Symp., Katata 1982, Prog. Math. 39, 399-463 (1983; Zbl 0523.14009)] observed that when \(f_0, \dots, f_n\) are homogeneous of degree \(d\) and \(g\) is homogeneous of degree \(\rho= (n+1) (d-1)\), the residue symbol has the following nice properties:
Quotient property. The map \(g \mapsto\text{Res}_0 \left( {g dx_0 \wedge\dots \wedge dx_n \over f_0\dots f_n} \right)\) induces an isomorphism \[ \mathbb{C} [x_0, \dots, x_n]_\rho/ \langle f_0,\dots, f_n\rangle_\rho \simeq \mathbb{C} \] (the subscript refers to the graded piece in degree \(\rho)\) uniquely characterized by the fact that the Jacobian determinant \(J=\text{det} (\partial f_i/ \partial x_j)\) maps to \(d^{n+1}\).
Trace property. Čech cohomology gives a cohomology class \([\omega_g] \in H^n (\mathbb{P}^n, \Omega^n_{\mathbb{P}^n})\) such that under the trace map \(\text{Tr}_{\mathbb{P}^n}: H^n (\mathbb{P}^n, \Omega^n_{\mathbb{P}^n}) \simeq \mathbb{C}\), we have \(\text{Res}_0 \left({g dx_0 \wedge \dots \wedge dx_n \over f_0\dots f_n} \right)= \text{Tr}_{\mathbb{P}^n} ([\omega_g])\).
In this paper, we will show how these properties of residues can be generalized to an arbitrary projective toric variety. The paper is organized into six sections as follows. In \(\S 1\), we define the cohomology class \([\omega_g]\in H^n(\mathbb{P}^n, \Omega^n_{\mathbb{P}^n})\), and then \(\S 2\) generalizes this to define toric residues in terms of a toric analog of the trace property. We recall some commutative algebra associated with toric varieties in \(\S 3\), and \(\S 4\) introduces a toric version of the Jacobian. In \(\S 5\), we show that the toric residue is uniquely characterized using a toric analog of the quotient property. Then \(\S 6\) explores different ways of representing the toric residue as an integral, and an appendix discusses the relation between the trace map and the Dolbeault isomorphism.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32A27 Residues for several complex variables
14F25 Classical real and complex (co)homology in algebraic geometry
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References:

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