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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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