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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:
 [1] Audin, M.,The Topology of Torus Actions on Symplectic Manifolds, Progress in Math.93, Birkhäuser Verlag, Basel-Boston, 1991. · Zbl 0726.57029 [2] Baily, W., The decomposition theorem forV-manifolds, Amer. J. Math.78 (1956), 862–888. · Zbl 0173.22705 · doi:10.2307/2372472 [3] Batyrev, V., Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori,Duke Math. J. 69 (1993), 349–409. · Zbl 0812.14035 · doi:10.1215/S0012-7094-93-06917-7 [4] Batyrev, V. andCox, D., On the Hodge structure of projective hypersurfaces in toric varieties,Duke Math. J. 75 (1994), 293–338. · Zbl 0851.14021 · doi:10.1215/S0012-7094-94-07509-1 [5] Bruns, W. andHerzog, J.,Cohen-Macaulay Rings, Cambridge Univ. Press, Cambridge, 1993. [6] Cattani, E., Cox, D. andDickenstein, A., Residues in toric varieties, Preprint, 1995. [7] Cox, D., The homogeneous coordinate ring of a toric variety,J. Algebraic Geom. 4 (1995), 17–50. · Zbl 0846.14032 [8] Danilov, V., The geometry of toric varieties,Uspekhi Mat. Nauk 33:2 (1978), 85–134. (Russian). English transl.: Russian Math. Surveys33 (1978), 97–154. [9] Fulton, W.,Introduction to Toric Varieties, Princeton Univ. Press, Princeton, 1993. · Zbl 0813.14039 [10] Gelfand, I., Kapranov, M. andZelevinsky, A.,Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser Verlag, Basel-Boston, 1994. [11] Goto, S. andWatanabe, K., On graded rings I,J. Math. Soc. Japan 30 (1978), 179–213. · Zbl 0378.13006 · doi:10.2969/jmsj/03020179 [12] Griffiths, P., On the periods of certain rational integrals I,Ann. of. Math. 90 (1969), 460–495. · Zbl 0215.08103 · doi:10.2307/1970746 [13] Griffiths, P. andHarris, J.,Principles of Algebraic Geometry, Wiley, New York, 1978. [14] Hartshorne, R.,Residues and Duality, Lecture Notes in Math.20, Springer-Verlag, Berlin-Heidelberg-New York, 1966. [15] Hochster, M., Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes,Ann. of Math. 96 (1972), 318–337. · Zbl 0237.14019 · doi:10.2307/1970791 [16] Kleiman, S., Toward a numerical theory of ampleness,Ann. of Math. 84 (1966), 293–344. · Zbl 0146.17001 · doi:10.2307/1970447 [17] Morrison, D. andPlesser, M. R., Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties,Nuclear Phys. B 440 (1995), 279–354. · Zbl 0908.14014 · doi:10.1016/0550-3213(95)00061-V [18] Oda, T.,Convex Bodies and Algebraic Geometry, Springer-Verlag, Berlin-Heidelberg-New York, 1988. · Zbl 0628.52002 [19] Peters, C. andSteenbrink, J., Infinitesimal variation of Hodge structure and the generic Torelli theorem for projective hypersurfaces, inClassification of Algebraic and Analytic Manifolds (Ueno, K., ed.). Progress in Math.39, pp. 399–463, Birkhäuser Verlag, Basel-Boston, 1983. [20] Sternberg, S.,Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, N. J., 1964. · Zbl 0129.13102 [21] Tsikh, A.,Multidimensional Residues and their Applications, Amer. Math. Soc., Providence, R. I., 1992. · Zbl 0758.32001
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