×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.