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Residues in toric varieties. (English) Zbl 0883.14029
Let \(\Sigma\) be a fan defining a complete, \(n\)-dimensional toric variety \(X_\Sigma\). In Ark. Mat. B 4, No. 1, 73-96 (1996), D. A. Cox has defined meromorphic \(n\)-forms \(\omega_F(H)\) on \(X_\Sigma\) which depend on the following parameters: \(F=(F_0,\dots,F_n)\) is a tuple of homogeneous polynomials in the homogeneous coordinate ring \(S\) of \(X_\Sigma\), and \(H\in S\) is homogeneous of degree \(\rho:=K_X+\sum_i\deg F_i\). If the polynomials \(F_0,\dots,F_n\) do not vanish simultaneously on \(X_\Sigma\), then this construction gives rise to the so-called toric residue \(\text{Res}_F:S_\rho/(F_0,\dots,F_n)_\rho\to\mathbb{C}\) via the definition \(\text{Res}_F(H):=\text{Tr}_X([\omega_F(H)]\in H^n(X,\Omega^n))\).
The first subject of the paper is to prove the global transformation law controlling a linear change of the parameter \(F\). An important application is the residue isomorphism theorem stating that, if \(\Sigma\) is simplicial and if the divisors \(\deg F_i\) are ample, then the toric residue map \(\text{Res}_F\) is an isomorphism.
Finally, the paper relates the toric residue construction to the usual local residues. There is a theorem showing that the first may be written as a sum of local ones. Moreover, if all polynomials \(F_i\) have the same degree, then this ample divisor gives rise to an affine cone over \(X_\Sigma\). Then, the toric residue equals the local one taken in the origin of that cone.
Reviewer: K.Altmann (Berlin)

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32A27 Residues for several complex variables
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