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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)

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 32A27 Residues for several complex variables
##### Keywords:
toric varieties; fan; toric residue map
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