Compositio Mathematica

Residues in toric varieties

a1 Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA; e-mail:
a2 Department of Mathematics and Computer Science, Amherst College, Amherst, MA 01002, USA; e-mail: dac@CS.AMHERST.EDU
a3 Departamento de Mattemática, F.C.E. y N., Universidad de Buenos Aires, Ciudad Universitaria–Pabellón I, 1428 Buenos Aires, Argentina; e-mail:

Article author query
cattani e   [Google Scholar] 
cox d   [Google Scholar] 
dickenstein a   [Google Scholar] 


We study residues on a complete toric variety $X$, which are defined in terms of the homogeneous coordinate ring of $X$. We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When $X$ is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent $X$ as a quotient $(Y\setminus\{0\})/C\ast$ such that the toric residue becomes the local residue at 0 in $Y$.

Key Words: toric varieties; residues; toric residues; homogeneous ideals; ample divisors; Global Transformation Law; orbifolds; residual currents.