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Torsion of differentials on toric varieties Klaus Altmann \Lambda
 

Summary: Torsion of differentials on toric varieties
Klaus Altmann \Lambda
Institut f¨ur reine Mathematik, Humboldt­Universit¨at zu Berlin
Ziegelstr. 13a, D­10099 Berlin, Germany.
E­mail: altmann@mathematik.hu­berlin.de
Abstract
We introduce an invariant for semigroups with cancellation property. When
the semigroup equals the set of lattice points in a rational, polyhedral cone,
then this invariant describes the torsion of the differential sheaf on the asso­
ciated toric variety.
Finally, as an example, we present the case of two­dimensional cones (corre­
sponding to two­dimensional cyclic quotient singularities).
1 An invariant for semigroups
(1.1) Let S be a commutative semigroup with 0 and cancellation property (i.e.
a + s = b + s implies a = b for a; b; s 2 S). In particular, S can be embedded into a
group, and the notion \Gammaa for a 2 S makes sense. Assume that (inside this group)
S `` (\GammaS) = f0g; then via
a – b :() a \Gamma b 2 S ;
S turns also into a partially ordered set.
For each ` 2 S we will define a certain abelian group T ` . Their direct sum T :=

  

Source: Altmann, Klaus - Fachbereich Mathematik und Informatik & Institut für Mathematik, Freie Universität Berlin

 

Collections: Mathematics