On Isotopies and Homologies of Subvarieties of Toric Varieties

Abstract

In C n we consider an algebraic surface Y and a finite collection of hypersurfaces {S i }. Froissart's theorem states that if Y and {S i } are in general position in the projective compactification of C n together with the hyperplane at infinity then for the homologies of Y \ S i we have a special decomposition in terms of the homology of Y and all possible intersections of S i in Y . We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of C n in which Y and {S i } are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in Y leaving invariant all hypersurfaces Y ∩ S k with the exception of one Y ∩ S i , which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface Y the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.

On the Asymptotic of Homological Solutions to Linear Multidimensional Difference Equations

“…Every k-cycle on V is homologous to a sum of iterated Leray coboundaries over the intersection of the closure V in C n with intersections of k coordinate planes T i = {z ∈ C n : z i = 0} (see [14]):…”