On Isotopies and Homologies of Subvarieties of Toric Varieties

In ℂⁿ 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 ℂⁿ 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 ℂn 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.