On isotopies and homologies of subvarieties of toric varieties

In $\mathbb C^n$ 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 $\mathbb C^n$ together with the hyperplane at infinity then for the homologies of $Y\setminus\bigcup S_i$ 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 $\mathbb C^n$ 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 $Y\cap S_k$ with the exception of one $Y\cap S_i$, 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.