On amoebas of algebraic sets of higher codimension
For any complex affine variety \(V\) one can define amoeba of \(V\) as a set \[ A(V)=\mathrm{Log}(V)\subset \mathbb R^n, \] where \(\mathrm {Log}(z)=(\log |z_1|,\ldots, \log |z_n|)\). Usually, the complement of amoeba \(\mathbb R^n\setminus A\) has a nicer behavior then the amoeba \(A\), and it is more convenient to describe its properties. The the paper studies the topology of \(\mathbb R^n\setminus A(V)\) for \(V\) complete intersection. The main result is the analog of Lefschetz hyperplane theorem and says that the natural map \[ H_{k-1}(\pi\cap \mathbb R^n\setminus A(V)) \to H_{k-1}(\mathbb R^n\setminus A(V)) \] is injective for \(V\) complete intersection of codimension \(k\) and \(\pi\subset \mathbb R^n\) a hyperplane. It allows to describe the higher homology of the complement of the initial complex variety \(\mathbb C^n\setminus V\). Also the paper studies relation between boundary of the amoeba (critical points of the \(\mathrm{Log}\) map above) and logarithmic Gauss map for algebraic sets.
scientific article; zbMATH DE number 6353216
scientific article