Generalized logarithmic Gauss map and its relation to (co)amoebas (Q2862562)

Let \(V\) be a hypersurface of the algebraic torus \((\mathbb{C}^\ast)^n\). Following [\textit{M. M. Kapranov}, Math. Ann. 290, No. 2, 277--285 (1991; Zbl 0714.14031)] the logarithmic Gauss map \(\gamma:V\rightarrow\mathbb{P}\big(T_1\big((\mathbb{C}^\ast)^n\big)\big)\) associates to a point \(z\in V\) the translation of its tangent space \(T_zV\) into \(T_1\big((\mathbb{C}^\ast)^n\big)\).NEWLINENEWLINEIn this article the authors generalize this construction to all algebraic subvarieties of \((\mathbb{C}^\ast)^n\). They obtain a map \(\gamma_G:V_{reg}\rightarrow\mathbb{G}_{n-k,k}\) from the set \(V_{reg}\) of regular points of \(V\) into the Grassmannian of \((n-k)\)-dimensional linear subspaces of \(\mathbb{C}^n\). Their main result is a classification of the critical points of \(\gamma_G\) in terms of certain real Schubert cells in \(\mathbb{G}_{n-k,n}\).NEWLINENEWLINEThis result is a generalization of a result of [\textit{G. Mikhalkin}, Ann. Math. (2) 151, No. 1, 309--326 (2000; Zbl 1073.14555)] treating only the hypersurface case.