The diminishing-action characterization of lower-dimensional amoebas

Let $V\subset({\mathbb C}^\times)^n$ be an irreducible subvariety, and let $\mathbb T\subset({\mathbb C}^\times)^n$ be a subtorus. Write

$$W:=({\mathbb T}\cdot V)/{\mathbb T}$$

for the image of $V$ in the quotient torus. The subtorus $\mathbb T$ has a diminishing action on $V$ if

$$\dim\mathbb T<2(\dim V-\dim W)\quad\text{and}\quad 2\dim W<n-\dim\mathbb T.$$

Diminishing-action conjecture. If

$$\dim_{\mathbb R}\mathscr{A}(V)<\min\{n,2\dim V\},$$

then there is a nontrivial proper subtorus $\mathbb T$ of $({\mathbb C}^\times)^n$ having a diminishing action on $V$. The preceding theorem proves the converse implication: a nontrivial proper subtorus with a diminishing action forces the amoeba to have dimension strictly below $\min\{n,2\dim V\}$.