The topology of the amoeba map and maximal volume

For a smooth complete intersection $V\subset(\C^*)^n$ of dimension $k$, $n\ge2k$, we bound the volume of its amoeba by $π^{2k}$ times a sum of mixed volumes, divided by the order of any finite group $Γ$ of symmetries commuting with the logarithmic map. For $n=2k\ge4$ almost every fibre of the amoeba map has a number of points divisible by $2|Γ|$. For $n=2k$ we characterise the varieties of maximal volume, show that for $k\ge2$ they are not real, and prove that no plane in general position in $(\C^*)^4$ has maximal volume. For $n>2k$ the bound is never attained, varieties of maximal volume have amoeba maps injective off a null set, and injectivity is generic. The amoeba map of almost every real $k$-plane is two-to-one, and that of the $k$-th symmetric power of a real line exactly $2^k$-to-one; conversely, real $k$-planes in general position in $(\C^*)^{2k}$ with $2^k$-to-one amoeba maps are symmetric powers. We also classify the real planes of $(\C^*)^4$ and study lines in $(\C^*)^3$.