An analogue of convexity for complements of amoebas of varieties of higher codimension An answer to a question asked by B. Sturmfels

Let V H ðC Þ be a variety and let Log denote the logarithmic moment map Log : ðC Þ ! R, ðz1; . . . ; znÞ 7! ðlogjz1j; . . . ; logjznjÞ. The amoeba A of the variety V H ðC Þ is its image under that map. Amoebas were introduced by Gelfand, Kapranov and Zelevinsky in [6]. They show that for varieties of codimension 1, the complement A of the amoeba A in R is a disjoint union of convex regions. If V is of codimension k þ 1 and A is its amoeba complement, we consider oriented ðk þ 1Þ-planes p in R. Let us call a class in the reduced homology group1 ~ HmðpV AÞ non-negative if its image in ~ HkðpnfpgÞFZ is non-negative for all p A pVA. We show that such a class is never sent to zero in ~ HkðAÞ, except if it is already zero. In other words, the maps ~ HkðpV AÞ ! ~ HkðAÞ induced by the inclusions pV A ! A never send non-zero non-negative classes to zero. The author expects these maps on homology to actually be injective, but this is only known for k 1⁄4 0. In that case, the result specializes to the one mentioned above and proven in [6]. Indeed, when k 1⁄4 0, we are looking at lines l and at the maps on ~ H0 induced by lV A ! A. Assuming our result, we want to show that A is a disjoint union of convex sets. Suppose by contradiction that X is a component of A that is not convex. Choose p; q A X such that the interval joining them is not contained in X , and let l be the oriented line pq !. It is then clear that 00 1⁄2q 1⁄2 p A ~ H0ðlV AÞ is a nonnegative class that is sent to zero in ~ H0ðAÞ. On the other hand, if all components of A are convex, then the map H0ðlV AÞ ! H0ðAÞ is always injective, and so is the map ~ H0ðlV AÞ ! ~ H0ðAÞ.