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By definition, a real algebraic variety $X$ is maximal if the sum of the mod 2 Betti numbers of the real points $X(R)$ is equal to the sum of the mod 2 Betti numbers of the complex points $X(C)$. In 2004 Bihan, Franz, McCrory, and van Hamel conjectured that every toric variety is maximal. I will discuss this conjecture for the case of projective toric varieties and give some beginning examples. I will state sufficient (but definitely not necessary) conditions for a projective toric variety to be maximal, and I will present a counterexample to the conjecture.
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