Suppose that $f: X \to \mathbb{A}^1$ is a toric degeneration in the sense of Nishinou-Siebert. In other words let X be a (possibly singular) toric variety equipped with a (not necessarily proper) toric morphism to $\mathbb{A}^1$. In this situation the fiber $Y = f^{-1}(0)$ is a union of toric divisors along their toric orbits. Let $i: T \to X$ be the inclusion of the dense torus.
Q: Has anyone studied the nearby cycle sheaf $\psi_f Ri_* \mathbb{C}_T$?
In the examples I am most interested in, $Y = \cup_i D_i$ is a normal crossing divisor and I expect that $\psi_f Ri_* \mathbb{C}_T [dim_{\mathbb{C}} T]$ has a filtration whose subquotients are Fourier transforms of sheaves of the form $R(j_i)_{*} \mathbb{C}_{T_i}$ where $j_i: T_i \to D_i$ is the inclusion of the maximal torus of $D_i$.
