We work over an algebraically closed field of characteristic zero. Let $X$ be a Fano variety, and $T\cong \mathbb{G}_m^r$ a torus acting faithfully on $X$.
I think we can canonically linearize the action of $T$ on $-K_X$. Is anything known about the GIT quotient for this action? Is the semi-stable locus non-empty? Is the quotient still Fano?
When the Fano variety is smooth, the field is $\mathbb{C}$ and the torus has complex dimension 1 (along with some the mild hypothesis that the (restriction of the action to $S^{1} \subset \mathbb{C}^{*}$) is semi-free), then the GIT quotient is a Fano variety. This is proven in Corollary 2 of "The Ricci curvature of symplectic quotients of Fano manifolds" by Futaki. This paper is written in the language of symplectic reduction but one can obtain the result for GIT quotients by applying the Kempf-Ness theorem.
