The canonical Q-Fano contraction conjecture for primitive varieties

Let $V$ be a smooth quasi-projective variety with an ample metrized invertible sheaf ${\cal L}$, and let $\overline{V}^{\cal L}$ be its ${\cal L}$-closure. Suppose that $V$ is ${\cal L}$-primitive and that $\alpha_{\cal L}(V)>0$. A canonical ${\bf Q}$-Fano variety is a normal variety with at worst canonical singularities and ample $K_W^{-1}$; its index is denoted $r(W)$.

Canonical ${\bf Q}$-Fano contraction conjecture. There exist a resolution of singularities

$$\rho:X\to\overline{V}^{\cal L}$$

and a birational projective morphism $\pi:X\to W$ to a canonical ${\bf Q}$-Fano variety $W$ such that

$$\pi^*K_W^{-1}\cong\rho^*(L)^{\alpha_{\cal L}(V)},$$

that is, $\alpha_{\cal L}(V)=r(W)$, and the support of $D$, where

$$\rho^*(L)^{\otimes r(W)}\otimes K_X={\cal O}(D),$$

is contained in the exceptional locus of $\pi$.

The conjecture asserts that primitive varieties arise from canonical ${\bf Q}$-Fano varieties by a suitable birational contraction. The source expects it to follow from the Minimal Model Program and notes that it holds for toric varieties.