The slope inequality conjecture for fibrations with mildly singular fibres

Let $f\colon X\longrightarrow B$ be a fibred $n$-dimensional variety, with general fibre $F$, and let $\omega_f$ denote its relative canonical sheaf. Assume that $\omega_f$ is relatively nef and ample on the general fibres, and that the general fibres have sufficiently mild singularities, for example they are log canonical or semi-log-canonical. The proposed slope inequality is

$$K_f^n\geq n\frac{K_F^{n-1}}{h^0(F,\omega_F)}\deg f_*\omega_f.$$

Slope inequality conjecture. Under these assumptions, the fibration satisfies the displayed slope inequality. The inequality is equivalent to the $f$-positivity of $\omega_f$ and follows from the Cornalba–Harris and Bost method when the general fibres have a Hilbert–Chow semistable canonical map. The conjecture proposes that sufficiently mild singularities should ensure the required stability or positivity in general.