Local inequalities and birational superrigidity of Fano varieties

We obtain local inequalities for log canonical thresholds and multiplicities of movable log pairs. We prove the non-rationality and birational superrigidity of the following Fano varieties: a double covering of a smooth cubic hypersurface in $\mathbb P^n$ branched over a nodal divisor that is cut out by a hypersurface of degree $2(n-3)\ge 10$; a cyclic triple covering of a smooth quadric hypersurface in $\mathbb P^{2r+2}$ branched over a nodal divisor that is cut out by a hypersurface of degree $r\ge 3$; a double covering of a smooth complete intersection of two quadric hypersurfaces in $\mathbb P^n$ branched over a smooth divisor that is cut out by a hypersurface of degree $n-4\ge 6$.