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 ℙⁿ branched over a nodal divisor that is cut out by a hypersurface of degree 2(n - 3) ≥, 10; a cyclic triple covering of a smooth quadric hypersurface in ℙ^2r+2 branched over a nodal divisor that is cut out by a hypersurface of degree r ≥ 3; a double covering of a smooth complete intersection of two quadric hypersurfaces in ℙⁿ branched over a smooth divisor that is cut out by a hypersurface of degree n - 4 ≥ 6.