The Lazić-Peternell Generalised Abundance Conjecture for projective klt pairs

Theorem 1.1: for a projective klt $\mathbb Q$-pair $(X,B)$ over an algebraically closed field of characteristic zero with $K_X+B$ pseudo-effective and $M$ nef $\mathbb Q$-Cartier on $X$, if $K_X+B+M$ is nef then it is numerically equivalent to a semiample $\mathbb Q$-Cartier divisor. Corollary: on a klt pair with $K_X+B\equiv0$ every nef $\mathbb Q$-Cartier divisor is numerically semiample. New ingredient: a nef divisor of full nef dimension on a klt Calabi-Yau pair is big. Kahler companion: for smooth compact Kahler $X$, snc $B$ with coefficients below one and nef $M$, $c_1(K_X+B+M)$ is represented in Bott-Chern cohomology by a semiample line bundle. Not shown: semiampleness of $K_X+B+M$ itself, lc pairs, or nef data living only on a higher model.

Theorem 1.1: for a projective klt $\mathbb Q$-pair $(X,B)$ over an algebraically closed field of characteristic zero with $K_X+B$ pseudo-effective and $M$ nef $\mathbb Q$-Cartier on $X$, if $K_X+B+M$ is nef then it is numerically equivalent to a semiample $\mathbb Q$-Cartier divisor. Corollary: on a klt pair with $K_X+B\equiv0$ every nef $\mathbb Q$-Cartier divisor is numerically semiample. New ingredient: a nef divisor of full nef dimension on a klt Calabi-Yau pair is big. Kahler companion: for smooth compact Kahler $X$, snc $B$ with coefficients below one and nef $M$, $c_1(K_X+B+M)$ is represented in Bott-Chern cohomology by a semiample line bundle. Not shown: semiampleness of $K_X+B+M$ itself, lc pairs, or nef data living only on a higher model.