Existence of minimal models and Mori fibre spaces for projective log canonical pairs, in generalized-pair form

Theorem 1.1: for a projective generalized lc $\mathbb Q$-pair $(X,B+M)$ over an algebraically closed field of characteristic zero there is a birational map $X\dashrightarrow Y$ extracting no divisors, with $(Y,B_Y+M_Y)$ generalized lc and log discrepancies not decreasing (strictly on contracted divisors), such that $K_Y+B_Y+M_Y$ is nef if $K_X+B+M$ is pseudo-effective, and otherwise $Y$ has a Mori fibre space structure with relative Picard number one. With $M=0$ this gives the ordinary lc case. Companion: smooth projective varieties with $\kappa_\sigma(K_X)=1$ have $\mathbb Q$-factorial terminal minimal models. Not shown: termination of arbitrary flip sequences, real coefficients, abundance (claimed in a separate release paper), or Q-factorial models in general.

Theorem 1.1: for a projective generalized lc $\mathbb Q$-pair $(X,B+M)$ over an algebraically closed field of characteristic zero there is a birational map $X\dashrightarrow Y$ extracting no divisors, with $(Y,B_Y+M_Y)$ generalized lc and log discrepancies not decreasing (strictly on contracted divisors), such that $K_Y+B_Y+M_Y$ is nef if $K_X+B+M$ is pseudo-effective, and otherwise $Y$ has a Mori fibre space structure with relative Picard number one. With $M=0$ this gives the ordinary lc case. Companion: smooth projective varieties with $\kappa_\sigma(K_X)=1$ have $\mathbb Q$-factorial terminal minimal models. Not shown: termination of arbitrary flip sequences, real coefficients, abundance (claimed in a separate release paper), or Q-factorial models in general.