Anticanonical nonvanishing for smooth projective varieties with smoothly semipositive anticanonical bundle

Theorem 1.1: if $X$ is smooth connected complex projective and $-K_X$ admits a smooth Hermitian metric of semipositive curvature, then $H^0(X,-mK_X)\ne0$ for some $m>0$. The key intermediate result (Theorem 1.2) gives sections of $aL+bD$ on varieties without holomorphic forms from a finite-volume adjoint metric. Not shown: an effective bound on $m$, the case $m=1$ (false in general, e.g. Enriques surface times $\mathbb P^1$), or nonvanishing when $-K_X$ is only nef or carries a singular semipositive metric.

Theorem 1.1: if $X$ is smooth connected complex projective and $-K_X$ admits a smooth Hermitian metric of semipositive curvature, then $H^0(X,-mK_X)\ne0$ for some $m>0$. The key intermediate result (Theorem 1.2) gives sections of $aL+bD$ on varieties without holomorphic forms from a finite-volume adjoint metric. Not shown: an effective bound on $m$, the case $m=1$ (false in general, e.g. Enriques surface times $\mathbb P^1$), or nonvanishing when $-K_X$ is only nef or carries a singular semipositive metric.