Minimal metrics and interior injectivity for nef adjoints

Let $(H,\Theta )$ be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H${}^1$-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.