An integral scalar curvature bound for real simplicial volume

We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant $a_n>0$, depending only on n, such that ${\int }_M({\mathrm{Scal}}_g^{-}{)}^{n/2}d{V}_g\ge a_n\Vert M\Vert$ for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g. Here ${\mathrm{Scal}}_g^{-}:=\max \{0,-{\mathrm{Scal}}_g\}$, and $\Vert M\Vert$ is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.