A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers

Let $\overline{Z}$ be a smooth projective variety with an simple normal crossing divisor $D$ such that $Ω_{\overline{Z}}^1(\log D)$ is nef. We prove that if $Y\subset Z:=\overline Z\setminus D$ is a smooth closed subvariety with nonzero Euler characteristic, and $P$ is a perverse sheaf on $Y$ with full support, then $χ(Y,P)>0$. This is a strict version of an inequality obtained in arXiv:2408.15788. Applying this to the trace-zero part of a finite direct image yields a logarithmic Riemann--Hurwitz inequality: if $Y$ has dimension $n$, any finite surjective morphism $f\colon X\to Y$ of degree $d$ with $X$ smooth satisfies $(-1)^nχ(X)\ge d\,(-1)^nχ(Y)$, the difference being an explicit sum of nonnegative intersection numbers. When $(-1)^nχ(Y)>0$ this forces any such $f$ with $χ(X)=χ(Y)$ to be an isomorphism. We verify the nef hypothesis for subvarieties of semiabelian varieties, and for $\overline{\mathscr M}_{g,n}$---the moduli of curves, obtaining in particular that every finite surjective self-morphism of a moduli space of curves with level structure is an isomorphism.