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

Let $\bar{Z}$ be a smooth projective variety with an simple normal crossing divisor $D$ such that ${\mathrm{\Omega }}_{\bar{Z}}^1(\log D)$ is nef. We prove that if $Y\subset Z:=\bar{Z}\setminus D$ is a smooth closed subvariety with nonzero Euler characteristic, and $P$ is a perverse sheaf on $Y$ with full support, then $\chi (Y,P)>0$. 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 $f\text{\hspace{0.17em}⁣}:X\to Y$ of degree $d$ with $X$ smooth satisfies $(-1{)}^n\chi (X)\ge d(-1{)}^n\chi (Y)$, the difference being an explicit sum of nonnegative intersection numbers. When $(-1{)}^n\chi (Y)>0$ this forces any such $f$ with $\chi (X)=\chi (Y)$ to be an isomorphism. We verify the nef hypothesis for subvarieties of semiabelian varieties, and for ${\bar{\mathcal{M}}}_{g,n}$—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.