A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers
Abstract
Let Z̅ be a smooth projective variety with an simple normal crossing divisor D such that ΩZ̅1(log D) is nef. We prove that if Y⊂ Z:=Z̅∖ 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: X→ Y of degree d with X smooth satisfies (−1)nχ(X)≥ 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 ℳ̅g,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.
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BibTeX
@misc{arapura2026higher,
title = {{A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers}},
author = {Donu Arapura and Chikako Mese and Deepam Patel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13081},
eprint = {2609.13081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13081},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers
AU - Donu Arapura
AU - Chikako Mese
AU - Deepam Patel
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13081
UR - https://arxiv.org/abs/2609.13081
AB - Let Z̅ be a smooth projective variety with an simple normal crossing divisor D such that Ω_Z̅^1(log D) is nef. We prove that if Y⊂ Z:=Z̅∖ 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: X→ Y of degree d with X smooth satisfies (−1)^nχ(X)≥ 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 ℳ̅_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.
ER -