A Riemann-Hurwitz Theorem for the Algebraic Euler Characteristic

TL;DR

This paper establishes a Riemann-Hurwitz type formula for calculating the Euler characteristic of sheaves under finite morphisms between smooth projective varieties, extending classical results to a broader algebraic setting.

Contribution

It introduces an analogue of the Riemann-Hurwitz theorem applicable to algebraic Euler characteristics with simple normal crossings conditions.

Findings

Derived a formula for Euler characteristics under finite maps

Extended classical Riemann-Hurwitz theorem to algebraic geometry

Applicable to varieties with simple normal crossing branch loci

Abstract

We prove an analogue of the Riemann-Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.