Grothendieck's proof of Hirzebruch-Riemann-Roch theorem

TL;DR

This paper provides a comprehensive, self-contained proof of the Hirzebruch-Riemann-Roch theorem for smooth projective varieties, connecting algebraic and geometric data through modern tools like K-theory and intersection theory.

Contribution

It offers a complete, accessible proof of the HRR theorem using Grothendieck's K-theory and Chow rings, bridging classical and modern algebraic geometry.

Findings

Complete proof of the HRR formula for smooth projective varieties.

Demonstrates how the formula recovers classical results for curves and surfaces.

Introduces the use of Chern character and Todd class as key tools.

Abstract

The Riemann-Roch Theorem is one of the cornerstones of algebraic geometry, connecting algebraic data (sheaf cohomology) with geometric ones (intersection theory). This survey paper provides a self-contained introduction and a complete proof of the Hirzebruch-Riemann-Roch (HRR) Theorem for smooth projective varieties over an algebraically closed field. Starting from the classical formulations for curves and surfaces, we introduce the two modern tools necessary for the generalization: the Grothendieck group $K_{0}(X)$ as the natural setting for the Euler characteristic, and the Chow ring $A_{\bullet}(X)$ as the setting for cycles and intersection theory. We then construct the fundamental bridge between these two worlds\textemdash the Chern character ($\mathrm{ch}$) and the Todd class ($\mathrm{td}$) \textemdash culminating in a full proof of the HRR formula: \[ \chi(X,\mathcal{E})=\int_{X}\mathrm{ch}(\mathcal{E})\cdot\mathrm{td}(X) \] We conclude by showing how this general formula recovers the classical theorems for curves and surfaces.