# Algebraic Atiyah-Singer index theorem

## TL;DR

This paper develops an algebraic weak version of the Atiyah-Singer index theorem and illustrates it through examples involving elliptic differential operators on smooth projective schemes over a field.

## Contribution

It introduces an algebraic approach to the Atiyah-Singer index theorem and applies it to specific elliptic operators derived from the Atiyah class.

## Key findings

- Computed examples of elliptic operators on smooth projective schemes
- Established an algebraic framework for the index theorem
- Connected Atiyah class to index computations

## Abstract

The aim of this work is to give an algebraic weak version of the Atiyah-Singer index theorem. We compute then a few small examples with the elliptic differential operator of order $\leq 1$ coming from the Atiyah class in $\mathrm{Ext}^1_{\mathcal{O}_X}(\mathcal{O}_X,\Omega^1_{X/k})$, where $X \longrightarrow \mathrm{Spec}(k)$ is a smooth projective scheme over a perfect field $k$.