# The Riemann-Roch Theorem on higher dimensional complex noncommutative tori

## TL;DR

This paper extends classical theorems like Riemann-Roch and Hodge to higher-dimensional complex noncommutative tori, offering new insights into their geometric and algebraic structures.

## Contribution

It establishes analogues of Riemann-Roch and Hodge theorems for complex noncommutative tori of any dimension, advancing the understanding of their geometric properties.

## Key findings

- Proved Riemann-Roch and Hodge theorems for noncommutative tori
- Discussed criteria to distinguish noncommutative abelian varieties
- Explored implications for non-algebraic noncommutative complex tori

## Abstract

We prove analogues of the Riemann-Roch Theorem and the Hodge Theorem for noncommutative tori (of any dimension) equipped with complex structures, and discuss implications for the question of how to distinguish "noncommutative abelian varieties" from "non-algebraic" noncommutative complex tori.