# A note on Flenner's extension theorem

## TL;DR

This paper presents a simplified proof of Flenner's extension theorem for differential forms on complex spaces, showing extension with possible logarithmic poles under certain codimension conditions, and discusses limitations in positive characteristic.

## Contribution

Provides a shorter, simpler proof of Flenner's extension theorem for p-forms on complex spaces, extending the known results to include forms with logarithmic poles.

## Key findings

- Extension of p-forms with logarithmic poles under codimension constraints
- Shorter, more accessible proof of Flenner's theorem
- Counterexamples in positive characteristic

## Abstract

We show that any $p$-form on the smooth locus of a normal complex space extends to a resolution of singularities, possibly with logarithmic poles, as long as $p \le \mathrm{codim}_X (X_{\mathrm{sg}}) - 2$, where $c$ is the codimension of the singular locus. A stronger version of this result, allowing no poles at all, is originally due to Flenner. Our proof, however, is not only completely different, but also shorter and technically simpler. We furthermore give examples to show that the statement fails in positive characteristic.