Stability and certain $\mathbb{P}^n$-functors

TL;DR

This paper demonstrates that Addington's $P^n$-functor preserves the stability of vector bundles from a K3 surface to its Hilbert scheme of points under certain conditions.

Contribution

It proves that a specific $P^n$-functor maintains the stability of vector bundles between a K3 surface and its Hilbert scheme, under particular numerical constraints.

Findings

Stable vector bundles are preserved under the functor.

The functor maps stable bundles to stable bundles under certain conditions.

The result applies to K3 surfaces and their Hilbert schemes.

Abstract

Let $X$ be a K3 surface. We prove that Addington's $\mathbb{P}^n$-functor between the derived categories of $X$ and the Hilbert scheme of points $X^{[k]}$ maps stable vector bundles on $X$ to stable vector bundles on $X^{[k]}$, given some numerical conditions are satisfied.