Transformation de Fourier-Mukai sur les Surfaces Hyperk\"ahl\'eriennes

TL;DR

This paper extends the Fourier-Mukai transform to hyperk"ahler surfaces, particularly K3 surfaces, and investigates how instantons are transformed under this new framework, generalizing known results from complex tori.

Contribution

It introduces a Fourier-Mukai transform for K3 surfaces and analyzes the transformation behavior of instantons on these surfaces, expanding the scope of the classical transform.

Findings

Transform maps instantons on K3 surfaces to instantons on dual surfaces.

Extension of Fourier-Mukai transform to hyperk"ahler surfaces.

Insights into the behavior of vector bundles under the transform.

Abstract

Given two compact hyperk\"ahler surfaces $X$ and $Y$ and a holomorphic vector bundle $Q$ on $X\times Y$, which is a generalized instanton, one can define a Fourier-Mukai transform, which, under suitable assumptions, maps vector bundles on $X$ to vector bundles on $Y$. If $X$ and $Y$ are dual complex tori, this transform maps instantons on $X$ to instantons on $Y$. After a quick review of these results, we define a Fourier-Mukai transform in the case when $X$ is a K3 surface, and study the behaviour of instantons on $X$ under this transform. Hard copies may be mailed on request