# Vector bundles on Fano threefolds and K3 surfaces

## TL;DR

This paper explores the relationship between vector bundles on Fano threefolds and K3 surfaces, demonstrating that certain bundles originating from the threefold form Lagrangian subvarieties within moduli spaces on the K3 surface.

## Contribution

It introduces a novel connection between vector bundles on Fano threefolds and Lagrangian subvarieties in moduli spaces on K3 surfaces, extending Tyurin's ideas.

## Key findings

- Vector bundles from Fano threefolds form Lagrangian subvarieties in moduli spaces.
- Concrete examples illustrating the Lagrangian property.
- Theoretical framework linking Fano threefolds and K3 surface moduli.

## Abstract

Let X be a Fano threefold, and let S be a K3 surface in X . Any moduli space M of simple vector bundles on S carries a holomorphic symplectic structure. Following an idea of Tyurin, we show that in some cases, those vector bundles which come from X form a Lagrangian subvariety of M . We illustrate this with a number of concrete examples.