The cotangent bundle of K3 surfaces of degree two

TL;DR

This paper investigates the geometry of the projectivised cotangent bundle of a general degree two K3 surface, revealing complex structures analogous to classical quartic surface features.

Contribution

It provides a detailed description of the geometry of a specific surface within the cotangent bundle of a general polarized K3 surface of degree two.

Findings

Describes the geometry of the surface D_S in the projectivised cotangent bundle.

Establishes an analogy between D_S and the surface of bitangents for quartic surfaces.

Enhances understanding of the positivity properties of the cotangent bundle of K3 surfaces.

Abstract

K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general polarised K3 surface $S$ of degree two. In particular, we describe the geometry of a surface $D_S \subset \mathbb{P}(\Omega_S)$ that plays a similar role to the surface of bitangents for a quartic in $\mathbb{P}^3$.