The Geometry and Moduli of K3 Surfaces

TL;DR

This paper introduces the theory of K3 surfaces, covering their moduli, degenerations, lattice polarizations, and geometric cones, providing foundational and explicit example-based insights into their complex structure.

Contribution

It offers a comprehensive overview of K3 surface theory, including new perspectives on their moduli spaces, degenerations, and lattice polarization structures.

Findings

Construction of K3 moduli space and its properties

Analysis of polarized K3 surface degenerations and compactifications

Explicit examples from lattice polarized K3 surfaces and elliptic fibrations

Abstract

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kahler cones of K3 surfaces, and give some of their applications.