# 1 – Intro | $K3$ surfaces & Lattices

When $X$ is $K3$ then

$$H^2(X,\ZZ) \simeq U\oplus U\oplus U \oplus E_{8}(-1) \oplus E_{8}(-1)$$thus endowing $H^2(X,\ZZ)$ with a lattice structure, called the

$K3$ lattice.

For any primitive hyperbolic sublattice $S$ of the $K3$ lattice, there exists a $K3$ surface $X$ such that

$S\simeq \NS(X)$.