The polarized degree of irrationality of $K3$ surfaces

Given a polarized variety $(X,L)$, we construct and study projections of low degree $ X\dashrightarrow \mathbb{P}(H^0(L^\vee)) \dashrightarrow \mathbb P ^n $ using the associated kernel bundles. As an application, we can show that the degree of irrationality of a very general $(1,6)-$polarized abelian surface, as well as that of a very general $K3$ surface of genus $6$ is $3$. We also give new upper bounds on the degree of irrationality of $K3$ surfaces of any genus. We study the family of projections of minimal degree of a very general $K3$ surface of genus $4,5,6$. As a different application of our construction, we exhibit new generically finite rational maps of low degree from some hyper-K\"ahler varieties and abelian varieties to projective spaces.