Compact moduli of K3 surfaces


We construct geometric compactifications of the moduli space $F_{2d}$ of polarized K3 surfaces in any degree $2d$. Our construction is via KSBA theory, by considering canonical choices of divisor $R\in |nL|$ on each polarized K3 surface $(X,L)\in F_{2d}$. The main new notion is that of a recognizable divisor $R$, a choice which can be consistently extended to all central fibers of Kulikov models. We prove that any choice of recognizable divisor leads to a semitoroidal compactification of the period space, at least up to normalization. Finally, we prove that the rational curve divisor is recognizable for all degrees.


Valery Alexeev

University of Georgia, Athens GA

Philip Engel

University of Georgia, Athens GA