Toroidal compactifications of the moduli space of polarised K3 surfaces
In this talk I would like to present the problem which is at the basis of my PhD project: toroidal compactifications of the moduli space of polarised K3 surfaces. The search for compactifications for moduli spaces is a central problem in algebraic geometry that has been widely studied and toroidal constructions provide a tool to compactify varieties arising as quotients of hermitian symmetric domains by the action of arithmetic groups. For the moduli space of principally polarised abelian varieties $\mathcal(A)_g$ some specific toroidal compactifications have been studied in detail, while for the case of K3 surfaces most of the work so far has been done thanks to the abstract knowledge of such toroidal compactifications. The goal of my research is to find and describe a specific toroidal compactification; the main idea at its basis is to mimic the construction of the perfect cone compactification for $\mathcal(A)_g$ and to apply that to the moduli space of $K3$ surfaces. Time permitting, I will add something regarding compactifications of other moduli spaces which involve lower rank lattices and therefore rely on more feasible computations.