Hodge theory of toroidal compactifications and Torelli theorems — K3 surfaces,abelian varieties, and IHSM

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In this project we want to investigate the interplay between compactifications of moduli spaces and Hodge theory. Our main focus are moduli spaces of abelian varieties, K3 surfaces, and irreducible holomorphic symplectic manifolds (IHSM). The classification of algebraic varieties is one of the most fundamental tasks in algebraic geometry. This typically leads to the construction of moduli spaces which parametrize the objects to be classified. K3 surfaces and abelian varieties are among the most thoroughly studied classes of algebraic varieties and in the last decades also IHSM, which are higher dimensional analogs of K3 surfaces, have become a recurrent topic in algebraic geometry. The moduli spaces of these classes of varieties are always quasi-projective, but usually not projective and it is a central question in moduli theory to ask for good compactifications of such moduli spaces.The relation between Hodge theory and algebraic geometry is classical and was built around the computation of periods of algebraic varieties. Even though Hodge theory is a fundamentally analytic theory, its main applications are to algebraic and arithmetic geometry. Torelli theorems establish a link between the geometry of moduli spaces and the images of period maps and in the cases of interest to us the period maps are even surjective. As a result one can identify such a moduli space with a quotient of the period domain by an arithmetic group. This endows these spaces with a rich structure and allows us to use techniques from Lie theory and modular forms. In particular, we have the theory of toroidal or semitoric compactifications due to Mumford, Looijenga, and others at our disposal. This has been exploited in the first part of this project and has led, among other things, to new semitoric compactifications of moduli spaces of K3 surfaces.In the current project we will use Torelli theorems as the guiding principle. More precisely, we want to relate degenerations of varieties (which should correspond to boundary points in a compactification of the moduli spaces) to degenerations of Hodge structures (which should be linked to specific semitoric compactifications) via Torelli theorems. The ultimate goal of this research project is the construction of modular compactifications of moduli spaces by means of Hodge theory.