Compact K3 moduli

Compact K3 moduli

Philip Engel, University of Georgia

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This is joint work with Valery Alexeev. A well-known consequence of the Torelli theorem is that the moduli space F_g of polarized K3 surfaces is the quotient of a 19-dimensional Hermitian symmetric space by the action of an arithmetic group. In this capacity, it admits a natural class of "semitoroidal compactifications". These are built from periodic tilings of 18-dimensional hyperbolic space, and were studied by Looijenga, who built on earlier work of Baily-Borel and Ash-Mumford-Rapaport-Tai. On the other hand, F_g also admits "stable pair compactifications": Choose canonically on any polarized K3 surface X an ample divisor R. Papers of Kollar-Shepherd-Barron, Alexeev, and others provide for the existence of a compact moduli space of "stable pairs" containing the K3 pairs (X,R) as an open subset.                                                                                                                                                                                                                                      

I will discuss two theorems in the talk: (1) There is a simple criterion on R, called "recognizability" ensuring that the normalization of a stable pair compactification is semitoroidal and (2) the rational curves divisor, generically the sum of geometric genus zero curves in the polarization, is recognizable. This gives a modular semitoroidal compactification for all genera g.