Compact K3 moduli

March 16, 2021 - 04:30 - March 16, 2021 - 05:30

Philip Engel, University of Georgia

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笔补蝉蝉肠辞诲别听谤别辩耻颈谤别诲

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.听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听听

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.

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Compact K3 moduli

Compact K3 moduli