Families of $K3$ surfaces over curves reaching the Arakelov-Yau type upper bounds and Modularity

Abstract

Let f : X → C be a family of semi-stable curves of genus g over a smooth projective C of genus q, and S ⊂ C the degeneration locus of f. The so-called Arakelov inequality states that deg f∗ωX/C ≤ g2 deg Ω 1 C(log S) = g 2 (2q − 2 + #S). When g ≥ 2 and #S = 0, the Miyaoka-Yau inequality for surfaces implies a much stronger inequality deg f∗ωX/C ≤ g − 1 6 deg Ω 1 C . In general, Tan [28] proved that the Arakelov inequality for a family f : X → C of semi-stable curves of genus ≥ 2 holds strictly. If g = 1, then deg f∗ωX/C can reach the upper bound in the inequality. Beauville has classified such families over C = P with #S = 4. More precisely, there are exactly 6 non-isotrivial families of semi-stable elliptic curves over P with 4 singular fibres. All of them are modular families of elliptic curves [2]. In this paper, we will consider the similar question for families of higher dimensional varieties. The Arakelov inequality is a special case of some more general inequalities for Hodge bundles. To state them, let V denote a polarized real variation of Hodge structure on a smooth projective curve C S such that the local monodromies around S are all unipotent, let (⊕p+q=kE, θ) denote the corresponding Hodge bundles. In [8] the following Arakelov-Yau type inequality was proven (also see [18] for a similar inequality): If k = 2l + 1, then