Families of \(K3\) surfaces over curves reaching the Arakelov-Yau type upper bounds and modularity
This paper considers non-isotrivial algebraic families of semi-stable \(K3\) surfaces over curves when the Arakelov-Yau inequality reaches the upper bound and becomes equality. Let \(f : X\to C\) be a family of semi-stable \(K3\) surfaces over a projective curve \(C\). The Arakelov-Yau inequality implies that \(\text{deg }(f_*\omega_{X/C})\leq \text{deg }\Omega_C^1(\log S)\) where \(S\) is the discriminant locus. This paper studies non-isotrivial families which satisfy the equality \(\text{deg}(f_*\omega_{X/C})=\text{deg\,}\Omega_C^1(\log\,S)\). They prove that the general fiber of \(f\) has Picard number at least 19, and after finite étale base change, the restriction of the family over the open subset admits a Nikulin involution whose quotient is the family of abelian surfaces isogenous to the symmetric square of a family of elliptic curves. (Such a family is called a family coming from Nikulin-Kummer construction of the square product of a family of elliptic curves.) Also they prove that if the second iterated Kodaira-Spencer map of the family is zero and if the Arakelov-Yau upper bound is reached, then the general fiber of \(f\) has the Picard number at least 18. In both cases, the monodromy representations are determined as subgroups of \(\text{PSL}_2({\mathbb Z})\). As a corollary, the authors restrict to the special case that \(X\) is a Calabi-Yau threefold. Then \(\# S\geq 4\). Let \(f: X\to {\mathbb{P}}^1\) be a Calabi-Yau threefold fibered by non-constant families of semi-stable \(K3\) surfaces, reaching the Arakelov-Yau bound. Then the following hold true: (1) If the iterated Kodaira--Spencer map of \(f\) is non-zero, then \(f\) has at least 4 singular fibers. If \(f\) has exactly 4 singular fibers, then \(X\) is a rigid Calabi-Yau threefold, the general fibers have Picard number at least 19, and birational to the Nikulin-Kummer construction of a square product of a family of elliptic curves \(g: E\to{\mathbb{P}}^1\). After passing to (if necessary) a double cover \(E^{\prime}\to E\), the family \(g^{\prime}: E^{\prime}\to{\mathbb{P}}^1\) is one of the six modular families of elliptic curves constructed by \textit{A. Beauville} [C. R. Acad. Sci. Paris Sér. I Math. 294, 657--660 (1982; Zbl 0504.14016)]. (2) If the iterated Kodaira-Spencer map of \(f\) is zero, then \(f\) has at least \(6\) singular fibers. If \(f\) has exactly 6 singular fibers over \(S\subset{\mathbb{P}}^1\), then \(X\) is a non-rigid Calabi-Yau threefold, the general fibers have Picard number at least 18 and \({\mathbb{P}}^1\setminus S\simeq {\mathbb{H}}/\Gamma\) where \(\Gamma\) is a subgroup of \(\text{SL}_2({\mathbb{Z}})\) of index 24. In this case, a stronger equality holds : \(\text{deg } (f_*\omega_{X/C}) ={{1}\over{2}} \text{deg }\Omega_C^1(\log S)\). Examples of non-rigid Calabi-Yau threefolds satisfying the property (2) have been found by \textit{R. Livné} and \textit{N. Yui} [J. Math. Kyoto Uni. 45, No. 4, 645--665 (2005; Zbl 1106.14025)].
scientific article; zbMATH DE number 2064162