Enriques Surfaces and Analytic Torsion

In a series of works [Bo3-5], Borcherds developed a theory of modular forms over domains of type IV which admits an infinite product expansion. Among such modular forms, Borcherds’s Φ-function ([Bo4]) has an interesting geometric background; It is a modular form on the moduli space of Enriques surfaces characterizing the discriminant locus. In his construction, Φ-function is obtained as the denominator function of the fake monster Lie algebra (the fake Conway Lie superalgebra of rank 10) ([Bo2, §14]), although Enriques surface itself plays no role. After Borcherds and a pioneering work by Jorgenson-Todorov ([J-T]), it is Harvey-Moore who discovered that the analytic torsion of an Enriques surface equipped with the normalized Ricci-flat Kähler metric coincides with Borcherds’s Φ-function at its period point ([H-M]). The goal of this article is to give a rigorous proof to Harvey-Moore’s observation and generalize their results to an interesting class of K3 surfaces. Let (X, ι) be a K3 surface with an anti-symplectic involution and κ be an ιinvariant Ricci-flat Kähler metric. Let M be a 2-elementary hyperbolic lattice. We call (X, ι) a 2-elementary K3 surface of type M if the invariant sublattice of H(X,Z) with respect to the ι∗-action is isometric to M . (When M = II1,9(2), X/ι is an Enriques surface. We denote by Ip,q (resp. IIp,q) the odd (resp. even) unimodular lattice of signature (p, q).) The moduli space of 2-elementary K3 surfaces of type M is isomorphic to the arithmetic quotient of the open subset of symmetric bounded domain of type IV via the period map. The period domain is denoted by ΩM , the modular group by ΓM , and the moduli space by MM = ΩM/ΓM . Here, ΩM = ΩM\HM and HM is the discriminant locus. Let Ci be the fixed locus of ι. Since only one of Ci can have positive genus g(M)(≥ 0) if M 6= II1,9(2) or II1,1⊕E8(−2), one can define a map jM : MM → Ag(M) to the Siegel modular variety by taking the period point of the Jacobi variety of Ci. (If M = II1,1⊕E8(−2), jM takes its value in A1 × A1.) Let LM be the sheaf of modular forms of weight 1 over ΩM and LS be the sheaf of Siegel modular forms of weight 1 over Ag(M). A section of the bundle LpM ⊗ j∗ ML q S (with a character of ΓM ) is called a modular form of weight (p, q) in this article whose Petersson norm is denoted by ‖ · ‖. We introduce the following: