K3 K3 K3 Surfaces with Involution 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. Such modular forms are said to be Borcherds's product in this paper. Among all Borcherds's products, 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 one of the fake monster Lie superalgebras ([Bo2, §14]), although En-riques surface itself plays no role. After Borcherds, Jorgenson-Todorov ([J-T2,3]) and Harvey-Moore ([H-M]) discovered that the Ray-Singer analytic torsion ([R-S]) of an Enriques surface equipped with the normalized Ricci-flat Kähler metric coincides with Borcherds's Φ-function at its period point. The goal of this paper is to give a rigorous proof to their observation and generalize it to an interesting class of K3 surfaces studied by Nikulin ([Ni4]). Let us briefly recall these surfaces. Let (X, ι) be a K3 surface with an anti-symplectic involution. Let M be a 2-elementary hyperbolic lattice. The pair (X, ι) is said to be a 2-elementary K3 surface of type M if the invariant sublattice of H 2 (X, Z) with respect to the action of ι is isometric to M. Since X/ι is an Enriques surface when M ∼ = II 1,9 (2), K3 surfaces of this class are a kind of generalizations of Enriques surfaces. (We denote by I p,q (resp. II p,q) the odd (resp. even) unimodular lattice of signature (p, q).) By the Torelli theorem ([P-S-S]) and surjectivity of the period map ([To1]), the moduli space of 2-elementary K3 surfaces of type M is isomorphic to an arithmetic quotient of an open subset of the symmetric bounded domain of type IV via the period map.