m at h . A G ] 3 1 A ug 1 99 8 GENERALIZED ENRIQUES SURFACES AND ANALYTIC TORSION

1.1 Background. Kronecker's limit formula states that the regularized determinant of Laplacian of an elliptic curve is represented by the norm of Jacobi's ∆-function at its period point. One of the striking property of Jacobi's ∆-function is that it admits the infinite product expansion. In a series of works [Bo1-3], Borcherds developed a theory of modular forms over domains of type IV which admits an infinite product expansion and constructed many modular forms generalizing Jacobi's ∆-function. Among such modular forms, Borcherds's Φ-function ([Bo2]) has an interesting geometric background; It is a modular form on the moduli space of En-riques surfaces whose zero locus is just the discriminant locus. In his construction, Φ-function is obtained as a denominator function of certain infinite dimensional Lie super algebra related to the transcendental lattice of an Enriques surface, although Enriques surface itself plays no role. On the other hand, there are a series of works by Jorgenson-Todorov ([J-T1-3]) which try to generalize Kronecker's limit formula for higher dimensional Calabi-Yau manifolds. They introduced an invariant of polarized Calabi-Yau manifold called analytic discriminant by using regularized determinant of Laplacian with respect to the Ricci-flat Kähler metric according to the polarization, and claim that Borcherds's Φ-function coincides with the analytic discriminant (see [J-T3] and §7.6 below). There are also some works by Harvey-Moore which treat Borcherds's work in the physical context. They found that, although analytic torsion of a K3 surface with Ricci-flat Kähler metric is trivial, Z/2Z-symmetry of (the universal cover of) an Enriques surface yields nontrivial Ray-Singer analytic torsion, and discovered that it coincides with Borcherds's Φ-function ([H-M, §5]). The purpose of this article is to generalize Kronecker's limit formula to a class of K3 surfaces ([Ni3], [V]).