Analytic torsion for log-Enriques surfaces and Borcherds product

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 den...

Analytic torsion of Hirzebruch surfaces

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces. 1

On the Complete Classification of Extremal Log Enriques Surfaces

We show that there are exactly, up to isomorphisms, seven rational extremal log Enriques surfaces Z and construct all of them; among them types D19 and A19 have been shown of certain uniqueness by M. Reid. We also prove that the (degree 3 or 2) canonical covering of each of these seven Z has either X3 or X4 as its minimal resolution. Here X3 (resp. X4) is the unique K3 surface with Picard numbe...

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 ad...

Cubic Surfaces and Borcherds Products