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K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space

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Abstract

In this paper, we introduce an invariant of a K3 surface with ℤ2-action equipped with a ℤ2-invariant Kähler metric, which we obtain using the equivariant analytic torsion of the trivial line bundle. This invariant is shown to be independent of the choice of the Kähler metric. It can be viewed as a function on the moduli space of K3 surfaces with involution. The main result of this paper is that this function can be identified with an automorphic form, which characterizes the discriminant locus. In particular, we show that the Ray–Singer analytic torsion of the trivial line bundle on an Enriques surface with Ricci-flat Kähler metric is given by the value of the norm of the Borcherds Φ-function at its period point.

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  1. Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, 153-8914, Tokyo, Japan

    Ken-Ichi Yoshikawa

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  1. Ken-Ichi Yoshikawa
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Correspondence to Ken-Ichi Yoshikawa .

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Mathematics Subject Classification (1991)

58G26, 14J28, 14J15, 32G20, 32N10, 32N15

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Yoshikawa , KI. K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space. Invent. math. 156, 53–117 (2004). https://doi.org/10.1007/s00222-003-0334-3

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