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K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: The structure of the invariant
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 10 / October 2020
- Published online by Cambridge University Press:
- 19 November 2020, pp. 1965-2019
- Print publication:
- October 2020
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- Article
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Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of
$K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such 
$K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds’ conjecture.
