Abstract 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.

中文翻译：

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模空间上具有对合、等变解析扭转和自守形式的 K3 曲面 IV：不变量的结构

摘要吉川在[发明。数学。156 (2004), 53-117] 介绍了具有对合的 $K3$ 表面的全纯扭转不变量。在本文中，我们将其结构完全确定为此类 $K3$ 曲面模空间上的自守函数。在模空间的每个组件上，它都表示为显式 Borcherds 提升和经典 Siegel 模形式的乘积。我们还介绍了它的扭曲版本。我们证明了它的模块化和对应于扭曲全纯扭转不变量的模块化形式的某种独特性。这用于研究 Borcherds 猜想的等变类似物。