Higher de Rham epsilon factors

This article is devoted to the study of a higher-dimensional generalisation of de Rham epsilon lines. To a holonomic $D$-module $M$ on a smooth variety $X$ and a generic tuple of $1$-form $(\nu_1,\dots,\nu_n)$, we associate a point of the $K$-theory space $K(X,Z)$. If $X$ is proper this $K$-theory class is related to the de Rham cohomology $R\Gamma_{dR}(X,M)$. The novel feature of our construction is that $Z$ is allowed to be of dimension $0$. Furthermore, we allow the tuple of $1$-forms to vary in families, and observe that this leads naturally to a crystal akin to the epsilon connection for curves. Our approach is based on combining a construction of Patel with a homotopy invariance property of algebraic $K$-theory with respect to $(\mathbb{P}^1,\infty)$. This homotopical viewpoint leads us naturally to the definition of an epsilon connection in higher dimensions. Along the way we prove the compatibility of Patel's epsilon factors with the graded lines defined by Deligne and Beilinson--Bloch--Esnault in the case of curves.