On the derivative of a normal function associated with a Deligne cohomology class
Let \(\pi: X \to S\) be a smooth family of varieties over \(\mathbb C\), and let \(\alpha \in H^i_{\mathcal D} (X, \mathbb Z (k))\) be a Deligne cohomology class. By restricting \(\alpha\) to the fibers, a family of Deligne cohomology classes \(\alpha_s \in H^i_{\mathcal D} (X_s, \mathbb Z (k))\) with \(X_s = \pi^{-1} (s)\) can be obtained, which determines a normal function. In this paper the author proves that the derivative of the normal function associated to \(\alpha\), when it exists, can be computed in a purely algebraic manner from the given cohomology class. He also obtains a similar result in syntomic cohomology. One of the consequences of this result is that, when \(\alpha\) has a motivic origin, the corresponding family of elements in syntomic cohomology has exactly the same derivative.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].
scientific article; zbMATH DE number 6077541