Fiber integration of gerbes and Deligne line bundles

Let $\pi : X \to S$ be a family of smooth projective curves, and let $L$ and $M$ be a pair of line bundles on $X$. We show that Deligne’s line bundle $\langle L,M \rangle$ can be obtained from the $\mathcal{K}_2$-gerbe $G_{L,M}$ constructed in [AR16] via an integration along the fiber map for gerbes that categorifies the well known one arising from the Leray spectral sequence of $\pi$. Our construction provides a full account of the biadditivity properties of $\langle L,M \rangle$. Our main application is to the categorification of correspondences on the self-product of a curve.

The functorial description of the low degree maps in the Leray spectral sequence for $\pi$ that we develop is of independent interest, and, along the way, we provide an example of their application to the Brauer group.

Keywords

algebraic cycle, gerbe, higher category

2010 Mathematics Subject Classification

14C25, 14F42, 55N15, 55P20

Full Text (PDF format)

Received 8 August 2021

Received revised 15 January 2022

Accepted 18 January 2022

Published 1 March 2023