Deligne-Beilinson cohomology of the universal K3 surface

Zhiyuan Li; Xun Zhang

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arXiv · 2201.02367

Deligne-Beilinson cohomology of the universal K3 surface

Zhiyuan Li·Xun Zhang

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Abstract

O'Grady's generalized Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarized K3 surfaces. In \cite{BL17}, this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on separated (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4-th DB-cohomology group of universal oriented polarized K3 surfaces with at worst an $A_1$-singularity and show that GFC for such family holds in DB-cohomology. In particular, this confirms O'Grady's original conjecture in DB cohomology.

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Zhiyuan Li, Xun Zhang. 2022-01-07. Deligne-Beilinson cohomology of the universal K3 surface. https://arxiv.org/abs/2201.02367

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