论文标题
通用K3表面的Deligne-Beilinson同谋
Deligne-Beilinson cohomology of the universal K3 surface
论文作者
论文摘要
O'Grady的广义Franchetta猜想（GFC）与通用极化K3表面上的Codimension 2代数循环有关。在\ cite {bl17}中，已经在贝蒂共同体学组中研究了这个猜想。经过Voisin的建议，我们在Deligne-Beilinson（db）的同一个研究小组中研究了这个问题。在本文中，我们开发了分离（平滑）磨难蒙福堆栈的Deligne-beilinson共同体组的理论。我们使用自动形态的提升组和Noether-Lefschetz理论，我们计算了第4- db-cohomology群体的通用偏光K3表面的第4- db-colomology群，最坏的是$ a_1 $ singularity，并显示了该家族在DB-COMOMOMOGOL中持有的GFC。特别是，这证实了O'Grady在DB共同体中的原始猜想。
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.
