A note about special cycles on moduli spaces of \(K3\) surfaces. (English) Zbl 1302.14030
Laza, Radu (ed.) et al., Arithmetic and geometry of \(K3\) surfaces and Calabi-Yau threefolds. Proceedings of the workshop, Toronto, Canada, August 16–25, 2011. New York, NY: Springer (ISBN 978-1-4614-6402-0/hbk; 978-1-4614-6403-7/ebook). Fields Institute Communications 67, 411-427 (2013).
Summary: We describe the application of the results of S. S. Kudla and J. J. Millson [Math. Ann. 274, No. 3, 353–378 (1986; Zbl 0594.10020); Math. Ann. 277, No. 2, 267–314 (1987; Zbl 0618.10022); Publ. Math., Inst. Hautes Étud. Sci. 71, 121–172 (1990; Zbl 0722.11026)] on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized \(K3\) surfaces. In this case, the special cycles can be interpreted as higher Noether-Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether-Lefschetz numbers as Fourier coefficients. Examples of such complete families associated to quadratic spaces over totally real number fields are constructed.
For the entire collection see [Zbl 1267.14002].
For the entire collection see [Zbl 1267.14002].
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
| 14J10 | Families, moduli, classification: algebraic theory |
| 14C25 | Algebraic cycles |
| 14G35 | Modular and Shimura varieties |
