Proceedings of the International Conference on Geometry, Integrability and Quantization

Enumerative Geometry on Quasi-Hyperbolic 4-Spaces with Cusps

Rolf-Peter Holzapfel

Abstract

We introduce orbital functionals $\int \beta$ simultaneously for each commensurability class of orbital surfaces. They are realized on infinitely dimensional orbital divisor spaces spanned by (arithmetic-geodesic real 2-dimensional) orbital curves on any orbital surface. We discover infinitely many of them on each commensurability class of orbital Picard surfaces, which are real 4-spaces with cusps and negative constant Kähler–Einstein metric degenerated along an orbital cycle. For a suitable (Heegner) sequence $\int \mathbf{h}_N, N \in \mathbb{N}$, of them we investigate the corresponding formal orbital $q$-series $\sum_{N=0}^{\infty} (\int \mathbf{h}_N)q^N$. We show that after substitution $q = {\rm e}^{2 \pi \mathbf{i} \tau}$ and application to arithmetic orbital curves Ĉ on a fixed Picard surface class, the series $\sum_{N=0}^{\infty}(\int_{Ĉ} {\mathbb{h}_N})e^{2 \pi \mathbf{i} \tau}$ define modular forms of well-determined fixed weight, level and Nebentypus. The proof needs a new orbital understanding of orbital heights introduced in [12] and Mumford–Fulton’s rational intersection theory on singular surfaces in Riemann–Roch–Hirzebruch style. It has to be connected with Zeta and Theta functions of hermitian lines, indefinite quaternionic fields and of a matrix algebra along a research marathon over 75 years represented by Cogdell, Kudla, Hirzebruch, Zagier, Shimura, Schoeneberg and Hecke. Our aim is to open a door to an effective enumerative geometry for complex geodesics on orbital varieties with nice metrics.

Article information

Source
Proceedings of the Fourth International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov and Gregory L. Naber, eds. (Sofia: Coral Press Scientific Publishing, 2003), 42-87

Dates
First available in Project Euclid: 12 June 2015

Permanent link to this document
https://projecteuclid.org/ euclid.pgiq/1434148596

Digital Object Identifier
doi:10.7546/giq-4-2003-11-41

Mathematical Reviews number (MathSciNet)
MR1977559

Zentralblatt MATH identifier
1039.53104

Citation

Holzapfel, Rolf-Peter. Enumerative Geometry on Quasi-Hyperbolic 4-Spaces with Cusps. Proceedings of the Fourth International Conference on Geometry, Integrability and Quantization, 42--87, Coral Press Scientific Publishing, Sofia, Bulgaria, 2003. doi:10.7546/giq-4-2003-11-41. https://projecteuclid.org/euclid.pgiq/1434148596


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