Special homogeneous curves


  • David Lindemann




We classify all special homogeneous curves. A special homogeneous curve $\mathcal {H}$ consists of connected components of the hyperbolic points in the level set $\{h=1\}$ of a homogeneous polynomial $h$ in two real variables of degree at least three, and admits a transitive group action of a subgroup $G\subset \mathrm {GL}(2)$ on $\mathcal {H}$ that acts via linear coordinate change.


Alekseevsky, D. V., Cortés, V., and Devchand, C., Special complex manifolds, J. Geom. Phys. 42 (2002), no. 1–2, 85–105. https://doi.org/10.1016/S0393-0440(01)00078-X

Cortés, V., Dyckmanns, M., and Lindemann, D., Classification of complete projective special real surfaces, Proc. Lond. Math. Soc. (3) 109 (2014), no. 2, 423–445. https://doi.org/10.1112/plms/pdu013

Cortés, V., Dyckmanns, M., Jüngling, M., and Lindemann, D.,A class of cubic hypersurfaces and quaternionic Kähler manifolds of co-homogeneity one, Asian J. Math. 25 (2021), no. 1, 1–30. https://doi.org/10.4310/AJM.2021.v25.n1.a1

Cortés, V., Han, X., and Mohaupt, T., Completeness in supergravity constructions, Comm. Math. Phys. 311 (2012), no. 1, 191–213. https://doi.org/10.1007/s00220-012-1443-x

Cortés, V., Nardmann, M., and Suhr, S., Completeness of hyperbolic centroaffine hypersurfaces, Comm. Anal. Geom. 24 (2016), no. 1, 59–92 https://doi.org/10.4310/CAG.2016.v24.n1.a3

de Wit, B., and Van Proeyen, A., Special geometry, cubic polynomials and homogeneous quaternionic spaces, Comm. Math. Phys. 149 (1992), no. 2, 307–333. http://projecteuclid.org/euclid.cmp/1104251224

Freed, D. S., Special Kähler manifolds, Comm. Math. Phys. 203 (1999), no. 1, 31–52. https://doi.org/10.1007/s002200050604

Demailly, J.-P., and Paun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247–1274. https://doi.org/10.4007/annals.2004.159.1247

Korchagin, A. B., and Weinberg, D. A., The isotopy classification of affine quartic curves, Rocky Mountain J. Math. 32 (2002), no. 1, 255–347. https://doi.org/10.1216/rmjm/1030539619

Lindemann, D., Properties of the moduli set of complete connected projective special real manifolds, Math. Z. 303 (2023), no. 2, Paper No. 37, 63 pp. https://doi.org/10.1007/s00209-022-03184-4

Lindemann, D., Limit geometry of complete projective special real manifolds, arXiv:2009.12956.

Lindemann, D., Special geometry of quartic curves, arxiv:2206.12524.

Lindemann, D., and Swann, A., Special homogeneous surfaces, arXiv:2303.18228.

Magnússon, G. Th., Cohomological expression of the curvature of Kähler moduli, arXiv:2004.06881.

Nomizu, K., and Sasaki, T., Affine differential geometry. Geometry of affine immersions, Cambridge Tracts in Mathematics, 111. Cambridge University Press, Cambridge, 1994.

Wilson, P. M. H., Sectional curvatures of Kähler moduli, Math. Ann. 330 (2004), no. 4, 631–664. https://doi.org/10.1007/s00208-004-0563-9



How to Cite

Lindemann, D. (2024). Special homogeneous curves. MATHEMATICA SCANDINAVICA, 130(2). https://doi.org/10.7146/math.scand.a-143045