2004/01/27 by Burt Totaro, Totaro, Burt · 1 citation
Mathematics · Physics and Astronomy · #53A15 (Primary) 15A72 #53C20 #58F07 (Secondary) #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.DG #msc:15A72 #msc:53A15 #msc:53C20 #msc:58F07
paper · pdf · doi:10.48550/arxiv.math/0401381
21 pages. To appear in Int. J. Math. Includes Dubrovin's counterexample to my original conjecture
openalex publication_date 2004/01/27 · arxiv created 2004/02/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Following P. M. H. Wilson's paper on sectional curvatures of Kahler moduli, we consider a natural Riemannian metric on a hypersurface f=1 in a real vector space, defined using the Hessian of a homogeneous polynomial f. We give examples to answer a question by Wilson about when this metric has nonpositive curvature. Also, we exhibit a large class of polynomials f on R3 such that the associated metric has constant negative curvature. Dubrovin found another such polynomial on R3, the Maschke sextic. We ask if these examples are the only ones with constant negative curvature, and we prove some partial results. This question is related both to the WDVV equations from string theory and to the "Clebsch covariant" from 19th-century invariant theory.