2025/12/04 by N. I. Shepherd‐Barron, N. I. Shepherd-Barron, Shepherd-Barron, N. I.
Mathematics · #14C30 #32G20 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:14C30 #msc:32G20
paper · pdf · doi:10.48550/arxiv.2512.04930
openalex publication_date 2025/12/04 · openalex created_date 2025/12/06 · arxiv created 2026/07/29 · arxiv updated 2026/07/30 · openalex updated_date 2026/08/01
Suppose that f:X→ C is a general Jacobian elliptic surface over the complex numbers. Then the primitive cohomology H1,1prim(X) has, up to a sign, a natural orthonormal basis (ηi)i∈ [1, N] given by certain meromorphic 2-forms ηi of the second kind, one for each ramification point of the classifying morphism ϕ from C to the stack of generalized elliptic curves. (Here N is any one of h1,1prim(X), the number of moduli of X and the degree of the ramification of ϕ; these numbers are equal.) A choice of local co-ordinate on the stack of elliptic curves provides, via the branch locus of ϕ, an étale local co-ordinate system (ti)i∈ [1, N] on the stack of Jacobian elliptic surfaces. The main result here is that truncation of the Gauss--Manin connexion yields the system \∂i H=(∂i ηi\wedgeηi)H\i∈ [1, N] of non-linear pde satisfied by H=[η1,…, ηN], where ∂i =∂/∂ ti and the skew tensor ∂i ηi\wedgeηi of rank 2 is the ecliptic of ηi (the plane in which the particle ηi is instantaneously moving with respect to ti). Moreover, after rigidification of the integral cohomology, H can be interpreted as providing a period map for these surfaces with values in the complex orthogonal group ON, and we prove a generic infinitesimal Torelli theorem for this map. For rational elliptic surfaces this can be calculated explicitly.