A non-linear differential equation for the periods of elliptic surfaces

Abstract

Suppose that $f:X\to C$ is a general Jacobian elliptic surface over the complex numbers. Then the primitive cohomology $H^{1,1}_{prim}(X)$ has, up to a sign, a natural orthonormal basis $(\eta_i)_{i\in [1, N]}$ given by certain meromorphic $2$-forms $\eta_i$ of the second kind, one for each ramification point of the classifying morphism $\phi$ from $C$ to the stack of generalized elliptic curves. (Here $N$ is any one of $h^{1,1}_{prim}(X)$, the number of moduli of $X$ and the degree of the ramification of $\phi$; these numbers are equal.) A choice of local co-ordinate on the stack of elliptic curves provides, via the branch locus of $\phi$, an {\'e}tale local co-ordinate system $(t_i)_{i\in [1, N]}$ on the stack of Jacobian elliptic surfaces. The main result here is that truncation of the Gauss--Manin connexion yields the system $$\{\partial_i H=(\partial_i \eta_i\wedge\eta_i)H\}_{i\in [1, N]}$$ of non-linear pde satisfied by $H=[\eta_1,\ldots, \eta_N]$, where $\partial_i =\partial/\partial t_i$ and the skew tensor $\partial_i \eta_i\wedge\eta_i$ of rank $2$ is the ecliptic of $\eta_i$ (the plane in which the particle $\eta_i$ is instantaneously moving with respect to $t_i$). 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 $O_N$, and we prove a generic infinitesimal Torelli theorem for this map. For rational elliptic surfaces this can be calculated explicitly.