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 $H_{prim}^{1,1}(X)$ 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 $\varphi$ from $C$ to the stack of generalized elliptic curves. (Here $N$ is any one of $h_{prim}^{1,1}(X)$, the number of moduli of $X$ and the degree of the ramification of $\varphi$; these numbers are equal.) A choice of local co-ordinate on the stack of elliptic curves provides, via the branch locus of $\varphi$, an {é}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 $H=[{\eta }_1,\dots ,{\eta }_N]$, 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.