The infinitesimal Torelli problem

Abstract

Let $g\ge 2$ and let the Torelli map denote the map sending a genus $g$ curve to its principally polarized Jacobian. We verify the well known fact that the map induced on tangent spaces by the Torelli map is dual to the multiplication map ${\mathrm{Sym}}^2{H}^0(C,{\omega }_C)\to H^0(C,{\omega }_C^{\otimes 2})$.