Clifford Multiplication on Spinor Abelian Varieties

We define a spinor Abelian variety $S_{\mathrm{\Delta }}$ to be a complex Abelian variety whose tangent space at the origin is a space of spinors for a suitable complex Clifford algebra ${\mathbb{C}}_q(V)$. We examine intrinsic properties of such varieties and the connection between Clifford multiplication and their endomorphism algebras. We then extend the analysis of Clifford multiplication to the dual torus $Pi{c}^0(S_{\mathrm{\Delta }})$.