of covariance, but in view the reductive nature of ${{\mathscr B}}$ it is not worthwhile to make the distinction.

This situation shows up when we consider a polarized abelian variety $A$, endowed with a homomorphism ${{\mathscr B}}\to  {\operatorname{End}}_{{\mathbb R}}(A)$ of algebras with involution. If we put $F:=H^0(A,\Omega_A)$, then ${\operatorname{End}}_{{\mathbb R}}(A):={{\mathbb R}}\otimes {\operatorname{End}}(A)$ (and hence ${{\mathscr B}}$) acts on $F$. As is well-known, the space of first order deformations of $A$ as a polarized variety can be identified with the complex dual of ${\operatorname{Sym}}^2F$. The first order deformations for which ${{\mathscr B}}$ still maps to the deformed object can be identified with the dual of the ${{\mathscr B}}$-coinvariants of ${\operatorname{Sym}}^2F$, i.e., the dual of $({\operatorname{Sym}}^2F)_{L({{\mathscr B}})}\cong ({\operatorname{Sym}}^2F)^{L({{\mathscr B}})}$. The more precise statement is that we have locally a universal deformation in the sense of Schlessinger with smooth base and whose cotangent space at the closed point can be identified with $({\operatorname{Sym}}^2F)^{L({{\mathscr B}})}$.\
We now turn to the proof of Theorem \[endo\]. We first reduce it to Lemma \[lemma:endo\] below. We write ${{\mathscr B}}$ for ${\operatorname{End}}_{{\mathbb R}}(J(C))$ and $F$ for $H^0(C, \Omega_C)$. The property that ${{\mathbb Q}}G\to {\operatorname{End}}_{{\mathbb Q}}(J(C))$ be surjective is equivalent to ${{\mathbb R}}G\to {{\mathscr B}}$ being surjective and since 