Extensions of homogeneous coordinate rings to $A_{\mathrm{\infty }}$-algebras

We study $A_{\mathrm{\infty }}$-structures extending the natural algebra structure on the cohomology of ${\oplus }_n{L}^n$, where $L$ is a very ample line bundle on a projective $d$-dimensional variety $X$ such that $H^i(X,L^n)=0$ for $0<i<d$ and all $n$. We prove that there exists a unique such nontrivial $A_{\mathrm{\infty }}$-structure up to homotopy and rescaling. In the case when $X$ is a curve we also compute the group of self-homotopies of this $A_{\mathrm{\infty }}$-structure.