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

Abstract

We study $A_{\infty}$-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 $A_{\infty}$-structure up to homotopy and rescaling. In the case when $X$ is a curve we also compute the group of self-homotopies of this $A_{\infty}$-structure.