# Extensions of homogeneous coordinate rings to Aµ-algebras

## A. Polishchuk

We study $A_{\infty}$-structures extending the natural algebra structure on the cohomology of $\oplus_{n\in\Z} L^n$, 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 $n \in \Z$. We prove that there exists a unique such nontrivial $A_{\infty}$-structure up to a strict $A_{\infty}$-isomorphism (i.e., an $A_{\infty}$-isomorphism with the identity as the first structure map) and rescaling.

In the case when $X$ is a curve we also compute the group of strict $A_{\infty}$-automorphisms of this $A_{\infty}$-structure.

Homology, Homotopy and Applications, Vol. 5(2003), No. 1, pp. 407-421

Available as: dvi dvi.gz ps ps.gz pdf