## Homology, Homotopy and Applications

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

A. Polishchuk

#### Abstract

We study $A_\infty$-structures extending the natural algebra structure on the cohomology of $\oplus_{n\in\mathbb{Z}} L^n$, 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 $n \in \mathbb{Z}$. 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.

#### Article information

Source
Homology Homotopy Appl., Volume 5, Number 1 (2003), 407-421.

Dates
First available in Project Euclid: 13 February 2006

Polishchuk, A. Extensions of homogeneous coordinate rings to $A_ \infty$-algebras. Homology Homotopy Appl. 5 (2003), no. 1, 407--421. https://projecteuclid.org/euclid.hha/1139839940