Contents Online

# Homology, Homotopy and Applications

## Volume 5 (2003)

### Number 1

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

Pages: 407 – 421

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n1.a17

#### Author

#### 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.

#### Keywords

$A_{\infty}$-algebra, $A_{\infty}$-isomorphism, homogeneous coordinate ring

#### 2010 Mathematics Subject Classification

18E30, 55P43