Weight structures and formality

TL;DR

This survey explores how weight structures in algebraic geometry, especially those with purity properties, can be used to establish formality results in differential graded algebras, linking geometric structures to algebraic formality.

Contribution

It provides a comprehensive overview of formality results derived from weight structures, emphasizing the role of purity properties in algebraic geometry.

Findings

Weight structures can induce formality in differential graded algebras.

Purity properties in weight structures are key to deducing formality.

Cohomology of algebraic varieties often exhibits structures like Hodge or Galois actions.

Abstract

This is a survey on formality results relying on weight structures. A weight structure is a naturally occurring grading on certain differential graded algebras. If this weight satisfies a purity property, one can deduce formality. Algebraic geometry provides us with such weight structures as the cohomology of algebraic varieties tends to present additional structures including a Hodge structure or a Galois action.