# Formality conjecture for minimal surfaces of Kodaira dimension 0

## TL;DR

This paper proves a formality conjecture for derived endomorphism DG-Lie algebras of polystable sheaves on minimal surfaces with Kodaira dimension zero, using equivariant L-infinity models without prior results.

## Contribution

It establishes the formality of the DG-Lie algebra of derived endomorphisms for polystable sheaves on certain minimal surfaces, introducing a new approach via equivariant L-infinity models.

## Key findings

- Proves formality for sheaves on Kodaira dimension zero surfaces.
- Develops a new method using equivariant L-infinity models.
- Does not rely on previous results, even for K3 surfaces.

## Abstract

Let F be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the DG-Lie algebra RHom(F,F) of derived endomorphisms of F is formal. The proof is based on the study of equivariant $L_{\infty}$ minimal models of DG-Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.