# Moscow Mathematical Journal

Volume 22, Issue 2, April–June 2022  pp. 239–263.

Deformations of Polystable Sheaves on Surfaces: Quadraticity Implies Formality

Authors:  Ruggero Bandiera (1), Marco Manetti (1), and Francesco Meazzini (1)
Author institution:(1) Università degli studi di Roma La Sapienza, Dipartimento di Matematica “Guido Castelnuovo”, P.le Aldo Moro 5, I-00185 Roma, Italy

Summary:

Abstract. We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf of a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.

2020 Math. Subj. Class. 14F08, 14D15, 16W50, 18N40.

Keywords: Deformation theory, polystable sheaves, formality, differential graded Lie algebras, $L_{\infty}$-algebras