Formality conjecture for K3 surfaces

Abstract

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the DG algebra RHom q (F, F ) is formal for any sheaf F polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

Formality conjecture for minimal surfaces of Kodaira dimension 0

“…The main goal of this paper is to provide an elementary proof of the following theorem, which extends an analogous result for K3 surfaces [BZ18].…”

“…The importance of the DG Lie algebra of derived endomorphisms relies on the fact that it controls the deformation theory of in the usual way via Maurer–Cartan equation modulus gauge action; cf. [AS18, BZ18, IM19, Mea18]. Moreover, for every .…”

“…Hence, by showing the formality of the DG Lie algebra one also proves the quadraticity of the Kuranishi map. Notice that since our approach involves techniques of -algebras we investigate derived endomorphisms as a DG Lie algebra, while the papers [KL07, BZ18] consider as an associative DG algebra and also Kaledin's refinement of the Massey products works in the associative setting [Kal07]. It is important to point out that the formality in the associative case is a stronger statement, but on the other hand the DG Lie formality is the one needed for applications to moduli spaces.…”

“…It is worth mentioning that formality is in general much stronger and harder to prove than the quadraticity property; from the point of view of derived algebraic geometry this can be easily understood since formality implies that the derived moduli space is locally quadratic. Nevertheless, formality of has been conjectured for polystable sheaves again by Kaledin and Lehn in [KL07], it has been studied in some cases by Zhang [Zha12], and finally completely solved by Budur and Zhang [BZ18] who proved that the conjecture holds true for any polystable sheaf using results about strong uniqueness of DG enhancements.…”

“…Later, Zhang showed that Kaledin's theorem may be applied to polystable sheaves with some constraints on the ranks of the corresponding stable summands [Zha12, Proposition 1.3], hence enlarging the class of polystable sheaves for which the formality conjecture holds. Eventually in [BZ18] Budur and Zhang established a very interesting result, namely that the formality of derived endomorphisms of any object in is preserved under derived equivalences; hence the formality conjecture follows since by [Yos09] any polystable sheaf can be mapped via a Fourier–Mukai transform to another polystable sheaf satisfying the hypothesis of [Zha12, Proposition 3.1].…”