Formality conjecture for minimal surfaces of Kodaira dimension 0

Abstract

Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ 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.

The geometry of antisymplectic involutions, I

“…Now we can argue exactly as in [BMM21] for the morphism f 1 . Indeed, this is étale and we can consider the algebra C := f 1, * O S of rank 2.…”

“…Lemma 2.4 is proved in [CPZ21, Section 2] in general, for all moduli spaces of Bridgeland semistable objects on the derived category of a K3 surface, by using the approach in [BMM21] (the case of a generic stability condition, which is not sufficient for our purposes, was proven earlier in [BZ19]). We give here a quick sketch of the proof, by following [CPZ21].…”

“…We claim this is formal. This can be checked either directly, or by reducing to the case of K3 surfaces; we sketch the latter argument, by using the ideas in [BZ19] and [BMM21].…”

Some remarks about deformation theory and formality conjecture

“…The conjecture has been proved in some cases by [ 17 ] and [ 30 ]. Then it has been completely solved by Budur and Zhang in [ 10 ], who showed this result more generally for every choice of the polarization and for polystable complexes in the bounded derived category with respect to a generic Bridgeland stability condition, and by Bandiera, Manetti and Meazzini in [ 7 ], who showed the conjecture for polystable sheaves on a smooth minimal projective surface of Kodaira dimension 0. A further generalization of this conjecture has been recently proved by Arbarello and Saccà in [ 5 ], for polystable objects in the bounded derived category of a K3 surface and in the Kuznetsov component of a cubic fourfold (without the genericity assumption on the stability condition).…”

“…Strategy of the proofs. In [ 7 ] Bandiera, Manetti and Meazzini prove an algebraic criterion to ensure the formality of a differential graded Lie algebra, which involves the notion of quasi-cyclic DG-Lie algebra. Then they apply this criterion to the DG-Lie algebra of derived endomorphisms of a coherent sheaf with linearly reductive automorphisms group on a smooth minimal surface of Kodaira dimension 0.…”