Derived automorphism groups of \(K3\) surfaces of Picard rank 1. (English) Zbl 1358.14019

Bayer, Arend; Bridgeland, Tom

Duke Math. J. 166, No. 1, 75-124 (2017).

Given a smooth projective variety \(X\) over \(\mathbb{C}\), we denote by \(D(X)\) the derived category of bounded complexes of coherent sheaves on \(X\) and by \(\text{Stab}(X)\) the set of full, locally finite, numerical stability conditions on \(D(X)\). The aim of this paper is to describe the group \(\text{Aut} D(X)\) of \(\mathbb{C}\)-linear, exact autoequivalences of \(D(X)\), where \(X\) is a \(K3\) surface of Picard rank one. To this end, it is enough to study the kernel \(\text{Aut}^0D(X)\) of the surjective map \(\text{Aut} D(X) \rightarrow \text{Aut}^+H^*(X)\), where \(\text{Aut}^+H^*(X)\) is the group of orientation preserving Hodge isometries of the Mukai lattice \(H^*(X)\). Conjecturally, \(\text{Aut}^0D(X)\) coincides with the foundamental group of \(\mathcal{P}_0^+(X):=\mathcal{P}^+(X) \setminus \bigcup_{\delta} \delta^{\perp}\), where \(\mathcal{P}^+(X)\) is a connected component of the open subset of elements \(\Omega \in N(X)\otimes \mathbb{C}\) whose real and imaginary parts span a positive definite two-plane, and \(\delta\) varies in the subset of \(N(X)\) of classes with negative self-intersection. Let \(\text{Stab}^{†}(X)\) be the connected component of \(\text{Stab}(X)\) containing the set of geometric stability conditions for which all skyscraper sheaves are stable of the same phase. We denote by \(\text{Stab}^*(X)\) the union of the connected components which are images of \(\text{Stab}^{†}(X)\) under an autoequivalence of \(D(X)\). The main result of this paper is Theorem 1.3, where they prove that \(\text{Stab}^*(X)\) is contractible for a \(K3\) surface \(X\) of Picard rank one; in particular, this implies that the above conjecture holds for \(X\). The proof is explained in Section 6; it requires the construction of a flow on \(\text{Stab}^*(X)\), whose properties are described in Section 4 and Section 5. Then, they deduce that \(\text{Aut}^0D(X)\) is the product of \(\mathbb{Z}\) acting by even shifts with the free group generated by the square of twist functors with respect to spherical objects (Theorem 1.4). Finally, in Section 7 they explain the relation of the above result with mirror symmetry and the Appendix is devoted to the study of Calabi-Yau autoequivalences.

Reviewer: Laura Pertusi (Milano)

MSC:

14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14J28	\(K3\) surfaces and Enriques surfaces
14J33	Mirror symmetry (algebro-geometric aspects)
18E30	Derived categories, triangulated categories (MSC2010)

Keywords:

derived category; autoequivalences; stability conditions; \(K3\) surfaces; mirror symmetrY