Derived categories of coherent sheaves and motives of \(K3\) surfaces (Q2906508)

Let \(X, Y\) be projective complex \(K3\) surfaces which are derived equivalent. By work of Mukai, Orlov and Huybrechts, there are several equivalent conditions for derived equivalence:NEWLINENEWLINE1. the transcendental lattices \(T(X), T(Y)\) are Hodge isometric;NEWLINENEWLINE2. the Mukai lattices \(\tilde H(X,\mathbb{Z}), \tilde H(Y,\mathbb{Z})\) are Hodge isometric;NEWLINENEWLINE3. either \(X\) and \(Y\) are isomorphic or either is a fine moduli space of \(\mu\)-stable vector bundles on the other.NEWLINENEWLINEThere is also a conjectural implication for motives (not an equivalence as the authors show with some specific \(K3\) surfaces). Namely Orlov conjectured (more generally for any pair of smooth projective varieties which are derived equivalent) that the motives \(M(X), M(Y)\) are isomorphic in Voevodsky's triangulated category of motives with \(\mathbb Q\)-coefficients. Here the authors prove Orlov's conjecture if \(X\) and \(Y\) have Picard number at least 5 using work of Nikulin and Kimura.NEWLINENEWLINEErroneously they state (Remark 4.4) that the condition \(\rho\geq 5\) is equivalent to the existence of an elliptic fibration. It is in fact true by classical theory of indefinite quadratic forms that the Picard lattice contains a non-zero divisor class \(D\) with \(D^2=-2\); this induces an elliptic fibration by an argument due to Piatetski-Shapiro and Shafarevich. However, the converse does certainly not hold, since any very general elliptic \(K3\) surface has Picard number 2.NEWLINENEWLINEThe paper concludes with considerations on \(K3\) surfaces with a symplectic involution, or Nikulin involution.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].