An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalent (Q2223516)

There are many different ways to try to classify algebraic varieties. Among those are \begin{itemize} \item birational equivalence: whether a birational map \(X \dashrightarrow Y\) exists; \item D-equivalence: whether there is an equivalence \(\mathcal D^b(\mathrm{coh}(X)) \cong \mathcal D^b(\mathrm{coh}(Y))\); \item L-equivalence: whether \(X\) and \(Y\) have the same class in \(K_0(\mathcal{V}ar)[\mathbb L^{-1}]\), where \(\mathbb L = [\mathbb A_1]\). \end{itemize} These three attempts are related, but in a mysterious way. The main result of this article is the following. Given a pair \(X,Y\) of two K3 surfaces of Picard number 1 and degree 2d, which are D- and L-equivalent. Then the Hilbert schemes of points \(X^{[n]}\) and \(Y^{[n]}\) are D- and L-equivalent. If \(n>2\) and if there are integer solutions to the equation \[ (n-1)x^2 - dy^2 = 1, \] then \(X\) and \(Y\) are birationally \emph{in}equivalent. For the proof, the author uses that a birational morphism \(\phi \colon X^{[n]} \dashrightarrow Y^{[n]}\) induces an isometry of Picard groups, hence preserves the movable cone. The equation of the main result comes now from the description of the movable cone, as obtained in [\textit{A. Bayer} and \textit{E. Macrì}, Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)]. To get examples, one can consider a very general K3 surface \(X\) of degree 12 and its Fourier-Mukai partner \(Y\), see [\textit{B. Hassett} and \textit{K.-W. Lai}, Compos. Math. 154, No. 7, 1508--1533 (2018; Zbl 1407.14010)] and [\textit{A. Ito} et al., Sel. Math., New Ser. 26, No. 3, Paper No. 38, 27 p. (2020; Zbl 1467.14051)]. If one chooses \(n=6y^2+2\) for an integer \(y\), then \((1,y)\) is an integer solution of the equation above. Hence, the corresponding hyperkähler varieties \(X^{[n]}\) and \(Y^{[n]}\) give examples of varieties, which are D-equivalent (already known by [\textit{D. Ploog}, Adv. Math. 216, No. 1, 62--74 (2007; Zbl 1167.14031)]) and L-equivalent, but not birationally equivalent. More results about birational (in)equivalence of such hyperkähler varieties are obtained in [\textit{C. Meachan} et al., Math. Z. 294, No. 3--4, 871--880 (2020; Zbl 1469.14011)] and [\textit{K. Yoshioka}, Math. Ann. 321, No. 4, 817--884 (2001; Zbl 1066.14013)].