On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold
Given a smooth cubic fourfold \(Y\) not containing a plane, it is well known that there is a moduli space \(M\) of generalized twisted cubics in \(Y\) and a morphism \(u\colon M\rightarrow Z\). The latter is the LLSvS symplectic eightfold. The first result of the paper is: The universal curve \(C\subset Y\times M\) induces a map \[ [C]_*\colon H^4(Y,\mathbb{Z})\rightarrow H^2(M,\mathbb{Z}) \] that restricts to a Hodge isometry \([C]_*\colon H^4_{\mathrm{prim}}(Y,\mathbb{Z})\rightarrow u^*(H^2_{\mathrm{prim}}(Z,\mathbb{Z}))\), with the intersection pairing on the left-hand side and the opposite of the Beauville-Bogomolov-Fujiki pairing on the right. Moreover, the authors prove that an analogous result holds in topological \(K\)-theory. That is, the Fourier-Mukai kernel \(I^\vee_C(-3h)\) induces a map \(K_{\mathrm{top}}(\mathcal{A})\rightarrow K_{\mathrm{top}}(M)\) that composed with the first Chern class restricts to a Hodge isometry. As usual, \(\mathcal{A}\) is Kuznetsov's \(K3\) category. In order to prove these two results, it is enough to prove them for one specific cubic fourfold. The first result is proved by taking \(Y\) not in any Hassett's Noether-Lefschetz divisor \(\mathcal{C}_d\). The result in \(K\)-theory is proved taking a Pfaffian cubic \(Y\) and a \(K3\) surface \(X\) parameterizing a complete family of quartic scrolls on \(Y\). The authors prove that if \(X\) has Picard rank \(1\), then there is a morphism \(j\colon Y\rightarrow X^{[4]}\). The last theorem of the article is a series of explicit conditions on \(d\) such that \(Y\in\mathcal{C}_d\) if and only if \(Z\) is birational to some moduli space of (twisted) sheaves on a \(K3\) surface, or to \(\mathrm{K3}^{[4]}\).
scientific article; zbMATH DE number 7655224
scientific article