O'Grady tenfolds as moduli spaces of sheaves (Q6536616)

This paper is concerned with determining when a manifold of OG10 type is birational to a moduli space of (twisted) sheaves on a \(K3\) surface. The authors obtain a numerical criterion for a manifold of OG10 type to be birational to a moduli space, and determine the Hassett divisors for which the associated Li-Pertusi-Zhao (LPZ) manifold is birational to a moduli space. As an application, the authors determine a criterion for a group of birational transformations to be induced on a manifold of OG10 type by a group of automorphisms of a \(K3\) surface. \N\NO'Grady constructed a resolution of the moduli space \(M_v(S,\theta)\) of \(\theta\)-semistable on a \(K3\) surface \(S\) with Mukai vector \(v=(2,0,-2)\) and \(\theta\) a primitive \(v\)-generic polarization, which is \textit{irreducible holomorphic symplectic} (ihs) of dimension \(10\). A manifold of \textit{OG10 type} is an ihs manifold which is deformation equivalent to the resolution of \(M_v(S,\theta)\). Perego and Rapagnetta obtained a resolution resolution \(\widetilde{M}_v(S,\theta)\), which is a manifold of OG10 type, of the moduli space \(M_v(S,\theta)\) where \(v=2w\) and \(w=(r,l,s)\) is a primitive Mukai vector such that \(w^2=2\), \(r\geq 0,l\in NS(S)\) and \(l\) is the first Chern class of a line bundle in case \(r=0.\)\N\NKuznetsov proved that the bounded derived category of a smooth cubic fourfold \(Y\) admits a semiorthogonal decomposition of the form \(D^b(Y)=\langle \mathcal{A}(Y),\mathcal{O}_Y,\mathcal{O}_Y(1),\mathcal{O}_Y(2)\rangle.\) The algebraic Mukai lattice of \(\mathcal{A}(Y)\) always contains an \(A_2\) lattice spanned by the classes \(\lambda_i:=\Pr[\mathcal{O}_L(i)]\) where \(L\subset Y\) is a line and \(\Pr:D^b(Y)\rightarrow \mathcal{A}(Y)\) is the natural projection functor. Li, Pertusi and Zhao provided a Bridgeland stability condition \(\sigma\) for which the moduli space \(X_Y:=M_\sigma(2(\lambda_1+\lambda_2),\mathcal{A}(Y))\) admits a resoluton \(\widetilde{X}_Y\) of OG10 type. The manifold \(\widetilde{X}_Y\) is called the \textit{Li-Pertusi-Zhao} (LPZ) manifold associated to \(Y\).\N\NIn Section 3, the authors define a \textit{numerical moduli space} as an ihs manifold \(X\) such that there is a primitive class \(\sigma\in H^{1,1}(X,\mathbb{Z})\) of divisibility \(3\) and \(\sigma^2=-6\) such that \(\sigma^{\perp}\) embeds a copy of the hyperbolic plane \(U\) in \(\Lambda^{1,1}_{24}\) via the unique embedding in the even unimodular lattice \(\Lambda_{24}\) of rank \(24\) with orthogonal complement of type \((1,1)\). The main result of the section is that an ihs manifold \(X\) of OG10 type is birational to \(\widetilde{M}_v(S,\theta)\) if and only if \(X\) is a numerical moduli space.\N\NIn Section 4, the authors generalize the result of Section 3 to the twisted case. The authors define a \textit{twisted numerical moduli space} in the same way as a numerical moduli space, where the hyperbolic plane can be replaced by the twisted one \(U(n)\) for \(n\in\mathbb{N}_{>0}\). They prove that an ihs manifold \(X\) of OG10 type is birational to the desingularization of the moduli space of twisted sheaves \(\widetilde{M}_v(S,\alpha,\theta)\), where \(S\) is a \(K3\) surface and \(\alpha\in Br(S)=H^2(S,\mathcal{O}_S^*)_{tor}\), if and only if \(X\) is a twisted numerical moduli space.\N\NIn Section 5, the authors give a criterion for determining when the LPZ manifold \(\widetilde{X}_Y\) associated to a cubic fourfold \(Y\) is birational to a (twisted) moduli space of sheaves on a \(K3\) surface. The irreducible Hassett divisor \(\mathcal{C}_d\) consists of cubic fourfolds \(Y\) having a primitive rank \(2\) lattice \(K\subset H^4(Y,\mathbb{Z})\) containing the square of a hyperplane class, such that \(K\) has discriminant \(d\). The authors prove that \(\widetilde{X}_Y\) is brational to a desingularized moduli space \(\widetilde{M}_v(S,\theta)\) if and only if \(Y\in\mathcal{C}_d\) such that \(d\) divides \(2n^2+2n+2\) for some \(n\in\mathbb{Z}\). They also prove that \(\widetilde{X}_Y\) is brational to a desingualrized twisted moduli space \(\widetilde{M}_v(S,\alpha,\theta)\) if and only if \(Y\in\mathcal{C}_d\) such that in the prime factorization \(\frac{d}{2}\), primes \(p\equiv 2(3)\) appear with even exponents.\N\NIn section 6, the authors prove that a group \(G\subset Bir(X)\) of birational transformations of a manifold \(X\) of OG10 type is induced by a group of automorphisms of the \(K3\) surface such that \(X\) is birational to \(\widetilde{M}_v(S,\theta)\) if and only if \(G\) is \textit{numerically induced}, i.e. \(X\) is a numerical moduli space and the class \(\sigma\) and the hyperbolic plane \(U\) are pointwise fixed by the action of \(G\). As a corollary, the authors determine that a birational symplectic involution \(\phi\in Bir(X)\) is induced if and only if the associated coinvarian lattice \(H^2(X,\mathbb{Z})_\phi\) is isometric to the lattice \(E_8(-2)\).