O'Grady tenfolds as moduli spaces of sheaves
type: Item
instance of: scholarly article
title: O'Grady tenfolds as moduli spaces of sheaves
zbMATH Open document ID: 1541.14057
author: Camilla Felisetti
author: Franco Giovenzana
author: Annalisa Grossi
published in: Forum of Mathematics, Sigma
publication date: 2024-05-13T00:00:00Z
review text: 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…
reviewed by: Simone Billi
Mathematics Subject Classification ID: 14J42
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 14J50
zbMATH DE Number: 7846248
zbMATH Keywords: irreducible holomorphic symplectic
zbMATH Keywords: cubic fourfolds
zbMATH Keywords: moduli spaces
zbMATH Keywords: birational tranformations
MaRDI profile type: Publication
cites work: On two rationality conjectures for cubic fourfolds
cites work: On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold
cites work: Hodge theory and derived categories of cubic fourfolds
cites work: Variétés kähleriennes dont la première classe de Chern est nulle
cites work: Stability conditions on \(K3\) surfaces
cites work: Hitchin fibrations, abelian surfaces, and the P=W conjecture
cites work: P=W conjectures for character varieties with symplectic resolution
cites work: On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties
cites work: Q4289665
cites work: Q3342879
cites work: Induced birational transformations on O'Grady's sixfolds
cites work: Nonsymplectic automorphisms of prime order on O'Grady's sixfolds
cites work: Symplectic birational transformations of finite order on O'Grady's sixfolds
cites work: Q4946819
cites work: The Kähler cone of a compact hyperkähler manifold
cites work: Q3622192
cites work: Lectures on K3 Surfaces
cites work: The K3 category of a cubic fourfold
cites work: Hodge Theory of Cubic Fourfolds, Their Fano Varieties, and Associated K3 Categories
cites work: Equivalences of twisted K3 surfaces
cites work: A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold
cites work: La singularité de O’Grady
cites work: Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations
cites work: Twisted cubics on cubic fourfolds and stability conditions
cites work: Algebraic geometric interpretation of Donaldson's polynomial invariants
cites work: A survey of Torelli and monodromy results for holomorphic-symplectic varieties
cites work: Birational geometry of singular moduli spaces of O'Grady type
cites work: Birational geometry of irreducible holomorphic symplectic tenfolds of O'Grady type
cites work: Monodromy and birational geometry of O'Grady's sixfolds
cites work: Induced automorphisms on irreducible symplectic manifolds:
cites work: Comparison of the Donaldson polynomial invariants with their algebro- geometric analogues
cites work: Symplectic structure of the moduli space of sheaves on an abelian or K 3 surface
cites work: Q4195021
cites work: Desingularized moduli spaces of sheaves on a K3
cites work: On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces
cites work: Factoriality Properties of Moduli Spaces of Sheaves on Abelian and K3 Surfaces
cites work: Deformation of the O'Grady moduli spaces
cites work: On the Beauville form of the known irreducible symplectic varieties
cites work: Autoequivalences of twisted K3 surfaces
cites work: Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds
cites work: Q4443551
DOI: 10.1017/FMS.2024.46
description: scientific article; zbMATH DE number 7846248