[Paper Review] Non-natural non-symplectic involutions on symplectic manifolds of K3^{[2]}-type

Hisanori Ohashi, Malte Wandel|arXiv (Cornell University)|May 28, 2013

Algebraic Geometry and Number Theory17 references16 citations

TL;DR

This paper classifies non-symplectic involutions on irreducible holomorphic symplectic manifolds of $K3^{[2]}$-type with 19 deformation parameters, identifying four conjugacy classes of cohomological actions. It constructs a new geometric example via moduli spaces of sheaves on degree-two $K3$ surfaces, realizing one class with a fixed locus of two smooth surfaces (branched covers of $\mathbb{P}^2$ of degrees six and ten), and proves deformation equivalence within three classes, while class 3 admits two distinguished deformation types related by Mukai flop.

ABSTRACT

We study non-symplectic involutions on irreducible symplectic manifolds of K3^{[2]}-type with 19 parameters, which is the second largest possible. We classify the conjugacy classes of cohomological representations into four different types and show that there are at most five deformation types, two of which are given by natural involutions and their flops. Next, we give a geometric realisation of one of the new types using moduli spaces of sheaves on K3 surfaces. The geometry of the manifold and the new involution is described in detail.

Motivation & Objective

• To classify non-symplectic involutions with 19 deformation parameters on $K3^{[2]}$-type manifolds using cohomological invariants.
• To determine deformation equivalence classes of such involutions and identify distinct types beyond the known natural involution and its flop.
• To geometrically realize a new deformation class using moduli spaces of sheaves on $K3$ surfaces of degree two.
• To describe the fixed locus of the new involution in detail, showing it consists of two smooth, connected surfaces with specific branched cover structures.

Proposed method

• Using lattice theory and the Torelli theorem to classify cohomological actions of involutions on $H^2(X,\mathbb{Z})$ via invariant sublattices.
• Applying the theory of marked manifolds and period maps to analyze deformation equivalence of involutions.
• Constructing the new involution as an induced automorphism on moduli spaces of sheaves over polarized $K3$ surfaces of degree two.
• Employing fibration structures and Jacobian fibrations over $|\mathcal{O}(H)|$ to analyze fixed points on fibers via hyperelliptic involutions.
• Using the function $r(\mathcal{L}) = \dim H^0(D,\mathcal{L})^{\iota_D}$ to distinguish fixed point components and prove disjointness of the two surfaces in the fixed locus.
• Analyzing degenerations in the compactified Jacobian to show connectedness and non-intersection of the fixed surface components.

Experimental results

Research questions

• RQ1 How many deformation classes exist for non-symplectic involutions with 19 parameters on $K3^{[2]}$-type manifolds, and how are they distinguished cohomologically?
• RQ2 Can a new 19-dimensional family of such involutions be geometrically realized beyond the known natural involution and its flop?
• RQ3 What is the structure of the fixed locus for the new involution, and how does it differ from that of the natural involution?
• RQ4 Are the natural involution and its Mukai flop deformation equivalent as automorphisms, and what does this imply for the classification of automorphisms?

Key findings

• There are exactly four conjugacy classes of cohomological actions for non-symplectic involutions with 19 parameters on $K3^{[2]}$-type manifolds, classified by the structure of the invariant sublattice $H^2(X,\mathbb{Z})^\iota$.
• Involutions in conjugacy classes No. 1, 2, and 4 are all deformation equivalent within their class, while those in class No. 3 are deformation equivalent to either the natural involution or its Mukai flop.
• A new 19-dimensional family of involutions is realized as automorphisms on moduli spaces of sheaves over $K3$ surfaces of degree two, exclusively in conjugacy class No. 1.
• The fixed locus of this new family consists of two smooth, connected surfaces, each a branched cover of $\mathbb{P}^2$ of degree six and ten, respectively.
• The two components of the fixed locus are disjoint, proven via the invariant section dimension function $r(\mathcal{L})$, which takes distinct values on each component.
• The geometric realization via moduli spaces provides the first known example of a non-natural, non-symplectic involution with 19 parameters, extending the classification beyond natural and flopped natural involutions.