HK manifolds of Type $K3^{[a^2+1]}$ as moduli spaces of projective bundles on HK manifolds of Type $K3^{[2]}$

Technical Summary: HK Manifolds of Type K3$[a^2+1]$ as Moduli Spaces of Projective Bundles on HK Manifolds of Type K3$[2]$

Problem Statement
The paper addresses the realization of hyperkähler (HK) manifolds as moduli spaces. While it is well-established that every known HK manifold is deformation-equivalent to a moduli space of semistable sheaves on a K3 or abelian surface, the specific geometric realization of arbitrary HK manifolds of Type K3$[a^2+1]$ (those deformation-equivalent to the Hilbert scheme of $a^2+1$ points on a K3 surface) as moduli spaces of bundles on a fixed type of HK manifold remains a central question. Specifically, the author investigates whether a projective HK manifold $W$ of Type K3$[a^2+1]$ can be realized as a moduli space of $\omega$-slope stable projective bundles (or twisted vector bundles) on a projective HK manifold $X$ of Type K3$[2]$ (deformation-equivalent to the Hilbert scheme of 2 points on a K3 surface).

Methodology
The proof strategy relies on a constructive approach involving the Hilbert scheme $S^{[2]}$ of a projective K3 surface $S$, followed by deformation arguments using Verbitsky's theory.

1. Construction on $S^{[2]}$:

   • The author considers a spherical vector bundle $E_1$ on a K3 surface $S$ with rank $2a$ and a Mukai vector $v_1$.
   • Let $E_2$ be a stable sheaf on $S$ with Mukai vector $v_2 = a v_1 - (0,0,1)$. The moduli space $M_{v_2}(S, h_S)$ is a HK manifold of Type K3$[a^2+1]$.
   • Using the Bridgeland-King-Reid (BKR) equivalence, the author constructs a vector bundle $G(E_1, E_2)$ on $S^{[2]}$ from the tensor product $E_1 \otimes E_2 \oplus E_2 \otimes E_1$.
   • The paper analyzes the stability of $G(E_1, E_2)$ with respect to polarizations on $S^{[2]}$. It is shown that for $E_2$ outside a specific determinantal divisor $D_{v_2} \subset M_{v_2}$, the bundle $G(E_1, E_2)$ is slope stable.
   • For $E_2 \in D_{v_2}$, $G(E_1, E_2)$ is unstable. The author performs a "semistable reduction" (via elementary modifications) to construct stable replacements. This involves analyzing the Brill-Noether stratification of $D_{v_2}$ and constructing specific extensions involving rigid bundles $E_1^{[2]\pm}$ and sheaves $B(H, U_k)$ or $B(p, W_a)$.
   • A key technical result is proving that these modified sheaves are locally free and slope stable for suitable polarizations, thereby defining a regular map $\psi: M_{v_2} \to M_w(S^{[2]})$, where $M_w$ is the moduli space of projective bundles on $S^{[2]}$ with a specific mock Mukai vector $w$.

2. Extension to General HK Manifolds:

   • The author utilizes Verbitsky's theory of projectively hyperholomorphic vector bundles. This theory allows the extension of stable bundles from a specific fiber (like $S^{[2]}$) to the entire twistor family of deformations.
   • By showing that the moduli space of these bundles deforms correctly along twistor paths, the results established for $S^{[2]}$ are extended to arbitrary HK manifolds $X$ of Type K3$[2]$.

3. Hodge Theory and Isometries:

   • The paper analyzes the second Chern class of the universal bundle on the product $X \times \widetilde{M}_w(X, \omega)$ to define a rational Hodge isometry between $H^2(X)$ and $H^2(\widetilde{M}_w(X, \omega))$.
   • This isometry is combined with Markman's results on the Shafarevich conjecture for HK manifolds.

Key Contributions and Results

• Theorem 1.1: For a HK manifold $X$ of Type K3$[2]$ and a suitable Kähler class $\omega$, the moduli space $M_w(X, \omega)$ of $\omega$-slope stable projective bundles with a specific mock Mukai vector $w$ contains an irreducible component $\widetilde{M}_w(X, \omega)^\bullet$. The normalization of this component is a HK manifold of Type K3$[a^2+1]$. If $X$ is projective and $\omega$ is ample, this normalization is isomorphic to a HK manifold of Type K3$[a^2+1]$.
• Theorem 1.3 (Surjectivity/Realization): Conversely, every projective HK manifold $W$ of Type K3$[a^2+1]$ is isomorphic to $\widetilde{M}_w(X, h)^\bullet$ for some polarized HK manifold $(X, h)$ of Type K3$[2]$. This establishes that Type K3$[a^2+1]$ manifolds are moduli spaces of projective bundles on Type K3$[2]$ manifolds.
• Theorem 1.7 (Shafarevich Conjecture Analogue): The paper proves an analogue of the Shafarevich conjecture for rational Hodge isometries between projective HK manifolds of Types K3$[a_1^2+1]$ and K3$[a_2^2+1]$. Specifically, any rational Hodge isometry $f: H^2(W_1) \to H^2(W_2)$ is induced by an algebraic cycle class in $CH^2(W_1 \times W_2)$.
• Modular Sheaves: The work demonstrates that the sheaves constructed (specifically $G(E_1, E_2)$ and their stable replacements) are modular but not atomic, distinguishing them from sheaves studied in recent works by Markman and Bottini.

Significance and Claims
The author claims that these results provide a concrete geometric realization of a broad class of HK manifolds (Type K3$[a^2+1]$) as moduli spaces of bundles on a lower-dimensional HK manifold (Type K3$[2]$). This parallels the classical result where HK manifolds are moduli spaces of sheaves on K3 surfaces, suggesting a structural resemblance between the theory of sheaves on Type K3$[2]$ manifolds and sheaves on K3 surfaces.

The paper explicitly notes that it does not resolve whether the moduli space $M_w(X, \omega)$ is irreducible, but it successfully identifies a specific irreducible component with the desired properties. The results rely heavily on the specific geometry of Type K3$[2]$ and the properties of modular sheaves, extending the known landscape of HK manifolds constructed via moduli problems. The proof of the Shafarevich conjecture analogue for these specific types relies on the existence of the moduli space realization and Markman's prior work on atomic sheaves.