On Fujita's conjecture for a general hyperkähler manifold in the standard series of examples

1. Problem Statement

The paper addresses Fujita's Conjecture, a central problem in algebraic geometry. The conjecture posits that for a compact complex manifold $X$ of dimension $n$ and an ample line bundle $L$, the bundle $K_X \otimes L^{\otimes m}$ is:

• Base point free for $m \ge n+1$.
• Very ample for $m \ge n+2$.

While the conjecture is known to hold for K3 surfaces and Abelian varieties (with tighter bounds), it remains widely open for higher-dimensional manifolds. The specific focus of this paper is to investigate these properties for general hyperkähler manifolds of two standard deformation types:

1. $K3^{[n]}$-type: Deformations of the Hilbert scheme of $n$ points on a K3 surface ($S^{[n]}$).
2. $Kum_n$-type: Deformations of the generalized Kummer variety associated with an Abelian surface ($Kum_n(T)$).

The goal is to determine explicit bounds on the square of the polarization ($2d$) and its divisibility ($t$) that guarantee the polarization (or its powers) is base point free or very ample for a general element in the corresponding moduli spaces $\Sigma^n_{d,t}$ and $\Upsilon^n_{d,t}$.

2. Methodology

The author employs a combination of lattice theory, deformation theory, and the geometry of Hilbert schemes to generalize previous results by Rieß, Varesco, and Debarre.

• Tautological Bundles on Hilbert Schemes:
  The paper utilizes the construction of tautological vector bundles $F^{[n]}$ on the Hilbert scheme $S^{[n]}$ associated with a line bundle $L$ on a surface $S$.

  • It establishes that if $L$ is $(n-1)$-very ample on $S$, then the tautological bundle $L^{[n]}$ is globally generated.
  • If $L$ is $n$-very ample, the map to the Grassmannian defined by $L^{[n]}$ is an embedding.
  • Crucially, the determinant line bundle $\det(L^{[n]})$ is shown to be base point free (if $L$ is $(n-1)$-very ample) or very ample (if $L$ is $n$-very ample).

• Lattice Structure and Divisibility:
  The author analyzes the Picard group structure of hyperkähler manifolds.

  • For $K3^{[n]}$-type: $Pic(S^{[n]}) \cong Pic(S) \oplus \mathbb{Z}\delta$, where $\delta$ is the exceptional divisor with $q(\delta) = -2(n-1)$.
  • For $Kum_n$-type: $Pic(Kum_n(T)) \cong Pic(T) \oplus \mathbb{Z}\delta$, with $q(\delta) = -2(n+1)$.
  • The divisibility $t$ of a polarization $H = a\lambda + b\delta$ is calculated using the Beauville-Bogomolov-Fujiki form, leading to the formula $div(H) = \gcd(a, 2b(n-1))$ (or $2b(n+1)$ for Kummer type).

• Openness and Connectedness:

  • Openness: The paper proves that base point freeness and very ampleness are open properties in families of hyperkähler manifolds (Corollary 4.5). This allows the author to construct a single "special" example with the desired properties and conclude that the general element in the same connected component shares these properties.
  • Connectedness: The author utilizes formulas from Debarre and Voisin ([2], [12]) to determine when the moduli spaces $\Sigma^n_{d,t}$ and $\Upsilon^n_{d,t}$ are connected (e.g., when $t=1, 2$ or $t=p^a$ with specific conditions). This ensures the results apply to the "general element" rather than just a specific component.

• Construction of Examples:
  The core proof strategy involves constructing a specific polarized manifold $(X, H)$ within the moduli space where $H$ satisfies the required ampleness conditions (by reducing the problem to the surface case via the tautological bundle logic) and then invoking the openness theorem.

3. Key Contributions and Results

The paper provides explicit inequalities involving $n$ (dimension), $d$ (half the square of the polarization), and $t$ (divisibility) that guarantee the desired properties. Let $\tau := \frac{t^2}{2(t-1)}$.

A. General Bounds for Arbitrary Divisibility ($t \ge 2$)

Theorem 5.8 establishes the existence of a connected component in the moduli space where the general element satisfies:

• Base Point Freeness:
  • $K3^{[n]}$-type: If $d \ge (\tau - 1)n + \tau + 1$.
  • $Kum_n$-type: If $d \ge (\tau - 1)n + 2\tau - 1$.
• Very Ampleness:
  • $K3^{[n]}$-type: If $d \ge (\tau - 1)n + 2\tau + 1$.
  • $Kum_n$-type: If $d \ge (\tau - 1)n + 3\tau - 1$.

B. Specific Case: Divisibility $t=1$

For the case where the polarization is primitive ($t=1$), the bounds are sharper and apply to the entire moduli space (which is connected):

• Base Point Freeness: Always holds for general elements.
• Very Ampleness:
  • $K3^{[n]}$-type: Holds if $d \ge n+1$.
  • $Kum_n$-type: Holds if $d \ge n+4$.

C. Characterization of Non-Empty Moduli Spaces

Proposition 5.10 provides a necessary and sufficient condition for the moduli spaces $\Sigma^n_{d,t}$ and $\Upsilon^n_{d,t}$ to be non-empty:

• $\Sigma^n_{d,t} \neq \emptyset \iff t \mid (2n-2)$ and $d \equiv t^2 - b^2(n-1) \pmod{\dots}$ for some $b$ coprime to $t$.
• $\Upsilon^n_{d,t} \neq \emptyset \iff t \mid (2n+2)$ and $d \equiv t^2 - b^2(n+1) \pmod{\dots}$ for some $b$ coprime to $t$.
  This allows the author to refine the inequalities in specific cases by ensuring the moduli space actually exists.

D. Generalization of Fujita's Conjecture

Corollary 5.13 confirms a version of Fujita's conjecture for these manifolds:

• For a general element $(X, H)$ in $\Sigma^1_{d,n}$ or $\Upsilon^1_{d,n}$, $H$ is base point free and $H^{n+2}$ is very ample.
• For $t \ge 2$, similar bounds involving $\tau$ ensure $H$ is base point free and $H^{n+2}$ is very ample within specific connected components.

4. Significance

• Extension of Known Results: The paper significantly generalizes previous work limited to low dimensions ($n=2,3,4$) or specific divisibilities ($t=1,2$) to arbitrary dimension $n$ and arbitrary divisibility $t$.
• Unified Framework: By formalizing the techniques of tautological bundles and lattice theory, the author provides a unified method to treat both $K3^{[n]}$ and $Kum_n$ types simultaneously.
• Explicit Bounds: The paper moves beyond qualitative existence to provide quantitative bounds on the polarization square ($2d$). This is crucial for concrete applications in the geometry of these manifolds, such as determining when they can be embedded into projective space.
• Refinement of Fujita's Conjecture: The results suggest that for hyperkähler manifolds, the bounds for very ampleness are comparable to, or slightly stronger than, the general conjecture ($m \ge n+2$), specifically showing that $H^{n+2}$ is very ample under the derived conditions.

In summary, Pilastro's work bridges the gap between the well-understood geometry of K3 surfaces and the complex geometry of higher-dimensional hyperkähler manifolds, providing a robust toolkit for analyzing the positivity of line bundles in these moduli spaces.