Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alessandro Pilastro

TL;DR

The paper advances Fujita-type questions for polarized hyperkähler manifolds of $K3^{[n]}$-type and $\mathrm{Kum}^n$-type by leveraging tautological bundles on Hilbert schemes, Grassmannian embeddings, and lattice-polarization theory. It develops explicit criteria for base point freeness and very ampleness of polarizations on general elements of the moduli spaces $\Sigma_{d,t}^n$ and $\Upsilon_{d,t}^n$, including sharp numerical bounds via the parameter $\tau=\frac{t^2}{2(t-1)}$. The work also establishes openness of these properties in families, enabling transfer of local results to generic elements within connected components, and provides necessary and sufficient non-emptiness conditions for the moduli spaces. Overall, it extends Fujita-type results to higher-dimensional hyperkähler manifolds, delivering concrete, computable thresholds and clarifying the polarization geometry in these moduli spaces.

Abstract

For the moduli spaces $Σ_{d,t}^n$ and $Υ_{d,t}^n$ of polarized hyperkähler manifolds of Hilb$^n$(K3)-type and Kum$^n$-type respectively, with polarization with square $2d$ and divisibility $t$, we study general base point freeness and very ampleness of the polarization. We provide cases where these moduli space are connected and a formula characterizing when these spaces are non-empty.

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

TL;DR

The paper advances Fujita-type questions for polarized hyperkähler manifolds of -type and -type by leveraging tautological bundles on Hilbert schemes, Grassmannian embeddings, and lattice-polarization theory. It develops explicit criteria for base point freeness and very ampleness of polarizations on general elements of the moduli spaces and , including sharp numerical bounds via the parameter . The work also establishes openness of these properties in families, enabling transfer of local results to generic elements within connected components, and provides necessary and sufficient non-emptiness conditions for the moduli spaces. Overall, it extends Fujita-type results to higher-dimensional hyperkähler manifolds, delivering concrete, computable thresholds and clarifying the polarization geometry in these moduli spaces.

Abstract

For the moduli spaces and of polarized hyperkähler manifolds of Hilb(K3)-type and Kum-type respectively, with polarization with square and divisibility , we study general base point freeness and very ampleness of the polarization. We provide cases where these moduli space are connected and a formula characterizing when these spaces are non-empty.
Paper Structure (5 sections, 28 theorems, 32 equations)

This paper contains 5 sections, 28 theorems, 32 equations.

Table of Contents

  1. Tautological vector bundles on $\mathop{\mathrm{Hilb}}\nolimits^n$ of a surface
  2. Embeddings of Hilbert schemes into Grassmannians
  3. Hyperkähler manifolds
  4. Moduli space of polarized hyperkähler manifolds
  5. Base point freeness and very ampleness of a general element

Key Result

Theorem 1

Let $n,t,d$ be positive integers with $n\geq 2$ and let $\tau:=\frac{t^2}{2(t-1)}$. Then the following holds:

Theorems & Definitions (40)

  • Theorem 1
  • Proposition 1.1: bini2022familiesbigstablebundles, (4.5)
  • Proposition 1.2: bini2022familiesbigstablebundles, Prop 4.5
  • Proposition 1.3: Lehn_1999, Lemma 3.7 - Thm 4.6
  • Lemma 1.4
  • proof
  • Lemma 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • ...and 30 more