Projective Hyper-Kähler Manifolds

Updated 29 January 2026

Projective hyper-Kähler manifolds are simply connected compact Kähler varieties of even complex dimension with a unique, nondegenerate holomorphic 2-form and a Beauville–Bogomolov–Fujiki lattice structure.
They exhibit rigid cohomological invariants, with period maps, Fujiki relations, and explicit wall-crossing in nef and movable cones guiding their birational geometry.
Key examples include the K3^[n], generalized Kummer, and O’Grady types, which serve as models for moduli spaces and illuminate deep interactions between algebraic, differential, and arithmetic geometry.

A projective hyper-Kähler manifold is a simply connected compact Kähler manifold of even complex dimension $2n$ that admits a nowhere-degenerate holomorphic 2-form, has holonomy group $\mathrm{Sp}(n)$ , and possesses an ample line bundle. Such manifolds, also known as projective irreducible holomorphic symplectic (IHS) manifolds, form a distinguished class within the landscape of compact Kähler and Calabi-Yau geometry due to their restrictive cohomological invariants and rigid geometric structures. The theory of projective hyper-Kähler manifolds interlaces lattice-theoretic Hodge theory, sophisticated birational geometry, explicit geometric moduli, and the arithmetic of their period maps.

1. Fundamental Structure and Lattice-Theoretic Invariants

Projective hyper-Kähler manifolds are characterized by the existence of a unique, up to scale, primitive integral quadratic form $q_X$ on $H^2(X,\mathbb{Z})$ called the Beauville–Bogomolov–Fujiki (BBF) form, with signature $(3, b_2(X)-3)$ . The Fujiki relation provides a global constraint on the intersection theory: $\int_X \alpha^{2n} = \lambda_X q_X(\alpha)^n$ for all $\alpha\in H^2(X)$ and for a positive rational constant $\lambda_X$ . The BBF form is positive on Kähler classes, and the existence of a positive-square class in $H^{1,1}(X, \mathbb{Z})$ is both necessary and sufficient for projectivity (Debarre, 2018, Bai, 2024).

A projective hyper-Kähler manifold $X$ possesses Hodge numbers determined by the symplectic form: $H^{2,0}(X) = \mathbb{C}[\sigma_X]$ , $H^{p,0}(X) = 0$ for $0 < p < 2n$ except $p=2$ . The period domain is a quadric in the projectivization $H^2(X, \mathbb{C})$ defined by $q_X$ .

2. Principal Examples and Deformation Types

Currently, up to deformation, only four infinite series of projective IHS manifolds are known (Denisi et al., 22 Jan 2026, Rapagnetta, 2015):

Series	Construction	Dimension $2n$	$b_2$	Fujiki Constant
$K3^{[n]}$	Hilbert schemes $S^{[n]}$ of K3 points	$2n$	23	$(2n)!/2^n n!$
Generalized Kummer $Kum^n$	$A^{[n+1]}$ for abelian surface $A$	$2n$	7	$(2n)! (n+1)/2^n n!$
O’Grady OG6	Desing. moduli sheaves on ab. surface	6	8	10
O’Grady OG10	Desing. moduli sheaves on K3 surface	10	24	60

These families encompass most known explicitly constructed projective hyper-Kähler manifolds. Every projective member in these families remains so under the minimal model program; no novel deformation types arise from the birational models (Rapagnetta, 2015).

Notable explicit examples include Fano varieties of lines on cubic fourfolds (of $K3^{[2]}$ -type), Debarre–Voisin fourfolds, and moduli space compactifications via Lagrangian fibrations on special hypersurfaces (Laza et al., 2016).

3. Ample, Nef, and Movable Cones; Wall Structure

In Néron–Severi space $\mathrm{NS}(X)_\mathbb{R}$ , the positive cone $\{x \mid q_X(x) > 0\}$ contains the Kähler, nef, and movable cones: $\mathrm{Nef}(X) \subset \mathrm{Mov}(X) \subset \mathrm{Pos}(X)$ The walls of $\mathrm{Mov}(X)$ and $\mathrm{Nef}(X)$ are determined by hyperplanes orthogonal to classes of negative square, typically $(-2)$ -classes, corresponding to divisorial contractions or reflections in the BBF lattice (Debarre, 2018, Galluzzi et al., 16 Jan 2025). The cone decomposition and wall-crossing phenomena, especially in Picard rank two, have been explicitly classified for K3 $[2]$ -type fourfolds. Possible types of divisorial contractions correspond to conic bundle structures over K3 surfaces, and their combinations are governed by lattice embedding restrictions and Pell equations (Galluzzi et al., 16 Jan 2025).

For Hilbert schemes and analogous moduli, explicit Diophantine conditions describe when certain classes are big and nef or ample. For example, on $S^{[n]}$ the class $H = aL_n - b\delta$ has

$q_{S^{[n]}}(H) = 2\Bigl( e a^2 - (n-1)b^2 \Bigr)$

and is ample or nef according to $(a, b)$ (Debarre, 2018).

4. Moduli Spaces, Period Maps, and Torelli Theorems

Projective hyper-Kähler manifolds admit coarse, quasi-projective moduli spaces parametrizing marked or polarized varieties. The period domain for fixed lattice and polarization is a quadric in projective space: $\mathcal{D}_h = \{ [\sigma] \mid q(\sigma) = 0,\; q(\sigma, \bar{\sigma}) > 0,\; \sigma \perp h \}$ The period map is an open embedding, and its image is the complement of finitely many “Heegner” divisors— Noether–Lefschetz loci corresponding to primitive negative square sublattices. This classification holds for $K3^{[n]}$ and generalizations, with explicit description in terms of lattice invariants. The global Torelli theorem asserts that a Hodge isometry preserving the BBF form and Kähler chamber is realized by a unique isomorphism of the underlying varieties (Debarre, 2018).

For the Hilbert schemes and Fano varieties of lines, the period map for cubic fourfolds can be identified as the complement of Heegner divisors $\mathcal{D}_2$ and $\mathcal{D}_6$ in the period domain (Debarre, 2018, Laza et al., 2016).

5. Surface Decomposability and Triangle Varieties

The concept of surface decomposability, central to the geometrization of cup product and cycle theory, is defined as follows: a projective hyper-Kähler $X$ of dimension $2n$ is surface-decomposable if there exist $n$ smooth projective surfaces $S_i$ and a generically finite cover $T \to X$ , together with morphisms to the product $\prod_{i=1}^n S_i$ , such that, for each holomorphic 2-form $\omega$ on $X$ , one can express $\phi^*\omega = \sum_{i=1}^n \psi_i^*\omega_i$ (Voisin, 2018). This splitting yields a factorization of the BBF intersection form as a product of surface intersection forms, geometrically realizing the Fujiki relation.

Triangle varieties further support this structure: a triangle variety is a Lagrangian subvariety $T \subset X \times X \times X$ dominating each factor, whose class yields geometric correspondences. For many explicit Picard rank one hyper-Kähler manifolds (e.g., Hilbert schemes, Fano varieties of cubic fourfolds, Debarre–Voisin, double EPW sextics), the existence of triangle varieties and corresponding surface decompositions has been demonstrated, providing evidence towards strong conjectures on the Chow ring and algebraic cycles (Voisin, 2018).

6. Birational Geometry and Invariants

Projective hyper-Kähler manifolds exhibit rich birational geometry with a fundamental role played by the BBF lattice and wall-crossing in the movable cone. Birational automorphism groups vary in families: Denisi, Onorati, Rizzo, and Viktorova proved that for nontrivial families, the birational automorphism group is locally constant (up to finite index) on co-finite open subsets, but for every known deformation type, infinitely many fibers admit infinite birational groups. Key birational invariants include the degree of irrationality, fibering gonality, and fibering genus. For very general projective hyper-Kähler manifolds, explicit lower bounds on the fibering genus in terms of dimension and $b_{2,\,tr}(X)$ have been established, confirming deep irrationality properties in the very general case (Bai, 2022, Bai, 2024).

Birational transformations such as Mukai flops induce isometries on the BBF lattice, preserving the Hodge structure and Fujiki constant. All minimal models of projective hyper-Kähler manifolds are again projective hyper-Kähler, showing stability under the minimal model program. However, moduli spaces of sheaves without symplectic resolutions cannot be birational to any smooth hyper-Kähler manifold, as such a birational map would induce, via the minimal model, a symplectic resolution, which is impossible in these cases (Rapagnetta, 2015).

7. Algebraic Cycles, Chow Theory, and Open Problems

The structure of the Chow ring of projective hyper-Kähler manifolds is governed by conjectures due to Beauville and Voisin. Beauville’s weak splitting conjecture posits that the cycle class map is injective on the subalgebra generated by divisor classes. Voisin extended this to include Chern classes of the tangent bundle. These conjectures are approached via geometric constructions such as surface decomposition: in many cases, surface decomposability reduces the weak splitting conjecture to the case of K3 surfaces, where it is known. Exceptions arise in Kummer varieties, where two Lagrangian cycles may be cohomologically equivalent but not rationally equivalent (Bai, 2024, Voisin, 2018).

Further, the Abel–Jacobi map and its triviality in families of Lagrangian subvarieties provide tools to verify Chow-theoretic injectivity, while exceptional families can provide counterexamples. For high-dimensional Fano varieties of planes in cubics, full verification of predicted Chow group behavior has been established (Bai, 2024).

The open landscape includes the conjecture that every projective hyper-Kähler manifold is surface-decomposable and swept out by algebraically coisotropic divisors; the existence of further deformation types; and a complete wall-and-chamber description of movable cones in all types. Explicit constructions such as the compactified intermediate Jacobian fibration over the moduli of cubic fourfolds illustrate that new high-dimensional projective hyper-Kähler manifolds can arise from holomorphic Lagrangian fibrations and modular compactification structures (Laza et al., 2016).

In summary, projective hyper-Kähler manifolds constitute a class of varieties with strict cohomological and geometric constraints, governed by the interplay of the Beauville–Bogomolov lattice, period domains, surface decompositions, birational invariants, and Chow-theoretic conjectures. Their study connects algebraic, differential, and arithmetic geometry, with numerous deep open questions regarding moduli, cycles, and geometric correspondences.