Beauvillebogomolov FormEdit

The Beauville–Bogomolov form is a central tool in the study of hyperkähler geometry, providing a canonical quadratic form on the second cohomology of a compact hyperkähler manifold. Born from the work of Beauville and Bogomolov and later clarified through the Fujiki relation, this form encodes both the topology and the rich geometric structure of the manifold. It serves as a bridge between lattice theory, deformation theory, and the global Torelli program for hyperkähler manifolds, and it underpins many concrete calculations, including the intersection numbers that govern the geometry of the Kähler cone and the moduli of these spaces.

Although the construction is intrinsic to the deformation class of the manifold, its formulation has a tidy history and several equivalent presentations. In some texts it is presented together with the Fujiki relation, highlighting the constant that links top-degree intersection numbers to powers of the form. The literature also reflects a handful of naming conventions, with some sources emphasizing the joint credit of Beauville and Bogomolov, and others using the longer phrase “Beauville–Bogomolov–Fujiki form” to acknowledge the role of Fujiki’s relation. Regardless of naming, the object is the same: a primitive integral quadratic form on H^2(M, Z) that controls the topology and the birational geometry of the manifold.

History and conceptual overview

The second cohomology group, H^2(M, Z), of a compact hyperkähler manifold M carries more than just a cohomology class structure; it inherits a canonical quadratic form q_M with deep geometric meaning. This form is characterized by its relation to top-degree intersection numbers through the Fujiki relation.
The form is named after Beauville and Bogomolov for their foundational observations, and it is often discussed together with Fujiki’s relation, which fixes a proportionality constant that depends on the deformation type. See Fujiki constant for more on this constant and its role.
The combination of Beauville–Bogomolov–Fujiki data provides a robust toolkit for distinguishing deformation types of hyperkähler manifolds and for understanding how their geometry changes (or remains invariant) under deformations and birational transformations.
In the broader landscape of complex geometry, the Beauville–Bogomolov form sits alongside structure theorems for K3 surfaces and their higher-dimensional analogues, where the second cohomology lattice becomes a central invariant.

Definition and key properties

Let M be a compact hyperkähler manifold of complex dimension 2n. The Beauville–Bogomolov form q_M is a primitive integral quadratic form on H^2(M, Z) that is compatible with the cup product and the complex structure.
The defining relation is the Fujiki formula: for any α ∈ H^2(M, R), ∫_M α^{2n} = c_M · q_M(α)^n, where c_M is the Fujiki constant attached to the deformation type of M. This equation expresses top-degree intersection numbers in terms of the Beauville form.
The Beauville form has signature (3, b_2(M) − 3). Here b_2(M) is the second Betti number, i.e., the rank of H^2(M, Z) as a lattice. This signature reflects the presence of a positive 3-dimensional subspace corresponding to the hyperkähler structure.
The lattice structure provided by q_M turns H^2(M, Z) into a quadratic lattice, and it interacts naturally with the Kähler cone and the positive cone inside H^2(M, R). The positivity properties of q_M on Kähler classes drive many geometric conclusions, such as the description of the ample and nef cones in deformation families.
For a concrete class of examples, consider the Hilbert scheme of points on a K3 surface, denoted Hilbert scheme of points on a K3 surface or K3^(n) in compact notation. The second cohomology lattice in this case splits as a direct sum of the K3 lattice with an extra class δ, with δ^2 negative and reflecting the exceptional divisor structure in the Hilbert scheme. This explicit lattice description makes the general theory tangible in higher-dimensional settings.

The Fujiki relation and the deformation picture

The Fujiki constant c_M is an invariant of the deformation type of M and plays a crucial role in relating the Beauville form to intersection theory. It is not fixed by cohomology alone but stabilizes under deformations within the same hyperkähler family.
The Fujiki relation shows that the top-degree intersection numbers are governed by a single quadratic form, turning many geometric questions into lattice-theoretic problems. This perspective underpins period maps and the global Torelli program for hyperkähler manifolds.
Deformation invariance means that, as M moves within its connected component of the moduli space of hyperkähler manifolds, the Beauville form q_M remains the same up to the natural identification of cohomology lattices. This makes q_M a powerful invariant for classifying and comparing different hyperkähler manifolds.

Examples and lattice consequences

For a K3 surface S, the Beauville form on H^2(S, Z) coincides with the classical intersection form on the K3 lattice, which is isomorphic to E_8(-1) ⊕ E_8(-1) ⊕ U^3 (the standard K3 lattice). The hyperkähler structure on a K3 surface is the starting point for higher-dimensional generalizations, where the same ideas are lifted to more intricate Hilbert schemes and generalized Kummer varieties.
In the case of Hilbert schemes of points on a K3 surface, M = Hilb^n(S), the second cohomology lattice has rank b_2(M) = 23 for n ≥ 2, and the lattice is well described as the sum of the K3 lattice with an extra class δ, subject to δ^2 = −2(n − 1). The Beauville form then governs the entire lattice structure that controls both the deformation theory and the birational geometry of these spaces.
The lattice viewpoint makes the global Torelli problem amenable to period-domain techniques: the period map records the Hodge structure together with q_M, and Torelli-type statements relate birational geometry to lattice-theoretic data.

Controversies and debates (historical and methodological)

Naming and attribution: In places the same quadratic form is referred to as the Beauville form, the Bogomolov form, or the Beauville–Bogomolov–Fujiki form. Debates about naming reflect historical credit and the desire to acknowledge the key contributions of multiple researchers. The mathematical content remains the same across these labels, but scholars sometimes prefer one label over another for clarity in combined references.
Extension to singular hyperkähler varieties: A live area of research involves extending the Beauville–Bogomolov framework to certain singular symplectic varieties and their crepant resolutions. Researchers debate which versions of the form persist, how to formulate the Fujiki-type relations in singular settings, and what the appropriate lattice-theoretic invariants should be in these broader contexts.
Normalization and sign conventions: While the core idea is robust, the precise normalization of q_M and the sign conventions used in various presentations can differ between sources. As with many objects defined up to isomorphism, this leads to minor but sometimes confusing discrepancies in explicit computations. The standard convention is chosen to align with the geometric interpretation of positivity on the Kähler cone and with the familiar signatures appearing in deformation theory.
Interplay with physics and moduli problems: The Beauville–Bogomolov form appears in questions about moduli spaces and, more broadly, in certain physical theories where hyperkähler manifolds arise as target spaces or compactifications. Depending on the formulation, there are discussions about which versions of the form best capture the relevant physical invariants, though these conversations are typically technical and context-dependent rather than ideological.