Global Torelli TheoremEdit

Global Torelli Theorem

The Global Torelli Theorem concerns whether the geometry of certain complex algebraic varieties is completely determined by their Hodge-theoretic data. In its most studied forms, it asks if a polarized Hodge structure on cohomology, together with the period information carried by a period map, suffices to recover the underlying variety up to isomorphism. The theorem has several precise incarnations, each tailored to a class of objects—most famously curves, abelian varieties, K3 surfaces, and, in a broader hyperkähler setting, irreducible holomorphic symplectic manifolds. It sits at the crossroads of geometry, topology, and arithmetic, translating geometric problems into lattice-theoretic and period-data problems.

The conceptual framework rests on a few core ideas. A complex variety carries a Hodge theory on its cohomology, which records how differential forms decompose under the complex structure. A choice of polarization turns the cohomology into a lattice with a bilinear form. The collection of all Hodge structures with fixed lattice data forms a period domain—an analytic parameter space for complex structures. The period map sends a family of varieties to their corresponding Hodge structures, and a marked variant uses a fixed reference lattice to keep track of isomorphisms. The central question is when this period data determines the variety itself, or at least its deformation class, and when extra geometric data beyond the period point is needed.

Conceptual framework

Hodge theory provides the decomposition H^n(X, C) = ⊕_{p+q=n} H^{p,q}(X) compatible with complex structure.
A polarization is an integral class that yields a positive definite form on primitive cohomology, turning the cohomology into a lattice.
The period domain parameterizes Hodge structures with fixed Hodge numbers and lattice, subject to the bilinear form given by the polarization.
A period map associates to each complex structure its Hodge structure, encoding the complex-analytic variation in a way that is amenable to arithmetic and lattice techniques.
A marking is an isomorphism between the second cohomology lattice of a variety and a fixed reference lattice, enabling a well-defined period map.

Key objects frequently discussed in this theory include K3 surface, irreducible holomorphic symplectic manifold (also called hyperkähler manifold in the compact setting), Jacobian varieties of curves, and general moduli spaces of complex structures.

Classical results

Curves and abelian varieties

For principally polarized curves of genus g ≥ 2, the classical Torelli theorem (the curve is determined by its Jacobian with its canonical polarization) provides a prototype: the polarized Hodge structure on H^1 determines the curve up to isomorphism.
For abelian varieties, the polarized Hodge structure on H^1 similarly recovers the variety, making the period data an effective classifier in this setting.

The basic paradigm—recovering geometry from Hodge-theoretic data—was first realized in these lower-dimensional, well-behaved cases and guided all later generalizations.

K3 surfaces and their global Torelli theorem

In dimension two, K3 surfaces provide a particularly sharp realization of the Global Torelli principle. The global Torelli theorem for K3 surfaces states that two K3 surfaces are isomorphic if and only if their second cohomology groups are related by a Hodge isometry that preserves the orientation of the positive cone and maps a suitable class (in practice, a Kähler class) to a Kähler class. This result connects the Hodge-theoretic data to birational and geometric structure in a concrete way.
The theorem is closely tied to the Beauville–Bogomolov–Fujiki form on H^2 and to the theory of moduli of K3 surfaces, reflecting how lattice-theoretic data control the geometry of these surfaces. For K3 surfaces, the period map is particularly well-behaved and injective on the appropriate moduli spaces after marking.

Throughout these cases, the central takeaway is that, for curves and K3-type objects, the period data and the associated lattice information carry enough geometric content to recover, or nearly recover, the underlying shape.

Hyperkähler world and beyond

Irreducible holomorphic symplectic manifolds

Beyond K3 surfaces, the natural higher-dimensional analogs are irreducible holomorphic symplectic manifolds (IHS), of which K3 surfaces are the two-dimensional prototype. The development of a global Torelli-type picture for IHS manifolds hinges on the period map for marked manifolds and the global structure of their H^2 lattices with the Beauville–Bogomolov–Fujiki form.

Verbitsky’s global Torelli-type results

A landmark achievement in this broader setting is the work of Misha Verbitsky and subsequent refinements, which establish that the period map for marked IHS manifolds is a local isomorphism and that a version of the global Torelli principle holds under suitable conditions. In particular: - If X and Y are marked IHS manifolds with a Hodge isometry H^2(X, Z) ≅ H^2(Y, Z) that respects the Kähler cones in a precise sense, then X and Y are closely related in the moduli sense—often birationally equivalent, and in many cases isomorphic after accounting for the birational Kähler cone. - The refinements emphasize the interplay between the H^2 lattice, the period point, and the birational geometry controlled by the Kähler (and birational) cones.

These results illustrate a high-water mark for a global Torelli-style statement in higher dimensions, showing how period data interacts with the richer birational geometry of IHS manifolds.

Limitations, counterexamples, and debates

General Calabi–Yau and other higher-dimensional varieties are not subject to a universal global Torelli principle. In many cases, two nonisomorphic varieties can share the same Hodge structure, and the period map fails to be injective without additional data. This highlights a fundamental limitation: lattice and period information can be necessary but not sufficient in full generality.
Even for hyperkähler manifolds, a complete global Torelli statement requires careful treatment of birational models, monodromy, and the structure of the Kähler cone. The clean “isomorphism iff H^2 isometry preserving the relevant cones” picture has to be qualified by birational equivalence classes and by the intricacies of the period domain.
Beyond period data, derived categories have emerged as influential invariants. Derived Torelli-type ideas suggest that equivalences of derived categories (D^b(X)) can encode deep geometric information, and in some settings (notably for K3 surfaces) there are strong links between derived equivalences and Hodge-theoretic data. This has sparked debate about whether derived invariants can or should augment classical Torelli-type statements, and about how these perspectives should be reconciled with moduli and birational classifications.
Critics of overreliance on period data emphasize the importance of explicit geometric and birational information, such as divisors, ample cones, and explicit moduli constructions. Proponents of the period-centric approach counter that a robust, lattice-based framework provides a unifying, computable, and structurally transparent handle on a wide range of geometric objects.

Contemporary perspective

The Global Torelli framework reflects a long-standing preference in parts of the mathematical community for invariants that survive deformations and carry arithmetic texture. The results for curves, abelian varieties, and K3/hyperkähler manifolds illustrate a spectrum: in some classes, period data almost completely determines geometry; in others, it must be complemented by additional birational or derived information. The ongoing dialogue between lattice-theoretic, Hodge-theoretic, and derived approaches continues to sharpen our understanding of when and how geometry can be reconstructed from cohomological fingerprints.