Symplectic Varieties

• Symplectic varieties are higher-dimensional analogues of holomorphic symplectic manifolds with controlled singularities that retain a nondegenerate holomorphic 2-form on their smooth locus.
• They are constructed via moduli spaces, quotients, and conical methods, and their geometry is characterized by the BBF quadratic form and deformation theory.
• Their rich structure links algebraic geometry, representation theory, and symplectic topology, enabling powerful dualities, Lagrangian fibrations, and deep classification results.

Symplectic varieties are higher-dimensional analogues of holomorphic symplectic manifolds, extending the concept from the smooth case to allow singularities while preserving the existence of a nondegenerate, holomorphic 2-form on the regular locus. Their structure is governed by strong cohomological and geometric properties, tight connections to moduli theory, and remarkable classification results linking algebraic geometry, representation theory, and symplectic topology.

1. Foundational Definitions and Invariants

A symplectic variety is a normal, projective variety $X$ of dimension $2n$ equipped with a reflexive holomorphic 2-form $\sigma \in H^0(X, \Omega_X^{[2]})$ that is nondegenerate on the smooth locus and extends across any resolution; such varieties have canonical (rational Gorenstein) singularities and vanishing canonical class $K_X \equiv 0$ (Brakkee et al., 2024).

The notion of primitive symplectic variety requires $H^1(X, \mathcal{O}_X) = 0$ and $H^0(X, \Omega_X^{[2]}) = \mathbb{C} \cdot \sigma$, with no finite quasi-étale covers splitting the symplectic form. For the irreducible symplectic case, one demands that all finite quasi-étale covers $g: Y \to X$ have exterior reflexive forms generated by $g^{[*]}\sigma$ and $\pi_1(X_{\text{reg}}) = 1$.

A key invariant is the Beauville–Bogomolov–Fujiki (BBF) quadratic form, $$q_X: H^2(X, \mathbb{Z}) \times H^2(X, \mathbb{Z}) \to \mathbb{Z}$$, characterized by the Fujiki relation: $\int_X \alpha^{2n} = c_X \cdot q_X(\alpha)^n$ for all $\alpha \in H^2(X,\mathbb{Z})$, with $c_X > 0$ rational (Brakkee et al., 2024, Bakker et al., 2018, Takamatsu, 2022).

2. Classification and Construction of Symplectic Varieties

Symplectic varieties arise in multiple contexts:

• Moduli Spaces of Sheaves: On K3 surfaces, the Beauville–Mukai moduli spaces of semistable sheaves provide symplectic varieties; when the Mukai vector is primitive and the polarization generic, these are smooth and irreducible symplectic of K3^[n]-type. Singular moduli spaces (including O’Grady’s 6- and 10-dimensional examples) yield terminal symplectic varieties (Bakker et al., 2018, Bakker et al., 2016).
• Quotients and Automorphisms: Quotients of smooth symplectic manifolds by involutions lead to $\mathbb{Q}$-factorial terminal symplectic models, important for moduli theory and birational geometry (Bakker et al., 2018).
• Relative Prym Construction: For a K3 surface with an anti-symplectic involution, the relative Prym variety associated to certain linear systems on the quotient surface produces normal projective symplectic varieties with canonically defined Lagrangian fibrations; criteria involving ampleness, transversality, and monodromy ensure primitivity and irreducibility. Infinite series of irreducible symplectic varieties in high even dimensions are thus constructed, generalizing classical examples (Brakkee et al., 2024).
• Complete Intersections: Homogeneous symplectic varieties embedded as complete intersections in affine space coincide with nilpotent cones of semisimple Lie algebras, inheriting the Kostant–Kirillov symplectic structure (Namikawa, 2012).
• Conical Symplectic Varieties: Graded (conical) symplectic varieties admitting symplectic resolutions are precisely normalizations of nilpotent orbit closures, with all symplectic resolutions of Springer type (cotangent bundles of flag varieties) (Brion et al., 2013).

3. Moduli, Deformation Theory, and Period Maps

Symplectic varieties admit unobstructed locally trivial deformations, with smooth Kuranishi space of dimension $h^{1,1}(X)$ (Bakker et al., 2018, Bakker et al., 2016). Marked moduli can be organized by fixing an integral lattice ($\Lambda, q$), inducing a period map from the moduli space to the period domain

$$\Omega_\Lambda = \{ [\sigma] \in \mathbb{P}(\Lambda_\mathbb{C}) : q(\sigma) = 0, q(\sigma, \overline{\sigma}) > 0 \}$$

The BBF form provides a global quadratic structure on $H^2$, and the period map is locally isomorphic by global Torelli theorems in both smooth and singular settings, leveraging Ratner–Verbitsky ergodicity and monodromy orbit closures (Bakker et al., 2018, Bakker et al., 2016).

The monodromy group $\mathrm{Mon}(\mathcal{M}) \subset O(\Lambda)$ generated by parallel transport under deformation is arithmetic of finite index; its orbits on the period domain stratify by rational rank, controlling the separation of points in moduli and the parametrization of bimeromorphic models.

Moduli of symplectic varieties can be further stratified by representation-theoretic data, stability conditions (θ-stability for quiver varieties), and the geometry of the Néron–Severi lattice. The cone conjecture gives rational polyhedral fundamental domains for automorphism and birational groups acting on nef and movable cones, yielding finiteness results for birational models, twists, and derived equivalence classes (Takamatsu, 2021).

4. Symplectic Singularities and Resolutions

Beauville defined symplectic singularities as normal varieties whose smooth locus carries a holomorphic symplectic form extending across all resolutions as regular 2-forms (Bakker et al., 2018, Bakker et al., 2016). The existence of symplectic resolutions is heavily constrained:

• Quiver Varieties: Nakajima quiver varieties are irreducible symplectic singularities, with projective symplectic resolutions classified by root data. Only for indivisible or very specific divisible roots (the O’Grady (2,2) case) do symplectic resolutions exist. Terminal, factorial singularities prevent the existence of crepant (hence symplectic) resolutions outside these cases (Bellamy et al., 2016, Schedler et al., 2018).
• Character Varieties: Moduli of representations of surface groups for $G$ of type A are symplectic singularities; symplectic resolutions occur only for genus 1 or genus 2 factors (Hilbert schemes or small blowups), with most cases being terminal, $\mathbb{Q}$-factorial and unobstructed for Poisson deformations (Bellamy et al., 2019).
• Conical Cases: Any symplectic resolution of a conical symplectic variety is essentially a Springer resolution of a nilpotent orbit closure (possibly composed with a linear projection), reflecting the rigid Lie-theoretic origins of such singularities (Brion et al., 2013).

5. Lagrangian Fibrations, Vanishing Theorems, and Hodge Theory

Symplectic varieties often exhibit Lagrangian fibrations: proper, surjective morphisms whose general fibers are abelian varieties and whose symplectic form vanishes when restricted to the fiber (Brakkee et al., 2024). Criteria for the existence and properties of such fibrations depend on geometric ampleness, monodromy, and connectedness of linear systems on the base.

In the singular setting, symplectic forms induce powerful dualities in the derived category and intersection cohomology. For isolated symplectic singularities:

• Enhanced vanishing theorems generalize both local and Steenbrink vanishing, with much larger ranges (Tighe, 2024).
• The map $$L_\sigma^{n-1}: \mathbb{D}_X(\underline{\Omega}_X^{2n-1}) \to \underline{\Omega}_X^{2n-1}$$ is a quasi-isomorphism, enabling "Hard Lefschetz" type symmetries at the derived and intersection cohomology level.
• Higher Du Bois and higher rational properties are tightly controlled, with symplectic singularities showing near-optimal behavior in the singular cohomological hierarchy.

For primitive symplectic 4-folds, all cohomology is pure, intersection cohomology coincides with singular cohomology, and exceptional components' odd cohomologies vanish (Tighe, 2024).

6. Symplectic Structures in Representation Theory and Geometry

Homogeneous symplectic varieties of complete intersection type correspond to nilpotent varieties of semisimple Lie algebras, inheriting Kostant–Kirillov Poisson structures (Namikawa, 2012). The study of Lagrangian subvarieties, moment maps, and symmetric degenerations ties symplectic geometry deeply to invariant theory and Hamiltonian group actions (Timashev et al., 2011).

The symplectic (type C) Grassmannian and flag varieties are linear slices of their classical (type A) analogues, realized by explicit Plücker equations and combinatorial criteria for complete intersections and smoothness. Tropical degenerations, PBW filtrations, and FFLV polytopes encode rich geometric structures and toric models for symplectic flag varieties (Balla et al., 2023, Xu et al., 2022).

7. Symplectic Topology and Normal-Crossings Degenerations

In symplectic topology, normal-crossings (“NC”) symplectic varieties model degenerations in terms of transverse unions of codimension-two submanifolds (Tehrani et al., 2014, Tehrani et al., 2021). Smoothability in the symplectic category is governed by a topological d-semistability criterion: triviality of the line bundle $\mathcal{O}_{X_\partial}(X_\emptyset)$. The multifold symplectic sum construction generalizes Gompf's two-fold sum, producing a vast class of symplectic manifolds via NC smoothings and enabling new techniques in relative and logarithmic Gromov–Witten theory.

The blowup and logarithmic tangent bundle machinery yields refined Chern class formulas, with direct applications in moduli theory, mirror symmetry, and enumerative geometry. These advances fulfill Gromov’s vision of a topological symplectic sum theory analogous to the algebraic case (Tehrani et al., 2014, Tehrani et al., 2021).

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Symplectic varieties thus represent a robust and unifying framework connecting complex, algebraic, and symplectic geometry, with profound implications for moduli theory, Hodge theory, and representation theory. Their combinatorial, birational, and deformation-theoretic properties continue to drive research in both pure mathematics and geometric representation theory.