Symplectic Resolutions: Geometry & Applications

Updated 23 October 2025

Symplectic resolutions are proper birational morphisms from smooth symplectic varieties to singular Poisson varieties, ensuring a global nondegenerate holomorphic 2-form.
They play a vital role in algebraic geometry, representation theory, and topology by resolving singularities in spaces like moduli spaces, quiver varieties, and quotient singularities.
Their structure organizes birational geometry through techniques such as Mukai flops and GIT variations, linking deformation theory and categorical dualities.

Symplectic resolutions are projective or proper birational morphisms from a smooth symplectic variety to a singular Poisson variety such that the pullback of the (degenerate) Poisson 2-form yields a global nondegenerate holomorphic 2-form on the resolution. This notion underlies major developments in algebraic geometry, representation theory, and geometric topology, especially where singular spaces with Poisson or symplectic structures arise, such as moduli spaces, quiver varieties, character varieties, and quotient singularities. The concept organizes much of the birational geometry of singular symplectic varieties, links deformation theory, and provides a bridge between algebraic, geometric, and representation-theoretic approaches.

1. Definitions and General Structure

A symplectic singularity $Y$ is a normal complex variety whose smooth locus admits a holomorphic symplectic form $\omega$ , with the property that for any resolution of singularities $\pi: X \to Y$ , the pullback $\pi^* \omega$ extends to a global (and generically nondegenerate) 2-form on $X$ . A symplectic resolution is a resolution $\pi: X \to Y$ such that $X$ is smooth, and the extended 2-form is both holomorphic and nondegenerate on all of $X$ .

Fundamental properties include:

Symplectic singularities are rational Gorenstein and their local geometry is controlled by the extension properties of the symplectic form.
Symplectic resolutions are automatically crepant (the canonical bundles satisfy $K_X = \pi^* K_Y$ ).

Typical sources include:

Quotient singularities $V/G$ for a symplectic vector space $V$ and $G < Sp(V)$ .
Conical symplectic varieties, i.e., conical affine varieties equipped with $\mathbb{C}^*$ -actions scaling the symplectic form (Brion et al., 2013).
Moduli spaces of sheaves, Higgs bundles, or local systems exhibiting Poisson/symplectic structures.

2. Classification and Existence Criteria

Linear Quotients and Symplectic Reflection Groups

The problem of classifying symplectic resolutions for linear quotients $V/G$ is nearly complete. For $G$ acting symplectically on $V$ :

In dimension 2, all symplectic resolutions are minimal resolutions of Kleinian singularities—ADE surface singularities.
In higher dimensions, almost all symplectic resolutions arise from so-called wreath product groups: $(\mathbb{C}^2)^n / (K \wr S_n)$ with $K < SL_2(\mathbb{C})$ . The reducibility, primitivity, and the nature of $G$ dictate the existence (Bellamy et al., 2013, Bellamy et al., 2020). For imprimitive or exceptional primitive groups, most do not admit symplectic resolutions—linear and representation-theoretic obstructions manifest in the structure of symplectic reflection algebras and the absence of sufficiently “large-dimensional” simple modules.

Quiver Varieties and Moduli Problems

Nakajima quiver varieties and their multiplicative counterparts (moduli of representations of multiplicative preprojective algebras) are thoroughly classified as symplectic singularities. The existence of a symplectic resolution hinges on a root-theoretic criterion:

A quiver variety admits a (projective) symplectic resolution if and only if the dimension vector is indivisible (prime) and lies in a specific region (e.g., the real roots or certain “fundamental regions” for the root system associated to the quiver). The only exceptions occur in special $(2,2)$ -cases, where a symplectic resolution can still exist (Bellamy et al., 2016, Schedler et al., 2018).
Multiplicative quiver varieties (and, by extension, tame character varieties for punctured surfaces) admit symplectic resolutions precisely under a q-indivisibility and non-decomposability criterion for the dimension vector.

Moduli of Higgs Bundles and Character Varieties

For moduli spaces of Higgs bundles and character varieties:

The moduli space $M_H(X, n)$ of semistable Higgs bundles of degree zero and rank $n$ on a compact Riemann surface of genus $g$ is a symplectic singularity (Tirelli, 2017). It admits a projective symplectic resolution if and only if $g=1$ for arbitrary rank, or $(g,n)=(2,2)$ .
More generally, for $G$ -character varieties of compact Riemann surfaces for connected complex reductive type A groups ( $GL_n$ , $SL_n$ , $PGL_2$ ), a symplectic resolution exists only for genus 1 and products of SL or PGL factors, or for genus 2 and all simple factors isomorphic to $SL_2$ (Bellamy et al., 2019).

3. Birational Geometry, Flops, and Movable Cone Structure

The birational geometry of symplectic resolutions is tightly controlled. In the particularly rich case of four-dimensional symplectic contractions, the following features appear (Andreatta et al., 2011):

All such contractions are Mori dream spaces: their movable cone of divisors $\mathrm{Mov}(X/Y)$ is polyhedral and decomposes into nef chambers, each corresponding to a symplectic resolution.
Any two symplectic resolutions are related by a finite sequence of Mukai flops: explicit local birational surgeries that replace families of Lagrangian planes (e.g., $\mathbb{P}^2$ ) with their duals, preserving the symplectic structure up to codimension 2.

The movable cone is often described as dual to the cone of essential curves (classes not contained in central fibers), encoding the geometry of transitions between birational models.

Families of symplectic resolutions for fixed quotient singularities (e.g., for a group of order 32 in $Sp(4, \mathbb{C})$ ) can be constructed via variation of GIT quotient or Cox ring constructions, with distinct chambers corresponding to different crepant/symplectic resolutions and transitions governed by enumerative and combinatorial invariants (e.g., root systems, Cartan matrices) (Donten-Bury et al., 2014).

4. Cohomology, Quantization, and Category $\mathcal{O}$

Symplectic resolutions are not just spaces but support deep categorical and representation-theoretic structures:

Poisson traces (zeroth Poisson homology) are subtle invariants: for $X$ a Poisson variety with finitely many symplectic leaves, the dimension of $O(X)/\{O(X), O(X)\}$ is finite and, when a symplectic resolution exists, is conjecturally dual to the top cohomology $H^{\dim X}(X̃)$ (Etingof et al., 2017).
Quantization: Conical symplectic resolutions admit canonical filtered quantizations (e.g., quantized algebras of functions, deformation quantizations). The associated category $\mathcal{O}$ is a highest weight category with standard, costandard, and Koszul properties, generalizing the BGG category $\mathcal{O}$ . This algebraic category relates to geometric category $\mathcal{O}^\mathfrak{g}$ (D-modules supported on attracting sets) via localization (Braden et al., 2014).
Symplectic duality: A powerful duality assigns to each symplectic resolution $M$ a dual resolution $M^!$ , paired at the level of fixed-point combinatorics, Weyl group data, cohomology, and derived Koszul dualities. This duality explains numerical and categorical correspondences in geometric representation theory (e.g., Nakajima quiver varieties—affine Grassmannian slices correspondence). T-structures, autoequivalences (twisting and shuffling functors), and centers of Yoneda algebras intertwine under duality (Braden et al., 2014, Kamnitzer, 2022).

5. Hodge Theory, Perverse Sheaves, and Filtrations

Recent advances show that the interplay between geometry, topology, and Hodge theory is illuminated by the existence of symplectic resolutions:

Perverse and Hodge structures: The symplectic Hard Lefschetz theorem extends to intersection complex Hodge modules for possibly singular holomorphic symplectic varieties admitting (global or étale-local) symplectic resolutions. This yields isomorphisms on graded de Rham complexes and symmetry of perverse-Hodge complexes for Lagrangian fibrations. The perverse numbers associated to the Hitchin fibration (or more generally, perverse or Hodge numbers) coincide under these circumstances (Xin, 19 Mar 2025).
PI=WI conjecture and P=W: For moduli spaces (e.g., Higgs bundle moduli, character varieties) admitting (resolved) symplectic structure, the perverse Leray filtration (arising from the Hitchin fibration) and the weight filtration (on mixed Hodge structures of the Betti moduli) coincide under the nonabelian Hodge correspondence (Felisetti et al., 2020). The existence of a symplectic resolution permits lifting this correspondence to resolutions and validating the conjectural equivalence.

6. Applications and Specialized Constructions

Group actions and orbifold resolutions: For symplectic orbifolds and quotients by finite (and even certain infinite) discrete symplectic groups, canonical smooth symplectic resolutions can be constructed via systematic resolution of singularities and desingularization of symplectic forms, sometimes produced equivariantly (Chen, 2017, Lassoued et al., 2021).
Quantum operations: Quantum Steenrod operations and equivariant quantum cohomology have new expressions in the context of conical symplectic resolutions, with enumerative geometry of holomorphic curves giving rise to quantum operations whose structural compatibility is closely linked to the symplectic resolution structure (Lee, 2023).
Birational rigidity and factorization: The structure of movable cones, exceptional loci, and flops allows profound control over the birational geometry and classification of symplectic resolutions, underlying the rigidity of symplectic singularities in dimensions $\geq 4$ and suggesting strong uniqueness phenomena for crepant resolutions (Andreatta et al., 2011, Yamagishi, 2017).

7. Open Problems and Future Directions

Classification gaps: Remaining open cases in the classification of linear quotient symplectic resolutions exist (notably certain symplectically primitive but complex imprimitive groups in dimension $4$ and a finite set of exceptional cases in higher dimensions) (Bellamy et al., 2020).
Generalization to moduli of objects in 2-Calabi–Yau categories: It is conjectured that the existence and geometry of symplectic resolutions in more general settings (e.g., character varieties of punctured surfaces, moduli of sheaves on Calabi–Yau varieties) are controlled by underlying categorifications and local models in 2-CY deformation theory (Schedler et al., 2018).
Hodge-theoretic and representation-theoretic dualities: Ongoing work seeks to further explicate the structure of perverse-Hodge complexes, categorified correspondences, and the role of symplectic duality in topological and enumerative invariants. The relationship between Poisson deformations, quantizations, and symmetry properties of moduli spaces is also a field of deep current investigation.

In sum, symplectic resolutions form a cornerstone in the modern landscape of algebraic and geometric representation theory, providing the geometric setting for deep theoretical advances ranging from the minimal model program, symplectic reflection algebras, and Coulomb branch constructions to highly nontrivial results in cohomological and enumerative geometry. These resolutions not only resolve singularities but also control birational, categorical, and Hodge-theoretic structures for a broad class of geometric and representation-theoretic objects.