In mathematics, particularly in representation theory, a symplectic resolution is a morphism that combines symplectic geometry and resolution of singularities.^[1]

Let be a morphism between complex algebraic varieties, where is smooth and carries a symplectic structure, and is affine, normal, and carries a Poisson structure. Then is a symplectic resolution if and only if is projective, birational, and Poisson.^[1]

A conical symplectic resolution is one that is equipped with compatible actions of on both and . Under these actions, contracts to a single point (denoted 0), the symplectic form is scaled with weight 2, and the morphism is compatible with these actions. The core of a conical symplectic resolution is defined as the central fiber . A conical symplectic resolution is Hamiltonian if it possesses Hamiltonian actions of a torus on both and . In this case, the morphism must be -equivariant, with the action commuting with the conical action. Additionally, the fixed point set must be finite.^[1]

The study of symplectic resolutions emerged as a natural generalization of classical techniques in representation theory. During the 20th century, mathematicians primarily investigated the representation theory of semisimple Lie algebras through geometric methods, focusing particularly on flag varieties and their cotangent bundles.^[1]

In the 21st century, this approach evolved into a more general framework where the traditional cotangent bundle of the flag variety was replaced by symplectic resolutions. This generalization led to significant developments in understanding the relationship between geometry and representation theory. The classical semisimple Lie algebra was correspondingly replaced by the deformation quantization of the affine Poisson variety.^[1]

• Coulomb branch
• Poisson variety
• Symplectic duality
• Symplectic geometry