In mathematics, particularly in representation theory, a symplectic resolution is a morphism that combines symplectic geometry and resolution of singularities.
Definition.
Let formula_1 be a morphism between complex algebraic varieties, where formula_2 is smooth and carries a symplectic structure, and formula_3 is affine, normal, and carries a Poisson structure. Then formula_4 is a "symplectic resolution" if and only if formula_4 is projective, birational, and Poisson.
A conical symplectic resolution is one that is equipped with compatible actions of formula_6 on both formula_3 and formula_2. Under these actions, formula_3 contracts to a single point (denoted 0), the symplectic form is scaled with weight 2, and the morphism formula_4 is compatible with these actions. The "core" of a conical symplectic resolution is defined as the central fiber formula_11. A conical symplectic resolution is Hamiltonian if it possesses Hamiltonian actions of a torus formula_12 on both formula_3 and formula_2. In this case, the morphism formula_4 must be formula_12-equivariant, with the formula_12 action commuting with the conical formula_6 action. Additionally, the fixed point set formula_19 must be finite.
History.
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.
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.