Mathematics Lecture Notes

Theorem8.19Derived McKay Correspondence

For a finite subgroup $G \subset SL(n, \mathbb{C})$ acting on $\mathbb{C}^n$ and the minimal resolution $Y \to \mathbb{C}^n/G$ : $D^b(\text{Coh}_G(\mathbb{C}^n)) \simeq D^b(\text{Coh}(Y))$

This establishes a derived equivalence between equivariant coherent sheaves on the quotient and coherent sheaves on the resolution.

Remark

The derived McKay correspondence shows that singularities can be understood via derived categories, connecting representation theory of finite groups to geometry of resolutions.

Theorem8.20Homological Mirror Symmetry

For a mirror pair $(X, \check{X})$ of Calabi-Yau manifolds, Kontsevich's Homological Mirror Symmetry conjecture states: $D^b(\text{Coh}(X)) \simeq D^{\pi} \text{Fuk}(\check{X})$

where the right side is the derived Fukaya category of Lagrangian submanifolds with $\pi$ -structure.

Remark

Homological Mirror Symmetry has profound implications for string theory and symplectic geometry, providing a categorical framework for the A-model/B-model duality.

Theorem8.21Bridgeland Stability Conditions

The space of Bridgeland stability conditions on $D^b(X)$ is a complex manifold $\text{Stab}(X)$ . For $X$ a K3 surface: $\dim \text{Stab}(X) = b_2(X) + 2$

Stability conditions provide a continuous generalization of classical slope stability.

ExampleTilting and Equivalences

A tilting object $T \in D^b(\mathcal{A})$ generates the category and satisfies $\text{Ext}^i(T, T) = 0$ for $i \neq 0$ . It induces a derived equivalence: $D^b(\mathcal{A}) \simeq D^b(\text{Mod}_{\text{End}(T)})$

This provides a powerful tool for relating different categories.

Theorem8.22Derived Equivalences and K-Theory

A derived equivalence $\Phi: D^b(X) \simeq D^b(Y)$ induces isomorphisms:

On algebraic K-theory: $K_0(X) \cong K_0(Y)$
On Hochschild homology: $HH_*(X) \cong HH_*(Y)$
On cyclic homology: $HC_*(X) \cong HC_*(Y)$

These are derived invariants preserved by categorical equivalences.

ExamplePerverse Sheaves

The category of perverse sheaves $\text{Perv}(X)$ is the heart of a t-structure on $D^b_c(X)$ (constructible derived category). These objects satisfy:

Sheaf-like properties (support conditions)
Cosheaf-like properties (vanishing cycles)

Perverse sheaves are fundamental in geometric representation theory and the geometric Langlands program.

Math Notes

Derived Categories - Applications