Let $E = \mathbb{C}/\Lambda$ be an elliptic curve and $\hat{E} = \mathbb{C}/\Lambda^\vee$ its dual. There exists a triangulated equivalence: $D^b(Coh(E)) \simeq D\mathcal{F}uk(\hat{E})$

The equivalence is constructed explicitly by:

1. Sending line bundles $\mathcal{L}_\alpha$ to Lagrangian tori $T_\alpha$
2. Sending skyscraper sheaves to Lagrangian circles
3. Verifying Ext groups match Floer cohomology

This was the first rigorous proof of HMS and established the feasibility of the categorical approach.

For a smooth quartic K3 surface $S \subset \mathbb{P}^3$ and its mirror $S^\vee$: $D^b(Coh(S)) \simeq D\mathcal{F}uk(S^\vee)$

The proof constructs the equivalence using:

• Spherical objects and spherical twists
• Lefschetz fibrations and vanishing cycles
• Symplectic topology techniques

This extended HMS beyond tori to higher-dimensional Calabi-Yau manifolds.

Let $X$ and $Y$ be smooth projective varieties. Every fully faithful exact functor: $\Phi: D^b(Coh(X)) \to D^b(Coh(Y))$

is representable by a Fourier-Mukai kernel $\mathcal{K} \in D^b(Coh(X \times Y))$: $\Phi(\mathcal{F}) = R\pi_{Y*}(\pi_X^*\mathcal{F} \otimes^L \mathcal{K})$

This provides a framework for understanding derived equivalences algebraically.

For a smooth toric Fano variety $X_\Sigma$ and its Landau-Ginzburg mirror $(M, W)$: $D^b(Coh(X_\Sigma)) \simeq \mathcal{F}S(M, W)$

where $\mathcal{F}S(M, W)$ is the Fukaya-Seidel category of the Landau-Ginzburg model.

Objects on the right are Lagrangian thimbles (vanishing cycles) associated to critical points of $W$.

The derived category $D^b(Coh(X))$ for a smooth projective variety $X$ is generated as a triangulated category by any spanning class, i.e., a collection $\{\mathcal{E}_i\}$ such that: $\text{Hom}(\mathcal{E}_i, \mathcal{F}[k]) = 0 \text{ for all } i, k \implies \mathcal{F} = 0$

For HMS, proving the Fukaya category is generated by specific Lagrangians reduces the conjecture to verifying Ext/Floer isomorphisms for generators.