Homological Mirror Symmetry - Examples and Constructions

Proving homological mirror symmetry requires sophisticated techniques from symplectic geometry, algebraic geometry, and category theory. We examine proven cases and construction methods that have established HMS in various contexts.

ExampleHMS for Elliptic Curves (Polishchuk-Zaslow)

For an elliptic curve $E = \mathbb{C}/\Lambda$ with dual $\hat{E} = \mathbb{C}/\Lambda^\vee$ :

B-side: $D^b(Coh(E))$ has generators:

$\mathcal{O}_E$ : structure sheaf
$\mathcal{O}_p$ : skyscraper at point $p$

A-side: $D\mathcal{F}uk(\hat{E})$ has generators:

Lagrangian tori (1-cycles)
Lagrangian circles

The equivalence sends line bundles to Lagrangian tori via the Abel-Jacobi map, and point sheaves to specific Lagrangian cycles. This was the first rigorous proof of HMS.

Quartic Surfaces and HMS

Seidel proved HMS for quartic K3 surfaces using symplectic techniques.

ExampleQuartic K3 Surfaces

A quartic K3 surface $S \subset \mathbb{P}^3$ defined by a degree 4 polynomial satisfies: $D^b(Coh(S)) \simeq D\mathcal{F}uk(S^\vee)$

where $S^\vee$ is the mirror K3 (another quartic with swapped lattice polarization).

Key technique: Use symplectic Khovanov homology and categorical localization to construct the equivalence explicitly.

The proof involves constructing a sequence of auto-equivalences (spherical twists) that transform the standard generators into the desired mirror generators.

Toric Varieties

HMS for toric varieties provides a framework where both sides can be computed explicitly.

DefinitionToric HMS

For a toric Fano variety $X_\Sigma$ defined by a fan $\Sigma$ : $D^b(Coh(X_\Sigma)) \simeq \mathcal{F}uk(M_\Sigma)$

where $M_\Sigma$ is the Landau-Ginzburg mirror, a space with superpotential $W: (\mathbb{C}^*)^n \to \mathbb{C}$ encoding toric data.

Objects on the A-side are thimbles: Lagrangian submanifolds defined as gradient flow lines of $\text{Re}(W)$ .

Homological Mirror Symmetry for Weighted Projective Spaces

Weighted projective spaces provide computable examples beyond the classical cases.

Example$\mathbb{P}^{1,1,2}$ Weighted Projective Line

The weighted projective line $\mathbb{P}^{1,1,2}$ is mirror to a Landau-Ginzburg model $(\ mathbb{C}^*, W)$ with superpotential: $W(x) = x + \frac{1}{x^2}$

The derived category $D^b(Coh(\mathbb{P}^{1,1,2}))$ (generated by line bundles $\mathcal{O}, \mathcal{O}(1), \mathcal{O}(2)$ ) is equivalent to the Fukaya-Seidel category of $(mathbb{C}^*, W)$ .

Critical points of $W$ correspond to exceptional collections of sheaves.

SYZ Construction and HMS

The Strominger-Yau-Zaslow conjecture provides a geometric approach to HMS via special Lagrangian fibrations.

Remark

If $X$ admits a special Lagrangian fibration $\pi: X \to B$ with fibers $T^n$ , the mirror $X^\vee$ is constructed as the moduli space of pairs $(L, \nabla)$ where:

$L$ is a fiber of $\pi$
$\nabla$ is a flat $U(1)$ -connection on $L$

The HMS equivalence relates:

Coherent sheaves on $X^\vee$ $\leftrightarrow$ Lagrangian sections of $\pi$

Derived Equivalences via Fourier-Mukai

Fourier-Mukai theory provides a framework for constructing derived equivalences.

DefinitionFourier-Mukai Kernel

An equivalence $\Phi: D^b(Coh(X)) \to D^b(Coh(Y))$ is Fourier-Mukai if: $\Phi(\mathcal{F}) = R\pi_{Y*}(\pi_X^*\mathcal{F} \otimes \mathcal{K})$

where $\mathcal{K} \in D^b(Coh(X \times Y))$ is the kernel.

Orlov's theorem states that all equivalences between derived categories of smooth projective varieties are Fourier-Mukai.

For HMS, the Fukaya-Mukai transform should similarly be expressible via a kernel in $D^b(Coh(X)) \times D\mathcal{F}uk(X^\vee)$ , though making this precise requires substantial technicalities.

Computational Tools

Proving HMS typically requires:

Semiorthogonal decompositions: Breaking categories into simpler pieces
Exceptional collections: Ordered sets of objects generating the category
Window categories: Subcategories described by inequalities
Matrix factorizations: For Landau-Ginzburg models

These tools allow explicit verification of categorical equivalences, turning abstract conjectures into computational theorems.