Editing Symplectic manifold (section)

=== Special Lagrangian submanifolds ===
In the case of [[Kähler manifold]]s (or [[Calabi–Yau manifolds]]) we can make a choice <math>\Omega=\Omega_1+\mathrm{i}\Omega_2</math> on <math>M</math> as a holomorphic n-form, where <math>\Omega_1</math> is the real part and <math>\Omega_2</math> imaginary. A Lagrangian submanifold <math>L</math> is called '''special''' if in addition to the above Lagrangian condition the restriction <math>\Omega_2</math> to <math>L</math> is vanishing. In other words, the real part <math>\Omega_1</math> restricted on <math>L</math> leads the volume form on <math>L</math>. The following examples are known as special Lagrangian submanifolds,
# complex Lagrangian submanifolds of [[hyperkähler manifold]]s,
# fixed points of a real structure of Calabi–Yau manifolds.
The [[SYZ conjecture]] deals with the study of special Lagrangian submanifolds in [[mirror symmetry (string theory)|mirror symmetry]]; see {{harv|Hitchin|1999}}. 

The [[Thomas–Yau conjecture]] predicts that the existence of a special Lagrangian submanifolds on Calabi–Yau manifolds in Hamiltonian isotopy classes of Lagrangians is equivalent to stability with respect to a [[Bridgeland stability condition|stability condition]] on the [[Fukaya category]] of the manifold.