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Some examples of special Lagrangian tori. (English) Zbl 0980.32006
Let $$M$$ be a Ricci-flat $$n$$-dimensional Kähler manifold, with Kähler form $$\omega\in A_+^{1,1}(M)$$ and a normalized parallel holomorphic volume form $$\Omega\in A^{n,0}(M)$$. The data $$(\omega,\Omega)$$ is called a Calabi-Yau structure on $$M$$. An oriented $$\omega$$-Lagrangian submanifold $$f:L \hookrightarrow M$$ with induced volume form $$dV_L$$ is said to be a special Lagrangian submanifold of $$M$$ if $$f^*(\Omega) =dV_L$$.
In this brief note, the author gives some elementary examples of special Langrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in projective space.
All of these examples are constructed as real slices of smooth hypersurfaces defined over $$\mathbb{R}$$, such as cubic curves in $$\mathbb{C} \mathbb{P}^2$$, quartic surfaces in $$\mathbb{C}\mathbb{P}^3$$, and particular quintic 3-folds in $$\mathbb{C} \mathbb{P}^4$$. Especially, the explicit description of such examples in which the special Lagrangian submanifold is a three-dimensional torus is of great interest and of importance in mirror symmetry theory and related quantum field theory.

##### MSC:
 32Q25 Calabi-Yau theory (complex-analytic aspects) 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 53D12 Lagrangian submanifolds; Maslov index 53C38 Calibrations and calibrated geometries
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