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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.

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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