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In this article at section 2. Toric geometry and Mirror Symmetry there is the statement that CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. Now, my questions refers to pages 3 and 4. Although I have some familiarity with fiber bundles I cannot figure out what this actually means.

Can you "dumb down" this for me? Explain what this fibration actually means (maybe using a familiar example) and tell me how they get this diagram of Fig. 1?

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There is a map $F:\mathbb C^3 \to \mathbb R^3$ given by

$$F (z_1, z_2, z_3) = (|z_1|^2 - |z_3|^2, |z_2|^2 - |z_3|^2, Im(z_1z_2z_3))$$

Then one can think of $\mathbb C^3$ as a "fibration" of $\mathbb R^3$:

$$\mathbb C^3 = \bigcup_{x\in \mathbb R^3} F^{-1}(x).$$

One can check that when $x$ does not lie in the "degenerate locus" in figure 1, then $F^{-1}(x)$ is diffeomorphic to $T^2 \times \mathbb R$. Note also that all $F^{-1}(x)$ are special Lagrangian. Special Lagrangian fibration are of great interest in mirror symmetry.

It is not true that all CY 3 folds are of this form locally. I do not know if this is true for toric CY 3 folds though.

Note that "fibration" in this case is not the same as a fiber bundle, as some of the fibers are not homotopic to each other.

I am no expert in this field, but to get an introduction to special Lagrangian fibration, you may take a look at "Calabi-Yau Manifolds and Related Geometries"

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  • $\begingroup$ Hello and thanks a lot for your answer. It is helpful. I am interested in understanding the philosophy of these fibrations at this point rather than become an expert. what is the "degenerate locus" in Figure 1? Can you elaborate a bit further onto the diagram? That would be very helpful, thanks a lot. $\endgroup$
    – Marion
    Apr 4, 2015 at 10:48

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