Example 1.1 . [03YJ]
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Example 1.1.
(Generic region) Suppose in some region only and dominate, so the hypersurface locally looks like . After normalising by powers of we may write this as in the coordinates on the algebraic torus . This model has -symmetry under the diagonal action on . The -orbits are the natural candidate for approximate SYZ fibres. Thus we naturally look for a Kähler metric with potential depending only on the logarithms . The holomorphic volume form on the hypersurface is up to a scale factor given by
namely . The complex Monge-Ampère equation naturally reduces to the real Monge-Ampère equation . One can further calculate that such regions take up most of the volume measure on , thus lending some evidence for the SYZ conjecture. We remark that the description only applies to local regions so the metrics are not complete.