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Conjecture 2.1. Given a compact Calabi-Yau manifold $X$ near the large complex structure limit, can we find a special Lagrangian torus fibration on $X$ ?

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In this chapter

Chapter overview

1 Overview
2 Metric SYZ conjecture
2.1 The genesis of the SYZ conjecture
2.2 Further motivations
2.3 Collapsing K3 surfaces with elliptic surfaces
2.4 Best hope on Calabi-Yau 3-folds
2.5 Strong vs. weak SYZ conjecture
3 Large complex structure limit
3.1 Volume asymptote and essential skeleton
3.2 The effect of blow up
3.3 Kontsevich-Soibelman conjecture
4 Analytical foundations
4.1 Yau’s solution to the Calabi conjecture
4.2 Complex pluripotential theory
4.2.1 Skoda inequality
4.3 Estimate on pluripotentials
4.4 Savin’s small perturbation theorem
4.5 Regularity theory for real Monge-Ampère
4.6 Special Lagrangian fibration
5 Nonarchimedean geometry
5.1 Berkovich space, hybrid topology
5.2 Model functions, metrics, positivity
5.3 Approximation by Fubini-Study metrics
5.4 NA Monge-Ampère measure
5.5 NA Calabi conjecture
5.6 Comparison property
6 Glimpse of proof strategy
6.1 Reduction to potential estimates
6.2 Strategy I: non-archimedean geometry
6.2.1 Motivation for NA geometry
6.2.2 Grafting the real MA solution
6.2.3 $C^{0}$ -convergence of the potential
6.3 Strategy II: a priori limit
6.3.1 Producing convex functions
6.3.2 $C^{0}$ -convergence of the potential and extension problem
6.3.3 Real MA metric
6.3.4 Relation to NA geometry
References

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