Abstract
We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.
Highlights
The Strominger–Yau–Zaslow conjecture [GW00, KS01, SYZ96] asserts that a Calabi–Yau manifold admitting a degeneration to a large complex structure limit admits a Lagrangian torus fibration
We focus on the problem of finding Lagrangian torus fibrations on affine manifolds W arising as X \ Y where X is a complex projective variety and Y is a simple normal crossing divisor
Remark 2.18. — the existence of a Lagrangian torus fibration on a contact manifold depends on the contact form α, Lemma 2.16 only depends on the contactomorphism type of the link
Summary
The Strominger–Yau–Zaslow conjecture [GW00, KS01, SYZ96] asserts that a Calabi–Yau manifold admitting a degeneration to a large complex structure limit admits a Lagrangian torus fibration. We focus on the problem of finding Lagrangian torus fibrations (not necessarily special) on affine manifolds W arising as X \ Y where X is a complex projective variety and Y is a simple normal crossing divisor. These should be thought of as local models for Lagrangian torus fibrations on compact Calabi–Yaus or other varieties (for example surfaces of general type). Such that the base is a 3-ball and the discriminant locus (set of singular fibres) is a Y-graph.
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