Assuming a canonical basis of the section ring satisfies valuative independence, the metric SYZ conjecture holds for polarised maximal degenerations of compact Calabi-Yau manifolds.
Valuative independence for Calabi--Yau varieties
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abstract
We construct valuatively independent bases for the space of sections of an ample line bundle on a log Calabi--Yau pair over a discretely valued field and the space of regular functions on an affine CY pair with maximal boundary. While the bases are not in general unique, they induce canonical functions on the respective skeletons and are expected to agree with tropicalizations of theta functions when they exist. The proof uses techniques from the study of higher rank degenerations in K-stability.
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A roadmap paper describing potential applications of numerical Ricci-flat Calabi-Yau metrics to heterotic string phenomenology and mathematical questions in special geometry.
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Valuative independence and metric SYZ conjecture
Assuming a canonical basis of the section ring satisfies valuative independence, the metric SYZ conjecture holds for polarised maximal degenerations of compact Calabi-Yau manifolds.
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What to do with a Ricci-flat Calabi--Yau metric?
A roadmap paper describing potential applications of numerical Ricci-flat Calabi-Yau metrics to heterotic string phenomenology and mathematical questions in special geometry.