Under a linear spectral gap assumption on the Kodaira Laplacian, the paper proves asymptotic expansions and algebra properties for Toeplitz operators on non-compact manifolds, plus geometric conditions ensuring the gap on classes like Kähler-Einstein and quasi-projective manifolds.
Some numerical results in complex differential geometry
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abstract
The first part of this paper discusses general procedures for finding numerical approximations to distinguished Kahler metrics, such as Calabi-Yau metrics, on complex projective manifolds. These procedures are closely related to ideas from Geometric Invariant Theory, and to the asymptotics of high powers of positive line bundles. In the core of the paper these ideas are illustrated by detailed numerical results for a particular K3 surface.
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2026 2verdicts
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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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Berezin-Toeplitz Quantization of non-compact manifolds
Under a linear spectral gap assumption on the Kodaira Laplacian, the paper proves asymptotic expansions and algebra properties for Toeplitz operators on non-compact manifolds, plus geometric conditions ensuring the gap on classes like Kähler-Einstein and quasi-projective 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.