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For a large class of maximally degenerate families of Calabi–Yau hypersurfaces of complex projective space, we study non- Archimedean and tropical Monge–Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting.
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Fourier restriction above rectangles
In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of the operator norms in these inequalities on the sidelengths of the rectangles, prove that this conjecture follows from (a slight reformulation of the) restriction conjecture for elliptic hypersurfaces, and prove that, if valid, the conjecture is essentially sharp. Such questions arise naturally in the study of restriction inequalities for degenerate hypersurfaces; we demonstrate this connection by using our positive results to prove new restriction inequalities for a class of hypersurfaces having some additive structure.
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- Award ID(s):
- 1653264
- PAR ID:
- 10281876
- Date Published:
- Journal Name:
- Mathematische Annalen
- ISSN:
- 0025-5831
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00208-021-02202-w
