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We show that the non-Archimedean skeleton of the d d -th symmetric power of a smooth projective algebraic curve X X is naturally isomorphic to the d d -th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of X X . The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under certain tropical genericity assumptions, to deduce a new tropical Riemann-Roch-Theorem for the tropicalization of linear systems.
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Tropical and non-Archimedean Monge–Ampère equations for a class of Calabi–Yau hypersurfaces
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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- PAR ID:
- 10516579
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Advances in Mathematics
- Volume:
- 439
- Issue:
- C
- ISSN:
- 0001-8708
- Page Range / eLocation ID:
- 109494
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.aim.2024.109494
