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The ellipsoidal capacity function of a symplectic four manifoldXmeasures how much the form onXmust be dilated in order for it to admit an embedded ellipsoid of eccentricityz. In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions,Xis said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the family of symplectic Hirzebruch surfacesH_{b}formed by blowing up the projective plane with weightb. We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parametersbthat are blocked, i.e. have no staircase, and an uncountable number of other values ofbthat do admit staircases. The remainingb-values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill–McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long asbis not one of these special rational values, any staircase inH_{b}has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations onH_{b}is echoed in the structure of the set of blockedb-intervals.
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Full ellipsoid embeddings and toric mutations
Abstract This article introduces a new method to construct volume-filling symplectic embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric and almost toric varieties. The construction uniformly recovers the full sequences for the Fibonacci Staircase of McDuff–Schlenk, the Pell Staircase of Frenkel–Müller and the Cristofaro-Gardiner–Kleinman Staircase, and adds new infinite sequences of ellipsoid embeddings. In addition, we initiate the study of symplectic-tropical curves for almost toric fibrations and emphasize the connection to quiver combinatorics.
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- Award ID(s):
- 1942363
- PAR ID:
- 10342848
- Date Published:
- Journal Name:
- Selecta Mathematica
- Volume:
- 28
- Issue:
- 3
- ISSN:
- 1022-1824
- 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/s00029-022-00765-3
