The Cantor-Bendixson Rank of Certain Bridgeland-Smith Stability Conditions

David Aulicino

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arXiv · 1512.02336

The Cantor-Bendixson Rank of Certain Bridgeland-Smith Stability Conditions

David Aulicino

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Abstract

We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor-Bendixson rank and give a tight bound. Finally, we present a family of surfaces realizing all possible Cantor-Bendixson ranks. The techniques in the proof of this result exclusively concern Abelian differentials on Riemann surfaces, also known as translation surfaces. The concept of a "slit translation surface" is introduced as the primary tool for studying meromorphic quadratic differentials with higher order poles.

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BibTeX RIS

David Aulicino. 2016-06-08. The Cantor-Bendixson Rank of Certain Bridgeland-Smith Stability Conditions. https://arxiv.org/abs/1512.02336

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