Bounds on Wahl singularities from symplectic topology

Evans, Jonathan David and Smith, Ivan (2020) Bounds on Wahl singularities from symplectic topology. Algebraic Geometry, 7 (1). pp. 59-85. ISSN 2313-1691

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Let X be a minimal surface of general type with positive geometric genus (b+>1) and let K2 be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length ℓ in a Q-Gorenstein degeneration, then ℓ≤4K2+7. This improves on the current best-known upper bound due to Lee (ℓ≤400(K2)4). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball Bp,1 embeds symplectically in a quintic surface, then p≤12, partially answering the symplectic version of a question of Kronheimer.

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Journal Article
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Algebraic Geometry
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03 Apr 2019 15:25
Last Modified:
21 Sep 2023 02:35