Bounds on Wahl singularities from symplectic topology

Jonathan David Evans; Ivan Smith

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arXiv · 1708.02268

Bounds on Wahl singularities from symplectic topology

Jonathan David Evans·Ivan Smith

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Abstract

Let X be a minimal surface of general type with positive geometric genus ($b_+ > 1$) and let $K^2$ 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 $\ell$ in a Q-Gorenstein degeneration, then $\ell \leq 4K^2 + 7$. This improves on the current best-known upper bound due to Lee ($\ell \leq 400(K^2)^4$). 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 $B_{p,1}$ embeds symplectically in a quintic surface, then $p \leq 12$, partially answering the symplectic version of a question of Kronheimer.

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Jonathan David Evans, Ivan Smith. 2017-08-07. Bounds on Wahl singularities from symplectic topology. https://arxiv.org/abs/1708.02268

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