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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Bounds on Wahl singularities from symplectic topology
AU - Evans, Jonathan David
AU - Smith, Ivan
PY - 2020/1/31
Y1 - 2020/1/31
N2 - 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.
AB - 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.
KW - symplectic topology
KW - vanishing cycles
KW - wahl singularities
U2 - 10.14231/ag-2020-003
DO - 10.14231/ag-2020-003
M3 - Journal article
VL - 7
SP - 59
EP - 85
JO - Algebraic Geometry
JF - Algebraic Geometry
SN - 2313-1691
IS - 1
ER -