Symplectic rational blow-ups on rational 4-manifolds

Park, Heesang; Shin, Dongsoo

doi:10.1090/proc/16519

Symplectic rational blow-ups on rational 4-manifolds
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by Heesang Park and Dongsoo Shin

Proc. Amer. Math. Soc. 152 (2024), 1309-1318

DOI: https://doi.org/10.1090/proc/16519

Published electronically: December 22, 2023

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Abstract:

We prove that if a symplectic 4-manifold $X$ becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold $X$ is originally rational. That is, a symplectic rational blow-up of a rational symplectic $4$-manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of Bădescu [J. Reine Angew. Math. 367 (1986), pp. 76–89] in algebraic geometry under a mild additional condition.

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